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Calculus of variations

The document provides a comprehensive overview of the calculus of variations, including its fundamental problems and historical context. Key topics covered include necessary conditions for extremum, the Euler-Lagrange equation, and classical problems like the brachistochrone problem. Furthermore, it discusses applications in geometrical optics and general principles governing the motion of conservative systems.

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1 Calculus of Variations SOLO HERMELIN "weak" neighbor ( )( )000 , txtA ( )( )fff txtB , ( )txx t "strong" neighbor ( )ε,tx http://www.solohermelin.com
2 Table of Content Calculus of VariationsSOLO . Introduction 1. General Formulation of the Simplest Problem of Calculus of Variations 2. Solution Method 2.1 Neighborhoods and Variations 3. Variations of the Functional J 4. Necessary Conditions for Extremum 4.4 Special Cases 4.5Examples 5. Boundary Conditions 6. Corner Conditions 7 Sufficient Conditions and Additional Necessary Conditions for a Weak Extremum 4.1 The First Fundamental Lemma of the Calculus of Variations 4.2The Euler-Lagrange Equation 4.3 The Second Fundamental Lemma of the Calculus of Variations 8 Legendre’s Necessary Conditions for a Weak Minimum (Maximum)
Table of Content (continue – 1) Calculus of VariationsSOLO 9. Jacobi’s Differential Equation (1837) and Conjugate Points 9.1 Conjugate Points 9.2 Fields Definition 10. Hilbert’s Invariant Integral 11. The Weierstrass Necessary Condition for a Strong Minimum (Maximum) Summary 12. Canonical Form of Euler-Lagrange Equations 12.1 Legendre’s Dual Transformation 12.2 Transversality Conditions (Canonical Variables ) 12.3 Weierstrass-Erdmann Corner Conditions (Canonical Variables) 12.4 First Integrals of the Euler-Lagrange Equations 12.5 Equivalence Between Euler-Lagrange and Hamilton Functionals 12.6 Equivalent Functionals 12.7 Canonical Transformations 12.8 Caratheodory's Lemma 12.9 Hamilton-Jacobi Equations Jacobi’s Theorem
Table of Content (continue – 2) Calculus of VariationsSOLO References Appendix 1: Implicit Functions Theorem Appendix: Useful Mathematical Theorems Appendix 2: Heine–Borel Theorem Appendix 3: Ordinary Differential Equations
5 HISTORY OF CALCULUS OF VARIATIONSSOLO “When the Tyrian princess Dido landed on the Mediterranean sea she was welcomed by a local chieftain. He offered her all the land that she could enclose between the shoreline and a rope of knotted cowhide. While the legend does not tell us, we may assume that Princess Dido arrived at the correct solution by stretching the rope into the shape of a circular arc and thereby maximized the area of the land upon which she was to found Carthage. This story of the founding of Carthage is apocryphal. Nonetheless it is probably the first account of a problem of the kind that inspired an entire mathematical discipline, the calculus of variations and its extensions such as the theory of optimal control.” (George Leitmann “The Calculus of Variations and Optimal Control – An Introduction” Plenum Press, 1981) Dido Maximum Area Problem
6 ( ) ∫∫ − == a a xy dxyAdJ maxmaxmax Given a rope of length P connected to each end of straight line of length 2 a < P find the shape of the rope necessary to enclose the maximum area between the rope and the straight line. The problem can be formulated as: Dido Maximum Area Problem HISTORY OF CALCULUS OF VARIATIONS ( ) ( ) ( ) ∫∫∫∫∫ −−− =+=      +=+== a a a a a a dxdxdx xd yd ydxdsdP θθ sectan11 2 2 22 subject o the isoperimetric constraint: where: θtan= xd yd SOLO Return to Table of Content Rope of length P ( )xθ x y a+a− y dx
7 1. General Formulation of the Simplest Problem of Calculus of Variations Given: (1) A Functional (function of functions) J [x (t)] Calculus of VariationsSOLO ( )[ ] ( ) ( ) ( ) ( )( ) ( ) ( )∫∫       == ⋅ff t t t t nn dttxtxtFdttxtxtxtxtFtxJ 00 ,,,,,,,, 11  ( ) ( ) ( )( )T n txtxtx ,,: 1 = ( ) ( ) ( )( ) ( ) ( ) T n T n tx dt d tx dt d txtxtx       == ⋅ ,,,,: 11  where: ( ) ( )      ⋅ txtxtF ,, ( ) ( )txtxt ⋅ ,, (2) shall be continuous and admit continuous partial derivatives of the first, second and third order in a domain which contains all points . .
8 General Formulation of the Simplest Problem of Calculus of Variations Calculus of VariationsSOLO . ( ) ( ) ( ) ( ) ( )( )T n tftftftftx ,,, 21 ==(1) The vector of functions where t0 ≤ t ≤tf, fi (t)i=1,n being single valued of t that minimizes (maximizes) the functional J in a weak neighborhood. (2) fi (t)i=1,n are continuous and consist of a finite number of arcs of continuously turning tangent, not parallel to the x axis; i.e. fi (t)€ D (1) (3) passes through two points (constant vectors), defined or not.( )tx ( )tx xt,(4) lies in a given region of the space. corner points( )( )000 , txtA ( )( )fff txtB , ( )txx t Find: Figure: A Possible Solution for the Problem of Calculus of Variations
9 General Formulation of the Simplest Problem of Calculus of Variations Calculus of VariationsSOLO Examples of Calculus of Variations Problems 1. Brachistochrone Problem A particle slides on a frictionless wire between two fixed points A(0,0) and B (xfc, yfc) in a constant gravity field g. The curve such that the particle takes the least time to go from A to B is called brachistochrone (βραχιστόσ Greek for “shortest“, χρόνοσ greek for “time). The brachistochrone problem was posed by John Bernoulli in 1696, and played an important part in the development of calculus of variations. The problem was solved by Johann Bernoulli, Jacob Bernoulli, Isaac Newton, Gottfried Leibniz and Guillaume de L’Hôpital. Let choose a system of coordinates with the origin at point A (0,0) and the y axis in the constant g direction x y V ( )tγ ( )fcfc yxB ,fcx fcy N  ( )0,0A
10 General Formulation of the Simplest Problem of Calculus of Variations Calculus of VariationsSOLO Examples of Calculus of Variations Problems 1. Brachistochrone Problem Since the motion of the particle is in a frictionless fixed gravitational field the total energy is conserved ( ) ygVyVygVV 2 2 1 2 1 2 0 22 0 +=→−= x y V ( )tγ ( )fcfc yxB ,fcx fcy N  ( )0,0A Second way to get this relation is: ( ) ygVVdygdVV sd yd ggV sd Vd td sd sd Vd td Vd =−→=→==== 2 0 2 2 1 sinγ where V0 is the velocity of the particle at point A and ( ) ( )22 ydxdsd += td xd xd yd td yd td xd td sd V 222 1       +=      +      == We have xd ygV xd yd xd V xd yd td 2 11 2 0 22 +       + =       + = The cost function is ∫∫       = +       + = cfcf xx xd xd yd yxFxd ygV xd yd J 00 2 0 2 ,, 2 1
11 HISTORY OF CALCULUS OF VARIATIONS The brachistochrone problem In 1696 proposed the Brachistochrone (“shortest time”) Problem: Given two points A and B in the vertical plane, what is the curve traced by a point acted only by gravity, which starts at A and reaches B in the shortest time. Johann Bernoulli 1667-1748 SOLO
12 The brachistochrone problem Jacob Bernoulli (1654-1705) Gottfried Wilhelm von Leibniz (1646-1716) Isaac Newton (1643-1727) The solutions of Leibniz, Johann Bernoulli, Jacob Bernoulli and Newton were published on May 1697 publication of Acta Eruditorum. L’Hôpital solution was published only in 1988. Guillaume François Antoine de L’Hôpital (1661-1704) SOLO HISTORY OF CALCULUS OF VARIATIONS
13 General Formulation of the Simplest Problem of Calculus of Variations Calculus of VariationsSOLO Examples of Calculus of Variations Problems 2. Problem of Minimum Surface of Revolution Given two points A (a,ya) and B (b, yb) a≠b in the plane. Find the curve that joints these two points with a continuous derivative, in such a way that the surface generated by the rotation of this curve about the x axis has the smallest possible area. x y ( )bybB , ( )ayaA , ( ) ( )22 ydxdsd += y Minimum Surface of Revolution The surface generated by the rotation of y (x) curve about the x – axis can be calculated using ( ) ( ) xd xd yd yydxdysdydS 2 22 1222       +=+== πππ Therefore ( )∫       +== b a xd xd yd xySJ 2 12: π 2 1,,       +=      xd yd y xd yd yxF We can see that F is not an explicit function of x.
14 General Formulation of the Simplest Problem of Calculus of Variations Calculus of VariationsSOLO Examples of Calculus of Variations Problems 3. Geometrical Optics and Fermat Principle The Principle of Fermat (principle of the shortest optical path) asserts that the optical length of an actual ray between any two points is shorter than the optical ray of any other curve that joints these two points and which is in a certain neighborhood of it. An other formulation of the Fermat’s Principle requires only Stationarity (instead of minimal length). ∫ 2 1 P P dsn An other form of the Fermat’s Principle is: Principle of Least Time The path following by a ray in going from one point in space to another is the path that makes the time of transit of the associated wave stationary (usually a minimum).
15 SOLO We have: constS = constdSS =+ sˆ ∫ 2 1 P P dsn 1 P 2 P ( ) ( ) ( )∫∫∫∫ =      +      +=== 2 1 2 1 2 1 ,,,, 1 1,, 1 ,, 1 0 22 00 P P P P P P xdzyzyxF c xd xd zd xd yd zyxn c dszyxn c tdJ  The stationarity conditions of the Optical Path using the Calculus of Variations ( ) ( ) ( ) xd xd zd xd yd zdydxdds 22 222 1       +      +=++= Define: xd zd z xd yd y ==  &: ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += General Formulation of the Simplest Problem of Calculus of Variations Calculus of Variations Examples of Calculus of Variations Problems 3. Geometrical Optics and Fermat Principle Paths of Rays Between Two Points
16 SOLO General Formulation of the Simplest Problem of Calculus of Variations Calculus of Variations Examples of Calculus of Variations Problems 4. Hamilton Principle for Conservative Systems The motion of a conservative system, from time t0 to tf is such that the integral ( )∫= ft t dtqqLJ 0 ,  has a stationary value ( δJ = 0), where ( ) ( ) ( )qVqqTqqL −=  ,:, qq , ( )qqT , ( )qV δ is the Lagrangian of the system are the generalized coordinate vector of the system and its derivatives kinetic energy of the system potential energy of the system the variation that will be defined in the next section. Since the system is conservative, the external forces acting on the system are given by ( ) ( )qVqQ ∇= For a non-conservative system the Extended Hamilton Principle is ( ) ( ) 0, 00 =+ ∫∫ ff t t t t dtqqQdtqqT δδ  The Hamilton Principle doesn’t require the minimization but only stationarity (vanishing of the first variation δJ = 0).
17 SOLO General Formulation of the Simplest Problem of Calculus of Variations Calculus of Variations Examples of Calculus of Variations Problems 5. Geodesics Suppose we have a surface specified by two parameters u and v and the vector .( )vur ,  The shortest path lying on the surface and connecting to points of the surface is called a geodesic. A B ( )vur ,  vdrv  udru  rd  The Shortest Path on a Surface The arc length differential is td td vd v r td ud u r td vd v r td ud u r td td rd td rd td td rd ds 2/12/1             ∂ ∂ + ∂ ∂ ⋅      ∂ ∂ + ∂ ∂ =      ⋅==  td td vd v r v r td vd td ud v r u r td ud u r u r 2/122 2                     ∂ ∂ ⋅ ∂ ∂ +                  ∂ ∂ ⋅ ∂ ∂ +            ∂ ∂ ⋅ ∂ ∂ = 
18 SOLO General Formulation of the Simplest Problem of Calculus of Variations Calculus of Variations Examples of Calculus of Variations Problems 5. Geodesics (continue) A B ( )vur ,  vdrv  udru  rd  The Shortest Path on a Surface The length of the path between the two points A and B is ∫               +            +      == B A r r td td vd G td vd td ud F td ud ESJ   2/122 2:       ∂ ∂ ⋅ ∂ ∂ = u r u r E  :       ∂ ∂ ⋅ ∂ ∂ = v r u r F  :       ∂ ∂ ⋅ ∂ ∂ = v r v r G  : where Return to Table of Content
19 SOLO Calculus of Variations 2. Solution Method ( )tx ( )ε,tx To find a candidate for the minimizing (maximizing) trajectory , construct variations (neighbors) of this trajectory and find the conditions under which those variations increase (decrease) the value of the functional J [x (t)]. The results of this method are known as the Calculus of Variations. "weak" neighbor ( )( )000 , txtA ( )( )fff txtB , ( )txx t "strong" neighbor ( )ε,tx Return to Table of Content
20 SOLO Calculus of Variations 2.1 Neighborhoods and Variations "weak" neighbor ( )( )000 , txtA ( )( )fff txtB , ( )txx t "strong" neighbor ( )ε,tx ( )tx( )ε,txLet define a function of the closeness of order k to Weak Neighborhood is a “weak” neighborhood of order k if:( )ε,tx ( ) ( ) ( ) ( ) ( ) ( )         ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ = → → → tx t tx t tx t tx t txtx k k k k ε ε ε ε ε ε ,lim ,lim ,lim 0 0 0  ( ) ( )( ) ( )( )00000 ,,, txtAtxt ∈εεε ( ) ( )( ) ( )( )fffff txtBtxt ,,, ∈εεε Strong Neighborhood If we only have (only for k = 0) then is called a “strong” neighborhood. If k > 0 then it is a “weak” neighborhood. ( ) ( )txtx = → ε ε ,lim 0 ( )ε,tx ( ) ( ) ( ) ( ) ( ) ( )32 0 2 2 0 , 2 1, ,:, εε ε ε ε ε ε εε εε Ο/+ ∂ ∂ + ∂ ∂ =−=∆ == d tx d tx txtxtxLet compute: ( ) 0lim 2 3 0 → Ο/ → ε ε ε w
21 SOLO Calculus of Variations First and Second Variations ( )ε,tx ( )txFirst Variation of ( ) ( ) ε ε ε δ ε d tx tx 0 , : =∂ ∂ = i.e. the differential of as a function of ε ( )txThe First Variation of is defined as ( )txSecond Variation of ( )txThe Second Variation of is defined as ( ) ( ) 2 0 2 2 2 , : ε ε ε δ ε d tx tx = ∂ ∂ = ( )( )fff txtB , x t ( )2,εtx ( )1,εtx ( )tx ft ( )1εft ( )2εft At the boundaries t0 and tf are functions of ε (see Figure)
22 SOLO Calculus of Variations First and Second Variations at the Boundary Therefore at the boundaries we have ( )( ) fiitx ,0 , = εε ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )xdxdxd d tx d tx txtxtx ii tt ii ii 32 32 0 2 2 0 2 1 , 2 1, ,:, Ο/++= =Ο/+ ∂ ∂ + ∂ ∂ =−=∆ == εε ε εε ε ε εε εεεε εε ε ε ε d x xd i i t t 0: = ∂ ∂ = 2 0 2 2 2 0 2 :&: ε ε ε ε εε d x xdd x xd i i i i t t t t == ∂ ∂ = ∂ ∂ = where: ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( )      ∂ ∂ +      ∂ ∂ +      ∂ ∂ = =      = εε εεε ε ε εε ε ε εε ε εε εε εε ε ,,, ,,2 2 i i i i i ii tx d d d dt d d tx td dt tx td d tx d d d d tx d d ( )       ∂ ∂ + ∂∂ ∂ + ∂ ∂ +      ∂∂ ∂ + ∂ ∂ = 2 22 2 22 2 2 εεεε ε εεε x d dt t x d td t x d dt t x d dt t x iiii 2 2 2 222 2 2 2 εεεεε ∂ ∂ + ∂ ∂ + ∂∂ ∂ +      ∂ ∂ = x d td t x d dt t x d dt t x iii ( )( ) ( )( ) ( ) ( )( )εε εε ε εεεε ε ,,, i i ii tx d dt tx t tx d d ∂ ∂ + ∂ ∂ = We have: ( )( )fff txtB , x t ( )fxd 3 Ο/ ( )ε,tx ( )tx ff dtt +( )0=εft fxδ fxd fx∆ fxd 2 ff dtx • fdt Variations at the Boundary tf
23 SOLO Calculus of Variations First and Second Variations at the Boundary ( )( )fff txtB , x t ( )fxd 3 Ο/ ( )ε,tx ( )tx ff dtt +( )0=εft fxδ fxd fx∆ fxd 2 ff dtx • fdt Variations at the Boundary tf ( )( ) ( ) ( )( ) εεε ε ε ε ε εε εεε dtxd d dt tx t xd i i ii 000 ,, === ∂ ∂ +      ∂ ∂ = ( )( ) ( )( )εεεε ε ,, 0 i tt i txx dt xd tx t ii •• = === ∂ ∂ ( ) i i dtd d dt = = ε ε ε ε 0 ( ) ( )( ) εεε ε δ ε dtxtx ii 0 ,: =∂ ∂ = ( ) ( ) fitxdttxxd iii ,0=+= • δ But Therefore we obtain:
( ) 2 0 2 2 2 : ε ε ε ε d d td td i i = =and define: 24 SOLO Calculus of Variations First and Second Variations at the Boundary ( )( )fff txtB , x t ( )fxd 3 Ο/ ( )ε,tx ( )tx ff dtt +( )0=εft fxδ fxd fx∆ fxd 2 ff dtx • fdt Variations at the Boundary tf ( ) ( ) ( ) ( ) ε ε ε ε ε ε ε ε εε εεεε d d dt d t tx d d dt t tx xd iiii i 00 2 00 2 2 2 , 2 , ====       ∂ ∂ ∂ ∂ +      ∂ ∂ = ( ) ( ) ( ) 2 0 2 2 2 0 2 2 0 ,, ε ε ε ε ε εε εεε d tx d d td t tx iii === ∂ ∂ + ∂ ∂ + ( ) ( ) ( )ii i txtx dt d t tx •• = == ∂ ∂ 2 2 0 2 2 , ε ε ( ) ( )i i txd t tx • = =      ∂ ∂ ∂ ∂ δε ε ε ε 0 , ( ) ( ) 2 0 2 2 2 , : ε ε ε δ ε d tx tx i i = ∂ ∂ = ( )( ) ( ) ( ) ( ) fitxtdtxdttxdttxxd iiiiiiiii ,02 2222 =+++= •••• δδ Also we have: But: Therefore Return to Table of Content
25 SOLO Calculus of Variations 3. Variations of the Functional J The value of the functional J in the neighborhood of is( )ε,tx ( )tx ( ) ( ) ( ) ( ) ( ) ∫       = • ε ε εεε ft t dttxtxtFJ 0 ,,,, ( ) ( )εε ,:, tx t tx ∂ ∂ = • where We can write: ( ) ( ) ( ) ( )JJJd d Jd d d dJ JJJ 3232 0 2 2 0 2 1 2 1 0: δδδεε ε ε ε εε εε Ο/++=Ο/++= =−=∆ == where ε ε δ ε d d dJ J 0 : = = the first variation of J 2 0 2 2 2 : ε ε δ ε d d Jd J = = the second variation of J ( ) 0lim 2 3 0 → Ο/ → ε ε ε
26 SOLO Calculus of Variations First Variation of the Functional J ( ) ( ) ( ) ( ) ( )               = ∫ • ε ε εε εε ε ft t dttxtxtF d d d dJ 0 ,,,, ( ) ( ) ( ) ( ) ( ) ( ) ε ε εε ε ε εε d dt txtxtF d dt txtxtF f fff 0 000 ,,,,,,,,       −      = •• ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫       ∂ ∂       + ∂ ∂       + ••• • ε ε ε ε εεε ε εε ft t x x dttxtxtxtFtxtxtxtF 0 ,,,,,,,,,, T n nn x x F x F x F x F x F x F F x x x xxtF       ∂ ∂ ∂ ∂ ∂ ∂ =                       ∂ ∂ ∂ ∂ ∂ ∂ =                       ∂ ∂ ∂ ∂ ∂ ∂ =      • ,,,:,, 21 2 1 2 1   T n x x F x F x F xxtF       ∂ ∂ ∂ ∂ ∂ ∂ =      • •    ,,,:,, 21 and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫             +      +       −      == ••• •• = • ft t T x T x ffff dttxtxtxtFtxtxtxtF dttxtxtFdttxtxtF d dJ J 0 ,,,, ,,,, 0000 0 δδ ε ε δ ε
27 SOLO Calculus of Variations Second Variation of the Functional J ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               ∂ ∂       + ∂ ∂       +             +      −             +      =    = ∫ ••• •• •• ε ε ε ε εεε ε εε ε ε ε ε εε ε ε εε ε ε ε ε εε ε ε εε εεεε ft t T x T x f fff f fff dttxtxtxtFtxtxtxtF d d d dt d d txtxtF d dt txtxtF d d d dt d d txtxtF d dt txtxtF d d d dJ d d d Jd 0 ,,,,,,,,,, ,,,,,,,, ,,,,,,,, 0 000 0 000 2 2  In this equation we have: ( ) ( ) fi x d dt t x F x d dt t x F d dt FtxtxtF d d it iT x iT x i tiii ,0,,,, =                 ∂ ∂ + ∂ ∂ +      ∂ ∂ + ∂ ∂ +=      •• • εεεεε εε ε  t F Ft ∂ ∂ =: ( ) ( ) 2 2 ε ε ε ε ε d td d dt d d ii =      ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       ∫       ∂ ∂       + ∂ ∂       •••ε ε ε ε εεε ε εε ε ft t T x T x dttxtxtxtFtxtxtxtF d d 0 ,,,,,,,,,,  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ε ε ε ε εεε ε εε d dt txtxtxtFtxtxtxtF f ffff T xffff T x       ∂ ∂       + ∂ ∂       = ••• ,,,,,,,,,,  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ε ε ε ε εεε ε εε d dt txtxtxtFtxtxtxtF T x T x 0 00000000 ,,,,,,,,,,       ∂ ∂       + ∂ ∂       − •••  ( ) ( ) td x F xx F xx F x F xx F xx F xx T xx T T x t t xx T xx T T x f              ∂ ∂         ∂ ∂ +      ∂ ∂         ∂ ∂ + ∂ ∂ +             ∂ ∂       ∂ ∂ +      ∂ ∂       ∂ ∂ + ∂ ∂ + ••••• ••••• ∫ εεεεεεεεεε ε ε 2 2 2 2 0 and
28 SOLO Calculus of Variations Second Variation of the Functional J (continue – 1) [ ] [ ]               =                 ∂ ∂ ∂ ∂ = ∂ ∂ =               ∂ ∂ ∂ ∂ = nnnn n n n xxxxxx xxxxxx xxxxxx xxx T x T xx FFF FFF FFF FFF x x F xx F x F      21 21212 12111 21 ,,,: 1 1 [ ] [ ]               =                 ∂ ∂ ∂ ∂ = ∂ ∂ =               ∂ ∂ ∂ ∂ = ••• nnnn n n n xxxxxx xxxxxx xxxxxx xxx T x T xx FFF FFF FFF FFF x x F x x F x F            21 21212 12111 21 ,,,: 1 1 [ ]               =                 ∂ ∂ ∂ ∂ =     ∂ ∂ =                 ∂ ∂ ∂ ∂ = •• • nnnn n n n xxxxxx xxxxxx xxxxxx xxx T x T xx FFF FFF FFF FFF x x F xx F x F          21 21212 12111 21 ,,,: 1 1 [ ]               =                 ∂ ∂ ∂ ∂ =     ∂ ∂ =                 ∂ ∂ ∂ ∂ = • ••• nnnn n n n xxxxxx xxxxxx xxxxxx xxx T x T xx FFF FFF FFF FFF x x F xx F x F             21 21212 12111 21 ,,,: 1 1
29 SOLO Calculus of Variations Second Variation of the Functional J (continue – 2) xx T xxxx ijji xx FFF x F xx F x F jiij =→=      ∂ ∂ ∂ ∂ =         ∂ ∂ ∂ ∂ =: xxxx T xxxx ijji xx FFFF x F xx F x F jiij   ≠=→=      ∂ ∂ ∂ ∂ =         ∂ ∂ ∂ ∂ =: xx T xxxx ijji xx FFF x F xx F x F jiij   =→=      ∂ ∂ ∂ ∂ =         ∂ ∂ ∂ ∂ =: Let integrate by parts the term ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∂ ∂       − ∂ ∂ =∫ ∂ ∂ ••• • ε ε ε ε ε ε εεε f f f t t T x t t T x t t T x dt x F dt dx Fdt x F 0 0 0 2 2 2 2 2 2 By using all those developments we get: 2 2 2 2 εεεεεεεε d td F d dtx d dt t x F x d dt t x F d dt F d Jd f t f t fT x fT x f t f f +                 ∂ ∂ + ∂ ∂ +      ∂ ∂ + ∂ ∂ += •• • 2 0 2 0000 0 0 εεεεεεε d td F d dtx d dt t x F x d dt t x F d dt F t t T x T xt +                 ∂ ∂ + ∂ ∂ +      ∂ ∂ + ∂ ∂ +− •• • ( ) ( )εε εεεεεεεε ff f t T x t T x t T x T x f t T x T x x F x F d dtx F x F d dtx F x F 0 0 2 2 2 2 0 ∂ ∂ − ∂ ∂ +         ∂ ∂ + ∂ ∂ −         ∂ ∂ + ∂ ∂ + •••• •• ( ) ( ) dt x F xx F xx F x F xx F xx F dt d F xx T xx T T x t t xx T xx TT x x f             ∂ ∂         ∂ ∂ +      ∂ ∂         ∂ ∂ + ∂ ∂ +             ∂ ∂       ∂ ∂ +      ∂ ∂       ∂ ∂ + ∂ ∂       −+ ••••• •••••• ∫ εεεεεεεεεε ε ε 2 2 2 2 0
30 SOLO Calculus of Variations Second Variation of the Functional J (continue – 3) ft fffT x ffT x f t d td F x d dtx d dt t x F d dtx d dt t x F d dt F d Jd         +         ∂ ∂ + ∂ ∂ +      ∂ ∂ +      ∂ ∂ + ∂ ∂ +      = •• • 2 2 2 2 22 2 2 22 εεεεεεεεεε 0 2 0 2 2 2 0 2 000 2 0 22 t T x T xt d td F x d dtx d dt t x F d dtx d dt t x F d dt F         +         ∂ ∂ + ∂ ∂ +      ∂ ∂ +      ∂ ∂ + ∂ ∂ +      − •• • εεεεεεεεε ( ) ( ) dt x F xx F xx F xx F xx F dt d F xx T xx T t t xx T xx TT x x f              ∂ ∂         ∂ ∂ +      ∂ ∂         ∂ ∂ +             ∂ ∂       ∂ ∂ +      ∂ ∂       ∂ ∂ + ∂ ∂       −+ •••• ••••• ∫ εεεεεεεεε ε ε0 2 2 ( ) ( ) ft fff T x ff T xft tdFxdtxdtxFdtxdtxFdtFd d Jd J       +      +++      ++== •••• = • 22222 0 2 2 2 22 δδδε ε δ ε ( ) ( ) ft fff T x ff T xft tdFxdtxdtxFdtxdtxFdtF       +      +++      ++− •••• • 2222 22 δδδ ( ) ( ) ( ) ( ) ( ) ( ) ∫                     +      +      ++      −+ •••• ••••• ε ε δδδδδδδδδ ft t xx T xx T xx T xx T T x x dtxFxxFxxFxxFxxF dt d F 0 2 Therefore
31 SOLO Calculus of Variations Second Variation of the Functional J (continue – 4) ( ) ( ) fitxdttxxd iii ,0=+= • δ But we found that: ( ) ( ) ( ) ( ) ( ) fitxtdtxdttxdttxxd iiiiiiiii ,02 2222 =+++= •••• δδ ( ) −      +      −+      −+= •• • ft ff T x ff T xft tdFtdxxdFdtdtxxdFdtFJ 22222 2δ ( ) −      +      −+      −+− •• • 0 0 2 0 22 00 2 0 2 t T x T xt tdFtdxxdFdtdtxxdFdtF ( ) ( ) ( ) ( ) ( ) ( ) ∫                     +      +      ++      −+ •••• ••••• ε ε δδδδδδδδδ ft t xx T xx T xx T xx T T x x dtxFxxFxxFxxFxxF dt d F 0 2 Hence: and the final result is: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫                     +      +      ++      −+ +            −+++      −−             −+++      −= •••• •• •• ••••• •• •• ε ε δδδδδδδδδ δ f f t t xx T xx T xx T xx T T x x t T x T x T x T xt t f T x f T x ff T xf T xt dtxFxxFxxFxxFxxF dt d F tdxFFxdFdtxdFdtxFF tdxFFxdFdtxdFdtxFFJ 0 0 2 0 2 0 2 00 2 0 2222 2 2
32 SOLO Calculus of Variations 4. Necessary Conditions for Extremum We found that: ( ) ( ) ( ) ( )JJJd d Jd d d dJ JJJ 3232 0 2 2 0 2 1 2 1 0: δδδεε ε ε ε εε εε Ο/++=Ο/++==−=∆ == For a Minimum Solution of the Functional J we must have: ΔJ ≥ 0 for any small dε (see Figure) ( )( )ε,txJ 0=ε ε Minimum of J as function of ε For a Maximum Solution of the Functional J we must have: ΔJ ≤ 0 for any small dε To prevent that the sign of dε to change the same of ΔJ the Necessary Condition for Extremum is 00 0 == = Jord d dJ δε ε ε This condition must be fulfilled for any admissible variation.
33 SOLO Calculus of Variations Necessary Conditions for Extremum (continue – 1) Suppose that is an extremal solution with the fixed end points and .( )tx* ( )0 * 0 * , xtA ( )0 * 0 * , xtB Let choose first all the variation that passes through those points (see Figure ). ( )( )000 * , txtA ( )( )fff txtB ,* ( )tx*x t ( )ε,1 tx ( )ε,2 tx Variations Passing through Fixed End Points 0&0 0 22 0 ==== tdtddtdt ff ( ) ( ) ( ) ( ) 0&0 0 22 0 ==== txtxtxtx ff δδδδ ( ) ( ) ( ) ( ) 0&0 0 22 0 ==== txdtxdtxdtxd ff Therefore: ( ) ( ) ( ) ( ) ( ) ( ) 0,,,, 0 =            +      = ∫ ••• • ft t T x T x dttxtxtxtFtxtxtxtFJ δδδ where: ( ) ( ) εε ε δ ε dtxtx 0 , =∂ ∂ = ( ) ( ) εε ε δ ε dtxtx 0 , = •• ∂ ∂ =
34 SOLO Calculus of Variations Necessary Conditions for Extremum (continue – 2) Transformation of the First Variation δ J by integration by parts (a) First way: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫       −      =∫       •••• ••• f f f t t T x t t T x t t T x dttxtxtxtF dt d txtxtxtFdttxtxtxtF 0 0 0 ,,,,,, δδδ Because we have( ) ( ) 00 == txtx f δδ ( ) ( ) ( ) ( ) ( ) 0,,,, 0 =            −      = ∫ •• • ft t T x x dttxtxtxtF dt d txtxtFJ δδ δ J must be zero for all admissible variations , where and dt0 = dtf = 0. ( )txδ ( ) ( ) 00 == txtx f δδ Note: •••••••       +      +      =      • xxxtGxxxtGxxtGxxtG dt d T x T xt ,,,,,,,, therefore integration by parts assumes however, that not only , but also exists and is continuous in (t0, tf ). • x •• x End Note Return to Table of Content
35 SOLO Calculus of Variations Necessary Conditions for Extremum (continue – 3) 4.1 The First Fundamental Lemma of the Calculus of Variations (Du Bois-Reymond-1879) If M(t) is a continuous function of t in (t0, tf ) and if ( ) ( ) 0 0 =∫ ft t tdtxtM δ for all functions that vanish at t0 and tf and which admit a continuous derivative in (t0, tf ), then ( )txδ ( ) fttttM ≤≤= 00 Paul David Gustav Du Bois-Reymond (1831-1889) Proof: Suppose M (t) ≠ 0, say greater than zero at a point t1 on the interval (t0, tf ). Because M(t) is continuous exists a neighbor of t1 say (t1-ζ, t1+ζ) in which we chose ( ) ( ) ( ) ( )   +−∈−−+− +−∉ = ζζζζ ζζ δ 11 2 1 2 1 11 , ,0 ttttttt ttt x kk ( )tM tft0t 1t ζ+1tζ−1t xδ admits a continuous derivative in (t0, tf ) and vanishes at t0 and t1 and nevertheless makes ( ) ( ) 0 0 >∫ ft t tdtxtM δ contrary to the hypothesis; therefore M (t) ≠ 0 is impossible for al t0≤ t ≤tf. q.e.d. Return to Table of Content
36 SOLO Calculus of Variations Necessary Conditions for Extremum (continue – 4) 4.2 The Euler-Lagrange Equation The Necessary Condition for an extremal is δ J = 0, where ( ) ( ) ( ) ( ) ( ) 0,,,, 0 =            −      = ∫ •• • ft t T x x dttxtxtxtF dt d txtxtFJ δδ For all variations satisfying and dt0 = dtf = 0.( )txδ ( ) ( ) 00 == txtx f δδ ( ) ( ) ( ) ( )( ) f T n ttttxtxtxtx ≤≤= 021 ,,, δδδδ  By choosing for i=1,…,n δ xi(t) ≠ 0 and δ xj(t) = 0 for all j ≠ i and using the First Fundamental Lemma, we can see that δ J= 0 for all admissible variations if and only if( )txδ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )           =      −      =      −      =      −      •• •• •• • • • 0,,,, 0,,,, 0,,,, 2 2 1 1 txtxtF dt d txtxtF txtxtF dt d txtxtF txtxtF dt d txtxtF n n x x x x x x 
37 SOLO Calculus of Variations Necessary Conditions for Extremum (continue – 5) The Euler-Lagrange Equation (continue – 1) As a matrix equation ( ) ( ) ( ) ( ) 0,,,, =      −      •• • txtxtF dt d txtxtF x x Euler-Lagrange Equation It was discovered by Euler in 1744. Later in 1760 Lagrange discussed this equation and introduced the notation δ and the notion of Variation. Leonhard Euler (1707-1783) Joseph-Louis Lagrange (1736-1813) By developing this equation we get: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,,,,,,,, =      −      +      +      ••••••• •••• txtxtFtxtxtFtxtxtxtFtxtxtxtF x txxxxx This is a Nonhomogeneous, Second Order, Differential Equation. ( ) ( )      • •• txtxtF xx ,,If is nonsingular on t0 ≤ t ≤tf, then ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )             −      +                  −= •••• − ••• •••• txtxtFtxtxtFtxtxtxtFtxtxtFtx x txxxxx ,,,,,,,, 1 The existence of is achieved if the matrix has an inverse for all t in (t0, tf ). If this condition is satisfied we have a Regular Problem. The problem is well defined if 2n boundary conditions are defined (see Appendix 3 ). ( )tx •• ( ) ( )      • •• txtxtF xx ,,
38 SOLO Calculus of Variations Necessary Conditions for Extremum (continue – 6) The Euler-Lagrange Equation (continue – 2) Leonhard Euler (1707-1783) Joseph-Louis Lagrange (1736-1813) Therefore the general solutions of the Euler-Lagrange Equations are therefore two vector parameters solutions( ) ( )T n T n βββααα ,,,,, 11  == ( ) ( )βαϕ ,,ttx = and those parameters are defined by the 2n boundary conditions.
39 SOLO Calculus of Variations Necessary Conditions for Extremum (continue – 7) (b) Second way: Du Bois-Reymond and Hilbert integrated the first, instead of the second, term of δ J by parts ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫ ∫ •••• •••               −      +            =             +      = • • f f f f f t t Tt t x x t t t t T x t t T x T x dttxdtxxtFxxtFtxdtxxtF dttxtxtxtFtxtxtxtFJ 0 0 0 0 0 ,,,,,, ,,,, δδ δδδ Paul David Gustav Du Bois-Reymond (1831-1889) David Hilbert (1862 – 1943) Because , we have:( ) ( ) 00 == txtx f δδ ( ) 0,,,, 0 0 =               −      = ∫ ∫ ••• • f ft t Tt t x x dttxdtxxtFxxtFJ δδ δ J must be zero for all admissible variations , such that .( )txδ ( ) ( ) 00 == txtx f δδ Return to Table of Content
40 SOLO Calculus of Variations Necessary Conditions for Extremum (continue – 8) 4.3 The Second Fundamental Lemma of the Calculus of Variations (Du Bois-Reymond-1879) Paul David Gustav Du Bois-Reymond (1831-1889) Proof: q.e.d. If N(t) is a continuous function of t in (t0, tf ) and if for all functions of classes C(1) that vanish at t0 and tf then N (t) must be constant in (t0, tf ). ( )txδ ( ) ( ) 0 0 =∫ •ft t dttxtN δ Let subtract from the previous equation the identity: ( ) ( ) ( )[ ] 00 0 =−=∫ • txtxCdttxC f t t f δδδ where C is a constant. ( )[ ] ( ) 0 0 =∫ − •ft t dttxCtN δ From all the possible variations let choose the following particular variation: ( ) ( )[ ] 0>−= • εεδ CtNtx For this variation we must have: ( )[ ] ( ) ( )[ ] 0 00 2 =∫ −=∫ − • ff t t t t dtCtNdttxCtN δ This is possible only if N (t) = C. Therefore N (t) = C is a necessary condition. The sufficiency condition is proven by substituting N (t) = C in the original equation.
41 SOLO Calculus of Variations Necessary Conditions for Extremum (continue – 9) The Second Fundamental Lemma of the Calculus of Variations (Du Bois-Reymond-1879) (continue – 1) Let apply the Second Fundamental Lemma of the Calculus of Variations to the equation: ( ) 0,,,, 0 0 =               −      = ∫ ∫ ••• • f ft t Tt t x x dttxdtxxtFxxtFJ δδ ( ) ( ) ( ) ( ) f T n ttttxtxtxtx ≤≤      = •••• 021 ,,, δδδδ where We obtain the following form of the Euler-Lagrange Equation: ∫       +=      •• • ft t x x dtxxtFCxxtF 0 ,,,, From this equation we can see that every solution of our problem with continuous first derivative – not only those admitting a second derivative – must satisfy the Euler-Lagrange Equation; i.e. the existence of is not necessary.( )tx •• Return to Table of Content
42 SOLO Calculus of Variations 4.4 Special Cases F doesn’t depend explicitly on the free variable t ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) xxxF td d xxF xxxF td d xxxFxxxFxxxFxxxFxxF td d T xx T x T x T x T x T x           −=       −−+=− ,, ,,,,,, For an extremal the Euler-Lagrange equation applies, and we have ( ) ( ) ( ) ( ) 0,, =      −      •• • txtxF dt d txtxF x x ( ) ( )[ ] 0,, =− xxxFxxF td d T x   Therefore ( ) ( ) constCxxxFxxF T x ==−   ,, Let perform the following:
43 SOLO Calculus of Variations Special Cases (continue – 1) F is not an explicit function of x In this case the Euler-Lagrange equation is: ( ) 0, =      • • txtF dt d x that can be integrated to give ( ) constCtxtF x ==      • • , F is not an explicit function of x In this case the Euler-Lagrange equation is: ( )( ) 0, =txtFx ( )( ) ( ) ( ) 0,..,0,det =∀≠ xtFtsxttxtF xxIf we can find that satisfies this equation.( )txx = According to Implicit Function Theorem this solution is unique..
44 SOLO Calculus of Variations Special Cases (continue – 2) F is an exact differential ( ) ( )( ) ( ) ( ) ( ) xxtVxtVxtV td d txtxtF T xt  ,,,,, +=≡ If this is true than ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )00 , , , , ,,,,, 000000 xtVxtVdtxtV td d dttxtxtF ff xtP xtP xtP xtP ffffff −=∫=∫  therefore the functional is independent on the integration path. Let find what conditions F must satisfy in order to be an exact differential. Let compute ( ) ( )( ) ( ) ( ) xxtVxtVtxtxtF xxxtx  ,,,, += ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) xxtVxtVtxtxtF td d xtVtxtxtF T txtxxxx   ,,,,,,, +=→= From those relations we can see that the condition that F is an exact differential if and only if the Euler-Lagrange equation is an identity. ( ) ( )( ) ( ) ( )( )txtxtFtxtxtF td d xx  ,,,, ≡ Return to Table of Content
45 SOLO Calculus of Variations 4.5 Example 1: Brachistocrone A particle slides on a frictionless wire between two fixed points A(0,0) and B (xfc, yfc) in a constant gravity field g. The curve such that the particle takes the least time to go from A to B is called brachistochrone. x y V ( )tγ ( )fcfc yxB ,fcx fcy N  ( )0,0A ∫∫       = +       + = cfcf xx xd xd yd yxFxd ygV xd yd J 00 2 0 2 ,, 2 1 We derived the cost function: xd yd y ygV y xd yd yxF = + + =      : 2 1 :,, 2 0 2   wher e F doesn’t depend explicitly on the free variable x, therefore if we replace and we use the result obtained for F not depending explicitly on x, we obtain ( ) ( )xtyx ,, → ( ) ( ) const ygVy y ygV y yyFyyyF y == ++ − + + =− α 212 1 ,, 2 0 2 2 2 0 2     const ygVy == ++ α 21 1 2 0 2  or
46 SOLO Calculus of Variations Example 1: Brachistocrone (continue – 1) x y V ( )tγ ( )fcfc yxB ,fcx fcy N  ( )0,0A const ygVy == ++ α 21 1 2 0 2  Let define a parameter τ such that τcos 1 1 2 =       + xd yd and const ygV ygV xd yd == + = +      + α τ 2 cos 21 1 2 02 0 2 From which ( ) ( )ττ αα τ 2cos12cos1 4 1 2 cos 2 22 22 0 +=+==+ r ggg V y Tacking the derivative of this equation with respect to τ we obtain τ τ 2sin2r d yd −=
47 SOLO Calculus of Variations Example 1: Brachistocrone (continue – 2) τ ττ τ ττ 2 22 2 2 cos /1 1 =       +            =       + d yd d xd d xd d xd d yd ( ) ( )ττ ττ τ τττ τ τττ ττ 2sin2 2cos12cos4 cossin16sin cos2sin4cos 0 2 4222 2 2222 22 +±=→ +±=±=→ =      → +      =      → rxx rr d xd r d xd r d xd d xd Let change variables to 2τ = θ – π, to get ( ) ( )θ θθ cos1 2 sin 2 0 0 −=+ −+= r g V y rxx θsinr θcosr θr x y 0x 0V g V 2 2 0 r r A B ),( yx θ We obtain the equation of a cycloid generated by a circle of radius r rolling upon the horizontal line and starting at the point g V y 2 2 0 −=         −− g V x 2 , 2 0 0
48 HISTORY OF CALCULUS OF VARIATIONS The brachistochrone problem ( ) ( )     −−= −+= g V ry rxx 2 cos1 sin 2 0 0 θ θθ Cycloid Equation ∫∫∫∫       = +       + === cfcfcf xxxt xd xd yd yxFxd ygV xd yd V sd tdJ 00 2 0 2 00 ,, 2 1 Minimization Problem Solution of the Brachistochrone Problem: SOLO Johann Bernoulli 1667-1748
49 SOLO Calculus of Variations Example 2: Minimum Surface of Revolution x y ( )bybB , ( )ayaA , ( ) ( )22 ydxdsd += y ( )∫       += b a xd xd yd xyJ 2 12π For this problem we derived the cost function: Given two points A (a,ya) and B (b, yb) a≠b in the plane. Find the curve that joints these two points with a continuous derivative, in such a way that the surface generated by the rotation of this curve about the x axis has the smallest possible area. We have ( ) ( ) ( ) ( ) xd yd xyxyxyyyxF =+= :12:,, 2  π F doesn’t depend explicitly on the free variable x, therefore we can apply the results for this special case, with ( ) ( )xtyx ,, → ( ) ( ) C y y yyyyyFyyyF y ππ 2 1 12,, 2 2 2 =         + −+=−     2 1 yCy +=or Separating variables, we obtain C xd C y C yd = −      1 2 1 2 −      = C y y
50 SOLO Calculus of Variations Example 2: Minimum Surface of Revolution (continue – 1) x y ( )bybB , ( )ayaA , ( ) ( )22 ydxdsd += y C xd C y C yd = −      1 2 Integration of this equation, gives         −      +=− 1ln 2 1 C y C y CCx from which 1exp 2 1 −      +=      − C y C y C Cx take the square 1exp211212122exp 1 222 1 −      − =−         −      +=−      +−      =      − C Cx C y C y C y C y C y C y C y C Cx From this equation we can compute 2 expexp 11       − −+      − = C Cx C Cx C y ( )       − = C Cx Cxy 1 coshor The solution is a curve called a catenary (catena = chain in Latin) and the surface of revolution which is generated is called a catenoid of revolution.
51 SOLO Example 3: Geometrical Optics and Fermat Principle We have: constS = constdSS =+ sˆ ∫ 2 1 P P dsn 1 P 2 P ( ) ( ) ( )∫∫∫∫ =      +      +=== 2 1 2 1 2 1 ,,,, 1 1,, 1 ,, 1 0 22 00 P P P P P P xdzyzyxF c xd xd zd xd yd zyxn c dszyxn c tdJ  Let find the stationarity conditions of the Optical Path using the Calculus of Variations ( ) ( ) ( ) xd xd zd xd yd zdydxdds 22 222 1       +      +=++= Define: xd zd z xd yd y ==  &: ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += Calculus of Variations
52 SOLO Necessary Conditions for Stationarity (Euler-Lagrange Equations) ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += 0= ∂ ∂ −      ∂ ∂ y F y F dx d  ( ) [ ] 2/122 1 ,, zy yzyxn y F    ++ = ∂ ∂ [ ] ( ) y zyxn zy y F ∂ ∂ ++= ∂ ∂ ,, 1 2/122  ( ) [ ] [ ] 01 1 ,, 2/122 2/122 = ∂ ∂ ++−         ++ y n zy zy yzyxn xd d    0= ∂ ∂ −      ∂ ∂ z F z F dx d  [ ] [ ] 0 11 2/1222/122 = ∂ ∂ −         ++++ y n zy yn xdzy d    Calculus of Variations Example 3: Geometrical Optics and Fermat Principle (continue – 1)
53 SOLO Necessary Conditions for Stationarity (continue - 1) We have [ ] 0 1 2/122 = ∂ ∂ −         ++ y n zy yn sd d   y n sd yd n sd d ∂ ∂ =      In the same way [ ] 0 1 2/122 = ∂ ∂ −         ++ z n zy zn sd d   z n sd zd n sd d ∂ ∂ =      Calculus of Variations Example 3: Geometrical Optics and Fermat Principle (continue –2)
54 SOLO Necessary Conditions for Stationarity (continue - 2) Using ( ) ( ) ( ) xd xd zd xd yd zdydxdds 22 222 1       +      +=++= we obtain 1 222 =      +      +      sd zd sd yd sd xd Differentiate this equation with respect to s and multiply by n      sd d 0=            +            +            sd zd sd d n sd zd sd yd sd d n sd yd sd xd sd d n sd xd sd nd sd zd sd nd sd yd sd nd sd xd sd nd =      +      +      222 sd nd and sd nd sd zd n sd d sd zd sd yd n sd d sd yd sd xd n sd d sd xd =            +            +            add those two equations Calculus of Variations Example 3: Geometrical Optics and Fermat Principle (continue – 3)
55 SOLO Necessary Conditions for Stationarity (continue - 3) sd nd sd zd n sd d sd zd sd yd n sd d sd yd sd xd n sd d sd xd =            +            +            Multiply this by and use the fact that to obtain xd sd cd ad cd bd bd ad = xd nd sd zd n sd d xd zd sd yd n sd d xd yd sd xd n sd d =            +            +      Substitute and in this equation to obtain y n sd yd n sd d ∂ ∂ =      z n sd zd n sd d ∂ ∂ =      xd zd z n xd yd y n xd nd sd xd n sd d ∂ ∂ − ∂ ∂ −=      Since n is a function of x, y, z x n xd zd z n xd yd y n xd nd zd z n yd y n xd x n nd ∂ ∂ = ∂ ∂ − ∂ ∂ −→ ∂ ∂ + ∂ ∂ + ∂ ∂ = and the previous equation becomes x n sd xd n sd d ∂ ∂ =      Calculus of Variations Example 3: Geometrical Optics and Fermat Principle (continue – 4)
56 SOLO Necessary Conditions for Stationarity (continue - 4) We obtained the Euler-Lagrange Equations: x n sd xd n sd d ∂ ∂ =      y n sd yd n sd d ∂ ∂ =      z n sd zd n sd d ∂ ∂ =      k sd zd j sd yd i sd xd sd rd kzjyixr ˆˆˆ ˆˆˆ ++= ++=   Define the unit vectors in the x, y, z directionskji ˆ,ˆ,ˆ The Euler-Lagrange Equations can be written as: n sd rd n sd d ∇=       This is the Eikonal Equation from Geometrical Optics. Calculus of Variations Example 3: Geometrical Optics and Fermat Principle (continue – 5) Return to Table of Content
57 SOLO Calculus of Variations 5. Boundary Conditions Until now we considered only the variations passing through the end points and . But those are not all the admissible variations. If or are not specified then if we consider all admissible variations (see Figure), then δ J will be given by: ( )* 0 * 0 * , xtA ( )*** , ff xtB ( )00 , xtA ( )ff xtB , ( )txδ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫             +      +      −      = ••••• • ft t T x xffff dttxtxtxtFtxtxtxtFdttxtxtFdttxtxtFJ 0 ,,,,,,,, 0000 δδδ ( )( )000 , txtA ( )( )fff txtB , ( )tx*x t ( )ε,1 tx ( )ε,2 tx Variations that Satisfy the Boundary Conditions Integrating by parts the second term of the integral as before we have: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫             −      +             +      −            +      = •• •••• • •• f f t t T x x t T x t T x dttxtxtxtF dt d txtxtF xtxtxtFdttxtxtFxtxtxtFdttxtxtFJ 0 0 ,,,, ,,,,,,,, δ δδδ
58 SOLO Calculus of Variations Boundary Conditions (continue – 1) ( )( )000 , txtA ( )( )fff txtB , ( )tx*x t ( )ε,1 tx ( )ε,2 tx Variations that Satisfy the Boundary Conditions ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫             −      +             +      −            +      = •• •••• • •• f f t t T x x t T x t T x dttxtxtxtF dt d txtxtF xtxtxtFdttxtxtFxtxtxtFdttxtxtFJ 0 0 ,,,, ,,,,,,,, δ δδδ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫             −      +             +            −      −             +            −      = •• •••• •••• • •• •• f f t t T x x t T x T x t T x T x dttxtxtxtF dt d txtxtF xdtxtxtFdtxtxtxtFtxtxtF xdtxtxtFdtxtxtxtFtxtxtFJ 0 0 ,,,, ,,,,,, ,,,,,, δ δ But ( ) ( ) ( ) iiiii dttxtxdtx • −= δ Therefore:
59 SOLO Calculus of Variations Boundary Conditions (continue – 2) We found before that the necessary conditions such that δ J is zero for those admissible solutions passing through the points and are the Euler-Lagrange Equation:( )* 0 * 0 * , xtA ( )*** , ff xtB ( ) ( ) ( ) ( ) 0,,,, =      −      •• • txtxtF dt d txtxtF x x For other admissible variations we shall need to add the additional necessary conditions, such that δ J is zero, called Transversality Conditions Equations: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) fitxdtxtxtFdttxtxtxtFtxtxtF iiii x iiiii T x iii ,00,,,,,, ==      +            −      •••• •• (a) Suppose that the following relation defines the boundary: ( ) ( ) ( ) ( ) ( ) fidttdtt dt d txdttx iitiiiii ,0=Ψ=Ψ=→Ψ= then the Transversality Conditions Equations are: ( ) ( ) ( ) ( ) ( ) ( ) fitxttxtxtFtxtxtF iitiii T x iii ,00,,,, ==      −Ψ      +      ••• •
60 SOLO Calculus of Variations Boundary Conditions (continue – 3) Geometric Interpretation of the Transversality Conditions Let plot as a function of . The hyper-plane tangent at( ) ( )      = • txtxtF ,,η • = xξ ( ) ( ) ( )      == •• iiiiii txtxtFtx ,,,ηξ is given by ( ) ( ) ( ) ( ) ( )      +      −      = ••• iiiiiii T x txtxtFtxtxtxtF ,,,, ξη  ( ) ( ) ( ) ( ) ( ) ( ) fitxttxtxtFtxtxtF iitiii T x iii ,00,,,, ==      −Ψ      +      ••• • We can see that for η = 0 the last equation is identical to the Transversality Conditions Equation. Geometric Representation of the Transversality Conditions ( ) ( ) ( ) ( ) ( ) fidttdtt dt d txdttx iitiiiii ,0=Ψ=Ψ=→Ψ=
61 SOLO Calculus of Variations Boundary Conditions (continue – 4) Transversality Conditions Suppose that ti and are not defined and is not a function of ti, then dti and are independent differentials and therefore both coefficients of dti and must be zero. ix ix ixd ixd ( ) ( ) ( ) ( ) ( ) 0,,,, =      −      ••• • iiii T x iii txtxtxtFtxtxtF ( ) ( ) 0,, =      • • iii x txtxtF Or, by using the second equation to simplify the first we get: ( ) ( ) ( ) ( ) fi txtxtF txtxtF iii x iii ,0 0,, 0,, =        =      =      • • • Those Equations are called Natural Boundary Conditions because they arise naturally when the original problem doesn’t specify boundary conditions.
62 SOLO Calculus of Variations Boundary Conditions (continue – 5) Example: Transversality Conditions for Geometrical Optics and Fermat’s Principle ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += For the Geometrical Optics we obtained: Assume that the initial and final boundaries are defined by the surfaces A (x0, y0, z0) and B (xf, yf, zf) respectively. The transversality conditions at the boundaries i=0,f are defined by ( ) ( ) ( )[ ] ( ) ( ) 0,,,,,,,, ,,,,,,,,,,,, =++ −− iziy izy dzzyzyxFdyzyzyxF dxzyzyxFzzyzyxFyzyzyxF     ( ) [ ] [ ] [ ] [ ] ( ) sd xd zyxn zy n zy zn z zy yn yzynFzFyzyzyxF zy ,, 1 11 1,,,, 2/122 2/1222/122 2/122 = ++ = ++ − ++ −++=−−         ( ) [ ] ( ) ( ) [ ] ( ) sd zd zyxn zy zzyxn z F F sd yd zyxn zy yzyxn y F F z y ,, 1 ,, ,, 1 ,, 2/122 2/122 = ++ = ∂ ∂ = = ++ = ∂ ∂ =         For are tangent to the boundary surfaces A (x0, y0, z0) and B (xf, yf, zf).fird i ,0=  From Transversality Conditions we can see that the rays are normal (transversal) to the boundary surfaces (see Figure). Transversality Conditions Return to Table of Content
63 SOLO Calculus of Variations 6. Corner Conditions In the development of the Euler-Lagrange Equation we assumed that not only is continuous, but also . However, there are a number of problems, for which this assumption is not true, for example problems of reflection or refraction. ( )tx ( )tx • We define such problems as follows: Find the curve that passes through the boundary points (given or not) and and extremizes the functional . This curve should reach the point after having been reflected by a given function (see Figure). ( )tx ( )00 , xtA ( )ff xtB , ( )[ ] ( ) ( )∫       = •ft t dttxtxtFtxJ 0 ,, ( )ff xtB , ( )tx Ψ= corner point ( )( )000 , txtA ( )( )fff txtB , ( )tx*x tct ( )tΨ The Corner Point of the Trajectories
64 SOLO Calculus of Variations Corner Conditions (continue – 1) corner point ( )( )000 , txtA ( )( )fff txtB , ( )tx*x tct ( )tΨ The Corner Point of the Trajectories Solution: Let define by tc the unknown time when the extremal is reflected. Then we can express the functional J in the form: ( )tx • ( )[ ] ( ) ( ) ( ) ( )∫∫       +      = •• f c c t t t t dttxtxtFdttxtxtFtxJ ,,,, 0 We suppose that is continuous in each of the intervals (t0, tc-), (tc+, tf). Then for both intervals we have: ( )tx • (1) The Euler-Lagrange Equation is: ( ) ( ) ( ) ( ) cf x x ttttttxtxtF dt d txtxtF ≠≤≤=      −      •• • 00,,,, ( )tx •• if exists, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ + • − •       +=            +=      •• •• f c c t t x x t t x x dttxtxtFCtxtxtF dttxtxtFCtxtxtF 0 0 0 ,,,, ,,,, ( )tx •• if doesn’t exist
65 SOLO Calculus of Variations Corner Conditions (continue – 2) corner point ( )( )000 , txtA ( )( )fff txtB , ( )tx*x tct ( )tΨ The Corner Point of the Trajectories Solution (continue – 1): (2) The Transversality Conditions at the initial (0) and final (f) points are: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) fitxdtxtxtFdttxtxtxtFtxtxtF iiii T x iiiii T x iii ,00,,,,,, ==      +            −      •••• •• Then: ( ) ( ) 00000 0000 =         +      −−         +      −= ++ • −− • + • + • − • − • c t T x c t T x c t T x c t T x txdFdtxFFtxdFdtxFFJ cccc δ But tc- = tc+ = tc and , thereforeccc xdxdxd == +− 00 ( ) 0 0000 =      −+               −−      −= + • − • + • − • •• c t T xt T x c t T x t T x txdFFdtxFFxFFJ cccc δ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,,,, ,,,, ,,,, 00 000 000 =            −      +          +      −          −      + • − • + • + • + • − • − • − • •• • • cccc x ccc x ccccc x ccc cccc x ccc txdtxtxtFtxtxtF dttxtxtxtFtxtxtF txtxtxtFtxtxtF The necessary conditions for extremal at the corners are: Those are the Weierstrass-Erdmann Corner Conditions. Those equations were developed independently by Weierstrass and Erdmann in 1877. Karl Theodor Wilhelm Weierstrass 1815-1897
66 SOLO Calculus of Variations Corner Conditions (continue – 3) corner point ( )( )000 , txtA ( )( )fff txtB , ( )tx*x tct ( )tΨ The Corner Point of the Trajectories Solution (continue – 2): (a) If they are a priori conditions at the corner like: ( ) ( ) ( ) ( ) ( ) cctccccc dttdtt dt d txdttx Ψ=Ψ=→Ψ= then the necessary conditions at the corner are: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )      −Ψ      +            −Ψ      +      + • + • + • − • − • − • • • 000 000 ,,,, ,,,, cctccc T x ccc cctccc T x ccc txttxtxtFtxtxtF txttxtxtFtxtxtF
67 SOLO Calculus of Variations Corner Conditions (continue – 4) Solution (continue – 3): (b) If they are not a priory conditions at the corner; i.e. the function is not a priori defined then dtc and are independent variables and ( ) ( )cc ttx Ψ= cxd ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )      =            −      =       −      + • − • + • + • + • − • − • − • •• • • 00 000 000 ,,,, ,,,, ,,,, ccc x ccc x cccc T x ccc cccc T x ccc txtxtFtxtxtF txtxtxtFtxtxtF txtxtxtFtxtxtF corner point ( )( )000 , txtA ( )( )fff txtB , ( )tx*x tct ( )tΨ The Corner Point of the Trajectories
68 SOLO Calculus of Variations Corner Conditions (continue – 5) Geometric Interpretation of the Corner Conditions Since the Corner Conditions where derived from the Transversality Conditions, we have a similar geometrical interpretation. Let plot as a function of .( ) ( )      = • txtxtF ,,η • = xξ Since the hyper-plane tangent at is given by( ) ( )+− = cc txtx ( ) ( ) ( )      == − • −− • − cccccc txtxtFtx ,,,ηξ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )− • − • − • − • − • − • − •       −      +      =       +      −      = cccc T xcccccc T x ccccccc T x txtxtxtFtxtxtFtxtxtF txtxtFtxtxtxtF ,,,,,, ,,,,   ξ ξη The hyper-plane tangent at is given by( ) ( ) ( )      == + • ++ • + cccccc txtxtFtx ,,,ηξ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )+ • + • + • + • + • + • + •       −       +      =       +      −      = cccc T x cccccc T x ccccccc T x txtxtxtF txtxtFtxtxtF txtxtFtxtxtxtF ,, ,,,, ,,,,    ξ ξη But according to the Corner Conditions the two tangent hyper-planes are the same (see Figure )
SOLO Calculus of Variations Corner Conditions (continue – 6) Example: Corner Conditions for Geometrical Optics and Fermat’s Principle ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += For the Geometrical Optics we obtained: Let examine the following two cases: 1. The optical path passes between two regions with different refractive indexes n1 to n2 (see Figure) In region (1) we have: In region (2) we have: ( ) ( ) 22 11 1,,,,,, zyzyxnzyzyxF  ++= ( ) ( ) 22 22 1,,,,,, zyzyxnzyzyxF  ++= ( ) ( ) ( )[ ]{ ( ) ( ) ( )[ ]} ( ) ( )[ ] ( ) ( )[ ] 0,,,,,,,, ,,,,,,,, ,,,,,,,,,,,, ,,,,,,,,,,,, 222111 222111 22222222222 11111111111 =−+ −+ −−− −− dzzyzyxFzyzyxF dyzyzyxFzyzyxF dxzyzyxFzzyzyxFyzyzyxF zyzyxFzzyzyxFyzyzyxF zz yy zy zy         The Weierstrass-Erdmann necessary condition at the boundary between the two regions is where dx, dy, dz are on the boundary between the two regions.
SOLO Calculus of Variations Corner Conditions (continue – 7) Example: Corner Conditions for Geometrical Optics and Fermat’s Principle (continue – 1) ( ) [ ] [ ] [ ] [ ] ( ) sd xd zyxn zy n zy zn z zy yn yzynFzFyzyzyxF zy ,, 1 11 1,,,, 2/122 2/1222/122 2/122 = ++ = ++ − ++ −++=−−         ( ) [ ] ( ) ( ) [ ] ( ) sd zd zyxn zy zzyxn z F F sd yd zyxn zy yzyxn y F F z y ,, 1 ,, ,, 1 ,, 2/122 2/122 = ++ = ∂ ∂ = = ++ = ∂ ∂ =         ( ) ( ) 0 21 21 =⋅        − rd sd rd n sd rd n rayray   where is on the boundary between the two regions andrd  ( ) ( ) sd rd s sd rd s rayray 2 :ˆ, 1 :ˆ 21  == are the unit vectors in the direction of propagation of the rays.
SOLO Calculus of Variations Corner Conditions (continue – 8) Example: Corner Conditions for Geometrical Optics and Fermat’s Principle (continue – 2) ( ) 0ˆˆ 2211 =⋅− rdsnsn  2211 ˆˆ snsn −Therefore is normal to .rd  Since can be in any direction on the boundary between the two regions (see Figure ) is parallel to the unit vector normal to the boundary surface, and we have rd  2211 ˆˆ snsn − 21 ˆ −n ( ) 0ˆˆˆ 221121 =−×− snsnn This the Snell’s Law of Geometrical Optics
SOLO Calculus of Variations Corner Conditions (continue – 9) Example: Corner Conditions for Geometrical Optics and Fermat’s Principle (continue – 3) 2. The optical path is reflected at the boundary. ( ) ( ) ( ) 0ˆˆ 21 21 =⋅−=⋅        − rdssrd sd rd sd rd rayray   n1 = n2 , we obtain i.e. is normal to , i.e. to the boundary where the reflection occurs. Also we can write 21 ˆˆ ss − rd  ( ) 0ˆˆˆ 2121 =−×− ssn ( ) ( ) ( ) 0ˆˆ 21 221121 =⋅−=⋅      − rdsnsnrd sd rd n sd rd n rayray   In this case, if we substitute in the equation Return to Table of Content
SOLO Calculus of Variations 7 Sufficient Conditions and Additional Necessary Conditions for a Weak Extremum We found that: ( ) ( ) ( ) ( )JJJd d Jd d d dJ JJJ 3232 0 2 2 0 2 1 2 1 0: δδδεε ε ε ε εε εε Ο/++=Ο/++==−=∆ == The Necessary Condition that Δ J ≥ 0 or ≤ 0 for all small dε is 00 0 == = Jor d dJ δ ε ε For Sufficient Conditions for a “Weak” Local Extremum we must add the following: ( ) ( )0000 2 0 2 2 ≤≥≤≥ = Jor d Jd δ ε ε for a minimum (maximum) solution. The expression for δ2 J is ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫                     +      +      ++      −+             −+++      −−             −+++      −= •••• •• •• ••••• •• •• ε ε δδδδδδδδδ δ f f t t xx T xx T xx T xx T T x x t T x T x T x T xt t f T x f T x ff T xf T xt dtxFxxFxxFxxFxxF dt d F tdxFFxdFdtxdFdtxFF tdxFFxdFdtxdFdtxFFJ 0 0 2 0 2 0 2 00 2 0 2222 2 2
SOLO Calculus of Variations Sufficient Conditions and Additional Necessary Conditions for a Weak Extremum (continue -1) Suppose first that the end points are fixed; i.e.: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 00 00 00 2 0 2 0 2 0 2 0 2 0 2 0 ==== ==== ==== ff ff ff txdtxdtxdtxd txtxtxtx tdtddtdt δδδδ In this case: ( ) ( ) ( ) ( ) ( ) ( ) ∫                     +      +      ++      −= •••• ••••• ε ε δδδδδδδδδδ ft t xx T xx T xx T xx T T x x dtxFxxFxxFxxFxxF dt d FJ 0 22 For an extremal solution the Euler-Lagrange Equation holds, therefore0=− • x x F dt d F ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫                               = ∫                     +      +      += • • •••• ••• • •••• ε ε ε ε δ δ δδ δδδδδδδδδ f f t t xxxx xx xxT T t t xx T xx T xx T xx T dt x x FF FF xx dtxFxxFxxFxxFxJ 0 0 2 We have the following properties of the derivatives T xxxx T xxxx T xxxx FFFFFF  ===
SOLO Calculus of Variations Sufficient Conditions and Additional Necessary Conditions for a Weak Extremum (continue -2) Let define: TT xxxx T xxxx TT xxxx RFFRFFQPFFP ======== •• :,:,: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ++= +++=                             = • •== == ε ε ε ε ε ε δδδδδδ δδδδδδδδ δ δ δδδ f f f ii ii t t TTT t t TTTTT t t T T T xdxd tddt dtxPxxQxxPx dtxPxxQxxQxxPx dt x x RQ QP xxJ 0 0 0 2 2 2 00 00 2   Therefore: Return to Table of Content
SOLO Calculus of Variations 8 Legendre’s Necessary Conditions for a Weak Minimum (Maximum) – 1786 Adrien-Marie Legendre 1752-1833 Proof: Let start from the necessary condition of an Minimal Optimal Trajectory, that ( ) ( ) ( ) 02 0 2 2 00 00 2 ≥++== ∫ == == ε ε δδδδδδδ f ii ii t t TTT xdxd tddt dtxPxxQxxPxJ  (The same reasoning applies for a Maximal Optimal trajectory where it is required that δ2 J ≤ 0) Suppose that R is not positive definite and we have a constant vector such that at some point t = τ, on our curve. Since R (t) was assumed continuous, this inequality will hold over some sufficiently small interval [τ-h, τ+h] . We now define the function so that it vanishes outside and at the end points of the interval, it has all the necessary derivatives; is sufficiently small in absolute value in the interval, but performs fairly rapid oscillations. v 0<vRvT ( )txδ ( )          +<<      − − ≤<−      − + = elsewhere ht h t h th h t h tx 0 1 1 ττν τ ε ττν τ ε δ ( )          +<<− ≤<− =⇒ elsewhere ht h th h tx 0 ττν ε ττν ε δ  The matrix must be Positive (Negative) Definite along a Minimal (Maximal) Optimal Trajectory xxFR =
SOLO Calculus of Variations Legendre’s Necessary Conditions for a Weak Minimum (Maximum) – 1786 (continue – 1) Adrien-Marie Legendre 1752-1833 Proof (continue – 1): ixδ t 1h−τ 2h−τ 2h+τ1h+τ hε          +<<      − − <<−      − + = elsewhere ht h t h th h t h x 0 1 1 ττν τ ε ττν τ ε δ          +<<− <<− = elsewhere ht h th h x 0 ττν ε ττν ε δ  t 1h−τ 2h−τ 2h+τ1h+τ h ε ixδ Since P (t) and Q (t) are continuous matrix functions in the interval t € [0, tf] we can find two positive numbers M1 and M2 such that: ( ) ( ) [ ]f TT ttMvtQvMvtPv ,0, 21 ∈∀<<
SOLO Calculus of Variations Legendre’s Necessary Conditions for a Weak Minimum (Maximum) – 1786 (continue – 2) Proof (continue – 2): ixδ t 1h−τ 2h−τ 2h+τ1h+τ hε          +<<      − − <<−      − + = elsewhere ht h t h th h t h x 0 1 1 ττν τ ε ττν τ ε δ          +<<− <<− = elsewhere ht h th h x 0 ττν ε ττν ε δ  t 1h−τ 2h−τ 2h+τ1h+τ h ε ixδ ( ) ( ) [ ]f TT ttMvtQvMvtPv ,0, 21 ∈∀<< 2 2 0 1 0 1 2 2 0 0 22 211 11 00 Mhdt h t dt h t M dtQv h t dtQv h t dtxQxdtxQx h h h h TT t t T t t T ff ε ττ ε ν τ εν τ εδδδδ τ τ τ τ ≤             ∫       − −+∫       − +≤ ∫ ∫       − −+      − +=∫≤∫ + < − < − +   1 22 0 1 2 0 1 2 1 2 0 0 2 2 2 2 211 11 00 Mhdt h t dt h t Mh dtPv h t hdtPv h t hdtxPxdtxPx h h h h TT t t T t t T ff ε ττ ε ν τ εν τ εδδδδ τ τ τ τ ≤             ∫       − −+∫       − +≤ ∫ ∫       − −+      − +=∫≤∫ + < − < − +  We have ( ) ( ) ( ) ( )[ ]{ } ( )[ ] 10212 21 2 22 0 ≤≤−+= +−+−=∫=∫ + − λλτ ε ντλτλ ε ν ε δδ τ τ someforhRv h hhhRv h dttRv h dtxRx T T h h T t t T f 
SOLO Calculus of Variations Legendre’s Necessary Conditions for a Weak Minimum (Maximum) – 1786 (continue – 3) ixδ t 1h−τ 2h−τ 2h+τ1h+τ hε          +<<      − − <<−      − + = elsewhere ht h t h th h t h x 0 1 1 ττν τ ε ττν τ ε δ          +<<− <<− = elsewhere ht h th h x 0 ττν ε ττν ε δ  t 1h−τ 2h−τ 2h+τ1h+τ h ε ixδ ( ) ( ) ( ) ( )[ ]{ } ( )[ ] 10212 21 2 22 0 ≤≤−+= +−+−=∫=∫ + − λλτ ε ντλτλ ε ν ε δδ τ τ someforhRv h hhhRv h dttRv h dtxRx T T h h T t t T f  Since by assumption and R (t) is a continuous matrix functions in the interval t € [0, tf] we can find a small h1 such that for all h ≤ h1 we have ( ) 0<vRvT τ ( )[ ] 021 2 <−≤−+ µνλτ hR Therefore 1 22 2 0 hhdtxRx ft t T ≤∀−≤∫ µεδδ  Using the previous results we can write ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 21 22 00 00 2 22 22 0000 2 2 hhMhMh dtxPxdtxQxdtxPxdtxPxxQxxPxJ ffff ii ii t t T t t T t t T t t TTT xdxd tddt ≤∀−+≤ ++≤++= ∫∫∫∫ == == µε δδδδδδδδδδδδδ ε ε ε ε ε ε ε ε  Since we can find a small h2 ≤ h1 such that for all h ≤ h1 ( ) 02 0 2 21 2 =−+ =h MhMh µ ( ) 2 2 21 222 022 hhMhMhJ ≤∀<−+≤ µεδ δ2 J turn out negative, which contradicts the before mentioned necessary condition for minimum; i.e. δ2 J ≥ 0. Therefore must be Positive Definite along the trajectory to have a minimum.xxFR = q.e.d.
SOLO Calculus of Variations Legendre’s Necessary Conditions for a Weak Minimum (Maximum) – 1786 (continue – 4) Example: Geometrical Optics and Fermat’s Principle ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += For the Geometrical Optics we obtained: ( ) [ ] [ ] [ ] ( ) [ ] 2/322 2 2/322 2 2/1222/1222 2 1 1 111 ,, zy zn zy yn zy n zy yzyxn yy F        ++ + = ++ − ++ =         ++∂ ∂ = ∂ ∂ ( ) [ ] [ ] 2/3222/122 2 11 ,, zy zyn zy zzyxn yzy F      ++ −=         ++∂ ∂ = ∂∂ ∂ ( ) [ ] ( ) [ ] 2/322 2 2/1222 2 1 1 1 ,, zy yn zy zzyxn zz F      ++ + =         ++∂ ∂ = ∂ ∂ From those equations we obtain: ( ) [ ] ( ) ( )        +− −+ ++ = 2 2 2/322 '' 1'' 1 1 ,, yyx zyz zy zyxn F XX    Let use Sylvester Theorem to check the positiveness of ''XXF [ ] ( ) ( ) [ ] ( )( )[ ] [ ] 0 1 11 11'' 1 det 1 det 2/122 2222 2/3222 2 2/322'' > ++ =−++ ++ =         +− −+ ++ = zy n zyyz zy n yyx zyz zy n F XX      1 ( ) 01 2 >+ z2 We can see that according to Sylvester Theorem is Positive Definite.''XXF James Joseph Sylvester 1814-1897 Return to Table of Content
SOLO Calculus of Variations 9. Jacobi’s Differential Equation (1837) and Conjugate Points Let start from the necessary condition of a Minimal Optimal Trajectory, that ( ) ( ) ( ) 02 0 2 2 00 00 2 ≥∫ ++= == == ε ε δδδδδδδ fii ii t t TTT xdxd tddt dtxRxxQxxPxJ  Define ( ) xRxxQxxPxxx TTT  δδδδδδδδ ++=Ω 2:, Therefore ( ) ( ) ( ) ( ) ∫Ω= == == ε ε δδδδ f ii ii t t xdxd tddt dtxxxJ 0 2 2 , 00 00 2  We have ( ) ( ) xRxQ x xx xQxP x xx T      δδ δ δδ δδ δ δδ 22 , 22 , += ∂ Ω∂ += ∂ Ω∂ We can see that ( ) ( ) ( ) x x xx x x xx xx TT     δ δ δδ δ δ δδ δδ       ∂ Ω∂ +      ∂ Ω∂ =Ω ,, ,2 Since ( ) ( ) ( ) ( ) ( ) ∫       ∂ Ω∂ −      ∂ Ω∂ =∫       ∂ Ω∂ = = f f f f t t Tt t T tx tx t t T T dtx x xx td d x x xx dtx x xx 0 0 0 0 ,,, 0 0 0 δ δ δδ δ δ δδ δ δ δδ δ δ           we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫       +−+= ∫       ∂ Ω∂ − ∂ Ω∂ = ∫               ∂ Ω∂ +      ∂ Ω∂ =∫ Ω= == == f f ffii ii t t TTTTT t t T t t TT t t xdxd tddt dtxRxQx dt d QxPx dtx x xx dt d x xx dtx x xx x x xx dtxxxJ 0 0 00 2 2 ,, 2 1 ,, 2 1 , 00 00 2 δδδδδ δ δ δδ δ δδ δ δ δδ δ δ δδ δδδδ ε ε       
SOLO Calculus of Variations Jacobi’s Differential Equation (1837) and Conjugate Points (continue – 1) we have ( ) ( ) ( )∫     +−+= ft t TTTTT dtxRxQx dt d QxPxxJ 0 2 δδδδδδδ  Gilbert Ames Bliss (1876 –1951) Gilbert A. Bliss (1876-1951) suggested to show that the minimum of δ2 J for all possible is non-negative. If this is true thanxδ ( ) ( ) 0min 22 ≥> xJxJ x δδδδ δ We obtain the following Secondary (Accessory) Variational Problem: ( ) ( ) ( ) ( ) ∫ Ω= == == ε εδδ δδδδ fii ii t tx xdxd tddtx dtxxxJ 0 2 2 ,minmin 00 00 2  The necessary conditions for a minimum are satisfaction of 1.Euler-Lagrange Equations and 2.Transversality 3.Weierstrass-Erdmann Corner Conditions Euler-Lagrange Equation for the Secondary Variational Problem: ( ) ( ) 0 ,, = ∂ Ω∂ − ∂ Ω∂ x xx dt d x xx   δ δδ δ δδ ( ) ( ) 0**** =+−+ xQxPxRxQ dt d T  δδδδor
SOLO Calculus of Variations Jacobi’s Differential Equation (1837) and Conjugate Points (continue – 2) ( ) ( ) 0**** =+−+ xQxPxRxQ dt d T  δδδδ Euler-Lagrange Equation for the Secondary Variational Problem: We can see that the extremal makes .*xδ ( ) 0* 00 00 2 2 2 == == = ii ii xdxd tddt xJ δδ Assume that det R ≠ 0 in t € [0,tf], i.e. R is non-singular and has an inverse, in this interval, then 0*** 11 =      −+      −++ −− xPQ dt d RxQQR dt d Rx TT δδδ  Jacobi’s Differential Equation Carl Gustav Jacob Jacobi 1804-1851 This is a Second Order Vectorial Homogeneous Linear Differential Equation with continuous coefficients.
SOLO Calculus of Variations Jacobi’s Differential Equation (1837) and Conjugate Points (continue – 3) 0*** 11 =      −+      −++ −− xPQ dt d RxQQR dt d Rx TT δδδ  Carl Gustav Jacob Jacobi 1804-1851 We apply the general existence and uniqueness theorems for linear differential equations and we obtain n solutions, for the initial conditions: U (t) is a nxn matrix and contains the n independent solutions of the Vectorial Homogeneous Linear Differential Equation: 011 2 2 =      −+      −++ −− uP td Qd R td ud QQ td Rd R td ud T T Where is a vector( ) ( ) ( ) ( )            = tu tu tu tu n  2 1 If are the n solutions of the Jacobi’s Vectorial Differential Equation, with initial conditions: ( ) ( ) ( )tututu n,,, 21  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1,,0,0:&0 0,,1,0:&0 0,,0,1:&0 0 2022 10101     === === === nnn etutu etutu etutu then define: ( ) ( ) ( ) ( )[ ]tutututU n,,,: 21 =
SOLO Calculus of Variations Jacobi’s Differential Equation (1837) and Conjugate Points (continue – 4) Carl Gustav Jacob Jacobi 1804-1851 Theorem For a weak minimum (maximum) it is necessary that: is Positive (Negative) Definite for t ϵ [0, tf ]( ) xxFtR = ( ) ( ) ( ) ( )[ ]tutututU n,,,: 21 =If is the n solution matrix of the Jacobi’s Homogeneous Linear Differential Equation: 011 2 2 =      −+      −++ −− uP td Qd R td ud QQ td Rd R td ud T T then U (t) must be nonsingular in t ϵ [0, tf ] . Proof Assume that for , then there exists n constants (c1, c2,…,cn) ≠ (0, 0,…,0), such that ( ) ( ) ( ) 0det,0 =→∈ cjcjfcj tUsingularistUtt ( ) ( ) ( ) 02211 =+++ cjnncjcj tuctuctuc  Define ( ) ( ) ( ) ( )     ≤< ≤≤+++ = fcj cjnn ttt ttttuctuctuc tx 0 :* 02211  δ We see that ( ) ( ) 0** 0 == cjtxtx δδ
SOLO Calculus of Variations Jacobi’s Differential Equation (1837) and Conjugate Points (continue – 5) Carl Gustav Jacob Jacobi 1804-1851 Proof (continue – 1) Let check the Weierstrass-Erdmann corner conditions at t - tcj ( ) ( ) +− == ∂ Ω∂ = ∂ Ω∂ cjcj tttt x xx x xx     δ δδ δ δδ ,, We have ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0222 , 0 ≠=+= ∂ Ω∂ −−−−−− − cjcjcjcjcjcj t txtRtxtRtxtQ x xx cj    δδδ δ δδ The expression is nonzero since R (tcj) is positive definite and (otherwise is uniquely defined by the terminal conditions ). ( ) 0≠−cjtxδ ( ) ( ) [ ]fttttxtx ,0 0∈∀== δδ ( ) ( ) [ ]fttttxtx ,0 0∈== δδ ( ) ( ) ( ) ( ) ( ) 022 , 00 =+= ∂ Ω∂ +−++ = +     cjcjcjcj tt txtRtxtQ x xx cj δδ δ δδ Therefore ( ) ( ) 0 ,, = ∂ Ω∂ ≠ ∂ Ω∂ +− == cjcj tttt x xx x xx     δ δδ δ δδ The Weierstrass-Erdmann corner conditions at t = tcj are not satisfied, hence is not the minimum of the second variation, therefore exists a variation such that . ( ) 0* 00 00 2 2 2 == == = ii ii xdxd tddt xJ δδ xδ ( ) 02 <xJ δδ q.e.d. Return to Table of Content
SOLO Calculus of Variations Jacobi’s Differential Equation (1837) and Conjugate Points (continue – 6) 9.1 Conjugate Points If U (t) is singular in t ϵ [0, tf ] we say that we have a conjugate point. In this case an optimal solution doesn’t exist. The geometric meaning of the conjugate points is as follows: The Second Order Euler-Lagrange Equation has a two-parameter family of solutions. Through any point here passes in general, a one-parameter family of extremals. Let denote this parameter by α and the solutions by . ( )0tx ( )α,tx The solution must satisfy the Euler-Lagrange Equation: ( ) ( ) ( ) ( ) 0,,,,,,,, =      −      •• • αααα txtxtF dt d txtxtF x x Let take the partial derivative with respect to t of previous equation ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )            +      −      +      = •••• αααααααααααα αααα ,,,,,,,,,,,,,,,,,,,,0 txtxtxtFtxtxtxtF dt d txtxtxtFtxtxtxtF xxxxxxxx   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )αααααα αααααααααααα αα αααα ,,,,,,,,,, ,,,,,,,,,,,,,,,,,,,, txtxtxtFtxtxtxtF dt d txtxtxtFtxtxtxtF dt d txtxtxtFtxtxtxtF xxxx xxxxxxxx           −            −       −            −      +      = •• ••••
SOLO Calculus of Variations Jacobi’s Differential Equation (1837) and Conjugate Points (continue – 7) Conjugate Points (continue – 1) Rearrange the previous equation ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,,,,,,,,, ,,,,,,,,,,,,, ,,,,, =             −            +             −      +            +       •• ••• • ααααα ααααααα ααα α α α txtxtxtFtxtxtF dt d txtxtxtFtxtxtFtxtxtF dt d txtxtxtF xxxx xxxxxx xx      Using TT xxxx T xxxx TT xxxx RFFRFFQPFFP ======== •• :&:&: we can write ( ) ( ) ( ) 0,,, =      −+      −++ ααα ααα txPQ td d txQQR td d txR TT  which is identical to the Jacobi Equation. Since is a solution of the Jacobi Equation if we have for any tcj ϵ [0, tf ] than we have ( )α,tx ( ) 0, =αα tx ( ) ( ) ( ) ( ) 0, 2211 =+++= cjnncjcjcj tuctuctuctx αα where were defined as the independent solutions of the Jacobi Equation. Therefore U (t) is singular if and according to Theorem the problem doesn’t have a minimum (maximum). ( ) ( ) ( ) ( )[ ]tutututU n,,,: 21 = ( ) [ ]fcjcj tttfortx ,0, 0∈=αα
SOLO Calculus of Variations Jacobi’s Differential Equation (1837) and Conjugate Points (continue – 8) Conjugate Points (continue – 2) Let tray to understand the meaning of .( ) [ ]fcjcj tttfort x ,0, 0∈= ∂ ∂ α α Suppose that two close solutions of the family intersect at tcj ϵ [0, tf ].( )α,tx ( ) ( )ααα ,, cjcj txdtx =+ In this case the family of solutions has an envelope (see Figure ) Extremal Trajectories Envelope of Extremal Trajectories G ( )0tx ftconjt 321 0t ( )ftx ( )conjtx Description of Conjugate Points We have ( ) ( ) ( ) [ ]fcj cjcj d cj tttfor d txdtx t x ,0 ,, lim, 0 0 ∈= −+ = ∂ ∂ → α ααα α α α If such a family has an envelope G, then a point of contact of an extremal with the envelope is called a conjugate point to on that extremal. In the Figure point is conjugate to between 0 and tf. ( )0tx ( )conjtx ( )0tx On a minimizing (maximizing) extremal curve connecting point and with nonsingular at each point of it, there can be no point conjugate to , between t0 and tf . ( )00 =tx ( )ftx xxFR = ( )conjtx ( )0tx
SOLO Calculus of Variations Jacobi’s Differential Equation (1837) and Conjugate Points (continue – 9) Examples of Conjugate Points: 1. The shortest path between two points A and B on the surface of a sphere is on that one great circle passing trough those two points. If the points are on an opposite diameter there are an infinity of great circles passing through, we don’t have one extremal and those two points are conjugate to each other. A' B' B A Poles as Conjugate Points on a Sphere 2. Rays from a point source refracted by a lens. The refracted rays forms an envelope called caustic. The point P’2 where the reflected ray touches the envelope is called a conjugate point. From the figure we can see that this point is reached by at least two rays with different optical paths. Field of rays passing through a lens and generating a caustic Return to Table of Content
91 SOLO Calculus of Variations Jacobi’s Differential Equation (1837) and Conjugate Points (continue – 10) 9.2 Fields Definition Let represent a one parameter family of solutions of the Euler-Lagrange equation in a simply connected region of . This family of solution defines a field if they are not conjugate points in this region. This means that through any point of this region passes one and only one curve of the family. ( )α,tx ( )xt, ( )xt, Field around a solution of Euler- Lagrange equation In Figure we can see a solution of the Euler-Lagrange equation passing trough and . A field of solutions are shown in a simply connected region that contains the solution. The conjugate point is shown outside this region. We say that the solution is Embedded in the Field. ( )0, =αtx ( )00 , xt ( )ff xt , ( )0, =αtx Return to Table of Content
92 SOLO Calculus of Variations 10. Hilbert’s Invariant Integral Suppose that defines a field of solutions of Euler-Lagrange equation; i.e.( )α,tx ( ) ( ) ( ) ( ) 0,,,,,,,, =      −      •• • αααα txtxtF dt d txtxtF x x Let define any curve C in the field region, that starts at and ends at , and passes through a point with a slope (instead of Field slope ). Trough and passes also the unique extremal solution . ( )00 , xt ( )ff xt , ( )xt, ( )xtX , ( )xtx , ( )00 , xt ( )ff xt , ( )0, =αtx Hilbert’s Integral ( )( ) ( )( ) ( ) ( )( )[ ]∫ −− ft t T xC tdxtXxtxxtxxtFxtxxtF 0 ,,,,,,,,   is invariant on the path C as long as this curve remains in the field of the unique extremal solution. is the field slope and is the path C slope at the point of C.( )xtx , ( )xtX , ( )xt, David Hilbert (1862 – 1943)
93 SOLO Calculus of Variations Hilbert’s Invariant Integral (continue – 1) Hilbert’s Invariant Integral David Hilbert (1862 – 1943) Proof Since on C we can write C td xd X = ( )( ) ( )( ) ( )[ ] ( )( )∫ +− ft t T x T xC xdxtxxtFtdxtxxtxxtFxtxxtF 0 ,,,,,,,,,,   ( ) ( )( ) ( )( ) ( ) ( ) ( )( )xtxxtFxtN xtxxtxxtFxtxxtFxtM x T x ,,,:, ,,,,,,,:,     = −=Define Rewrite ( ) ( )∫ + ft t T C xdxtNtdxtM 0 ,, This integral is path independent if there exists a function ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )xtVxtN xtVxtM xdxtNdtxtMxdxtVdtxtVxtVd x t TT xt ,, ,, ,,,,, = = +≡+= The following condition must be satisfied ( ) ( ) ( ) ( )xtN t xtM x xtV t xtV x xt ,,,, ∂ ∂ ≡ ∂ ∂ → ∂ ∂ ≡ ∂ ∂
94 SOLO Calculus of Variations Hilbert’s Invariant Integral (continue – 2) Hilbert’s Invariant Integral David Hilbert (1862 – 1943) Proof (continue – 1) The following condition must be satisfied ( ) ( )xtN t xtM x ,, ∂ ∂ ≡ ∂ ∂ Let check that this is satisfied ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )xtxxtF x xtx xtx x xtx xtxxtFxtxxtxxtF xtxxtF x xtx xtxxtFxtM x x TT xxxx x T x ,,, , , , ,,,,,,, ,,, , ,,,,           ∂ ∂ − ∂ ∂ −− ∂ ∂ += ∂ ∂ ( ) ( )( ) ( )( ) ( ) t xtx xtxxtFxtxxtFxtN t xxtxt ∂ ∂ += ∂ ∂ , ,,,,,,,    Let compute ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( ) 0,,,,,,,,,,, ,, ,,, , , ,,,,,,,,,, , ,,,,,,,, =−++      ∂ ∂ + ∂ ∂ = ∂ ∂ ++− ∂ ∂ += ∂ ∂ − ∂ ∂ xtxxtFxtxxtFxtxxtxxtFxtx x xtx t xtx xtxxtF xtx x xtx xtxxtFxtxxtxxtFxtxxtF t xtx xtxxtFxtxxtFxtM x xtN t xtxxx T xx T xxxxx xxtxt            But ( ) ( ) ( ) ( ) ( )xtx td xtxd xtx x xtx t xtx T , , , ,,     == ∂ ∂ + ∂ ∂ Since satisfies the Euler-Lagrange equation, that is given by( )xtx , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,,,,,,,, =      −      +      +      ••••••• •••• txtxtFtxtxtFtxtxtxtFtxtxtxtF x txxxxx we can see that along C we have ( ) ( ) 0,, = ∂ ∂ − ∂ ∂ xtM x xtN t t q.e.d.
95 SOLO Calculus of Variations Hilbert’s Invariant Integral (continue – 3) David Hilbert (1862 – 1943) 10.1 Example: Geometrical Optics and Fermat’s Principle ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += For the Geometrical Optics we obtained: The Hilbert’s Invariant Integral is ( ) ( )( ) ( ) ( )[ ] ( ) ( )( ){ ( ) ( ) ( ) ( )[ ] ( ) ( )( )} xdzyxzzyxyzyxFzyxZzyxz zyxzzyxyzyxFzyxYzyxyzyxzzyxyzyxF z zyxP zyxP yC ffff ,,,,,,,,,,,, ,,,,,,,,,,,,,,,,,,,, ,, ,, 0000     −− ∫ −− This is known as Hilbert’s Invariant Integral because it is invariant on the path C as long as this curve remains in the field of the unique extremal solution. ( ) ( ) ( ) ( )zyx x z zyxzzyx x y zyxy ,,,,,,,,, ∂ ∂ = ∂ ∂ =  is the field slope and ( ) ( ) CC x z zyxZ x y zyxY ∂ ∂ = ∂ ∂ = :,,,:,,  is the path C slope at the point (x,y,z) of C we have on path C ( ) ( ) dx x z dxzyxZzddx x y dxzyxYyd C C C C ∂ ∂ == ∂ ∂ == ,,,,, 
96 SOLO Calculus of Variations Hilbert’s Invariant Integral (continue – 4) David Hilbert (1862 – 1943) Example: Geometrical Optics and Fermat’s Principle (continue – 1) ( ) ( ) ( )[ ]{ ( ) ( ) ( ) ( ) }zdzyzyxFydzyzyxFxdzyzyxFzzyzyxFyzyzyxF zy zyxP zyxP zyC ffff   ,,,,,,,,,,,,,,,,,,,, ,, ,, 0000 −−−−∫ The Hilbert’s Invariant Integral is We can write ( ) [ ] [ ] [ ] [ ] ( ) sd xd zyxn zy n zy zn z zy yn yzynFzFyzyzyxF zy ,, 111 1,,,, 2/1222/1222/122 2/122 = ++ = ++ − ++ −++=−−        ( ) [ ] ( ) ( ) [ ] ( ) sd zd zyxn zy zzyxn z F F sd yd zyxn zy yzyxn y F F z y ,, 1 ,, ,, 1 ,, 2/122 2/122 = ++ = ∂ ∂ = = ++ = ∂ ∂ =         Now we can write the Hilbert’s Invariant Integral as ( ) ( ) ( ) ( ) ∫∫ ⋅=⋅ ffffffff zyxP zyxP zyxP zyxP ray rdsnrd sd rd n ,, ,, ,, ,, 1000010000 ˆ   This is the Lagrange’s Invariant Integral from Geometrical Optics. Joseph-Louis Lagrange (1736-1813) Integration Path through a Ray Bundle Return to Table of Content
SOLO Calculus of Variations 11. The Weierstrass Necessary Condition for a Strong Minimum (Maximum) –1879 Karl Theodor Wilhelm Weierstrass 1815-1897 11.1 Derivation from Hilbert’s Invariant Integral Along the unique extremal path (denoted as C* - see Figure ), that passes through and we have and the Hilbert Integral becomes ( )00 , xt ( )ff xt , ( ) ( )xtxxtX ,,  = ( )( ) ( )( ) ( ) ( )( )[ ] ( )( ) [ ]xtJtdxtxxtF tdxtXxtxxtxxtFxtxxtF f f t t C t t T xC ,*,,, ,,,,,,,, 0 0 * == −− ∫ ∫    where is the minimum of the functional[ ]xtJ ,* ( )[ ] ( ) ( )( )∫= ft t C dttXtxtFtxJ 0 ,,  Suppose that the extremal is a strong minimum and •C* represents the strong minimum curve •C represents a strong neighbor of C* We can compute ( )[ ] ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( )[ ] ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( )[ ]∫ ∫∫ ∫∫ −−−= −−−= −=∆ f ff ff t t T xC t t T xC t t C t t C t t C tdxtxxtXxtxxtFxtxxtFtXtxtF tdxtXxtxxtxxtFxtxxtFdttXtxtF dttxtxtFdttXtxtFtxJ 0 00 00 ,,,,,,,,,, ,,,,,,,,,, ,,,, *     
SOLO Calculus of Variations The Weierstrass Necessary Condition for a Strong Minimum (Maximum) (continue – 1) Karl Theodor Wilhelm Weierstrass 1815-1897 Derivation from Hilbert’s Invariant Integral (continue – 1) Let define the Weierstrass E-function: ( ) ( ) ( ) ( ) ( )xXxxtFxxtFXxtFXxxtE T x   −−−= ,,,,,,:,,, Therefore the strong minimum condition is ( )[ ] ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( )[ ]∫ −−−=∆ ft t T xC tdxtxxtXxtxxtFxtxxtFtXtxtFtxJ 0 ,,,,,,,,,,   ( )[ ] ( ) ( ) ( )( ) 0,,, 0 ≥∫=∆ ft t C dttXtxtxtEtxJ  Weierstrass Necessary Conditions for a Strong Minimum (Maximum) is: ( ) ( )00,,, ≤≥XxxtE  for every admissible set ( )Xxt ,,
SOLO Calculus of Variations The Weierstrass Necessary Condition for a Strong Minimum (Maximum) (continue – 2) Karl Theodor Wilhelm Weierstrass 1815-1897 11.2 Weierstrass Derivation Weierstrass, that first derived this necessary condition for a Strong Minimum (Maximum) used the following derivation: 1t δ+1tδε+1t ( )txx = ( ) ( ) ( )[ ]111 txtXtx  −+ δε 1 2 3 0t ft ( ) ( ) [ ] [ ] ( ) ( ) ( ) ( )[ ] [ ] ( ) ( ) ( ) ( )[ ] [ ]       ++∈− − −+ + +∈−−+ +∈ = δδε ε δε δε δ ε 1111 1 11111 110 , 1 , ,, , ttttxtX tt tx ttttxtXtttx ttttttx tx f    Weierstrass Strong Variation Suppose is a candidate trajectory passing trough points 1 and 2, such that it contains no points of discontinuity of and no conjugate points between those points. Let take an arbitrary curve through point 1 such that . Let point 3 a movable point on at t = t1 + εδ . Let connect points 3 with point 2 on The arc 1, 3, 2 constitutes a strong variation (by δ tacking as small as we want). This variation , has a discontinuous derivative at point 1. ( )txx = x ( )tXx = ( ) ( )11 tXtx  ≠ ( )tXx = ( )δ+= 1txx ( )ε,txx =
SOLO Calculus of Variations The Weierstrass Necessary Condition for a Strong Minimum (Maximum) (continue – 3) Weierstrass Derivation (continue – 1) 1t δ+1tδε+1t ( )txx = ( ) ( ) ( )[ ]111 txtXtx  −+ δε 1 2 3 0t ft ( ) ( ) [ ] [ ] ( ) ( ) ( ) ( )[ ] [ ] ( ) ( ) ( ) ( )[ ] [ ]       ++∈− − −+ + +∈−−+ +∈ = δδε ε δε δε δ ε 1111 1 11111 110 , 1 , ,, , ttttxtX tt tx ttttxtXtttx ttttttx tx f    Weierstrass Strong Variation ( ) ( ) [ ] [ ] ( ) ( ) ( ) ( )[ ] [ ] ( ) ( ) ( ) ( )[ ] [ ]       ++∈− − −+ + +∈−−+ +∈ = δδε ε δε δε δ ε 1111 1 11111 110 , 1 , ,, , ttttxtX tt tx ttttxtXtttx ttttttx tx f    This variation fails to lie within any weak neighborhood of , no matter how small is δ. ( )txx = Since is a strong minimum, we have:( )[ ]txJ ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ]( ) ( ){ } ( ) ( ) ( ) ( )[ ] ( )∫∫ + + +       −      − − −+−−+= −≤ δ δε δε ε ε εε ε 1 1 1 1 ,, 1 ,,,,,,,, ,0 111111 t t t t dtxxtFtxtXtxtxtFdtxxtFtxtXtxtxtF txJtxJ  Let assume δ→0 ( )[ ] ( )[ ] ( ) ( )( ) ( ){ } ( ) ( ) ( ) ( )[ ] ( ) ε ε ε ε ε ε δε ε δ −       −      − − − +−= − ≤ → 1 ,, 1 ,,, ,,,,, , lim0 111 1 0 xxtFtxtXtxtxtF xxtFtXtxtF txJtxJ  
SOLO Calculus of Variations The Weierstrass Necessary Condition for a Strong Minimum (Maximum) (continue – 4) Weierstrass Derivation (continue – 2) 1t δ+1tδε+1t ( )txx = ( ) ( ) ( )[ ]111 txtXtx  −+ δε 1 2 3 0t ft ( ) ( ) [ ] [ ] ( ) ( ) ( ) ( )[ ] [ ] ( ) ( ) ( ) ( )[ ] [ ]       ++∈− − −+ + +∈−−+ +∈ = δδε ε δε δε δ ε 1111 1 11111 110 , 1 , ,, , ttttxtX tt tx ttttxtXtttx ttttttx tx f    Now let take ε→0 ( ) ( )( ) ( ){ } ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )[ ] ( )          XxxtE x T x txtXxxtFxxtFtXtxtF xxtFtxtXxxtFxxtF xxtFtXtxtF ,,, 111 2 11 0 1 ,,,,,, 1 ,,,, 1 ,, lim ,,,,0 −−−= −       −    Ο/+− − − + −≤ → ε ε ε ε ε ε This Inequality is the Weierstrass Necessary Conditions for a Strong Minimum (Maximum) ( ) 0,,, ≥XxxtE or Since the Weierstrass condition directly concerns minimality, rather than stationarity as did Euler-Lagrange condition, it entails no further supporting statements analogous to the Legendre and Jacobi conditions that support the Euler-Lagrange stationary condition. A weak variation is included in the strong variations, therefore a condition that is necessary for a weak local minimum (maximum) is also necessary for a strong local minimum (maximum).
SOLO Calculus of Variations The Weierstrass Necessary Condition for a Strong Minimum (Maximum) (continue – 5) 11.3 Geometric Interpretation of Weierstrass Conditions Let plot as a function of (see Figure). The hyper-plane tangent at is given by ( ) ( )      = • txtxtF ,,η • = xξ ( ) ( ) ( )      == •• iiiiii txtxtFtx ,,,ηξ ( ) ( ) ( ) ( ) ( )      +      −      = ••• iiiiiii T x txtxtFtxtxtxtF ,,,, ξη  The E function will be given by the difference between and the tangent hyper-plane. We can see that the condition for minimality is that the tangent hyper-plane remains bellow the surface . ( ) ( )       == • XtxtxtF ,,η ( ) ( )      = • txtxtF ,,η Geometric Representation of Weierstrass Condition
SOLO Calculus of Variations The Weierstrass Necessary Condition for a Strong Minimum (Maximum) (continue – 5) 11.4 Example: Geometrical Optics and Fermat’s Principle ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += For the Geometrical Optics we obtained: Weierstrass E Function is defined as ( ) ( ) ( ) ( ) ( ) ( ) ( )zyzyxFzZzyzyxFyYzyzyxFZYzyzyxFZYzyzyxE zy   ,,,,,,,,,,,,,,,,,,:,,,,,, −−−−−= [ ] [ ] ( ) [ ] ( ) [ ] [ ] [ ] ( ) ( )[ ] [ ] [ ] ( ) [ ] [ ] [ ] ( )InequalitySchwarz zyZY zZyY ZYn zZyY zy n ZYn zzZyyYzy zy n ZYn zy zn zZ zy yn yYzynZYn 0 11 1 11 1 1 1 1 1 1 1 '' 1 11 2/1222/122 2/122 2/122 2/122 22 2/122 2/122 2/1222/122 2/1222/122 ≥         ++++ ++ −++= ++ ++ −++= −+−+++ ++ −++= ++ −− ++ −−++−++=               According to Weierstrass Condition if the Jacobi Condition (no conjugate points between and ) is satisfied every extremal is a strong minimum. ( )( )0',',',',',',,, ≥ZYXzyxzyxE Return to Table of Content
SOLO Calculus of Variations Summary Necessary Conditions for a Weak Relative Minimum (Maximum) Satisfies Boundary Conditions ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) fitxdtxtxtFdttxtxtxtFtxtxtF iiii T x iiiii x iii ,00,,,,,, ==      +            −      •••• •• Satisfies Weierstrass-Erdmann Corner Conditions ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,,,, ,,,,,,,, 00 000000 =            −      +          +         −      −      + • − • + • + • + • − • − • − • •• •• cccc T x ccc T x ccccc T x ccccccc T x ccc txdtxtxtFtxtxtF dttxtxtxtFtxtxtFtxtxtxtFtxtxtF Satisfies the Euler-Lagrange Equation [ ]f x x tttforF dt d F ,0 0∈=− • 1 Satisfies Legendre (Clebsh) Condition2 is Positive (Negative) Definite for( ) xxFtR = [ ]fttt ,0∈ It contains no Conjugate Point for3 [ ]fttt ,0∈ Necessary Conditions for a Strong Relative Minimum (Maximum). Satisfies (1), (2) and (3) and additionally: Weirestrass Necessary Conditions for a Strong Minimum (Maximum) is that: ( ) ( ) ( ) ( ) ( ) ( )00,,,,,,:,,, ≤≥−−−= xXxxtFxxtFXxtFXxxtE T x   4 for every admissible set ( )Xxt ,, Return to Table of Content
SOLO Calculus of Variations 12. Canonical Form of Euler-Lagrange Equations We found that the first variation of the cost function J is given by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫             −      +             +            −      = •••••• ••• ff t t T x x t t T x T x dttxtxtxtF dt d txtxtFxdtxtxtFdtxtxtxtFtxtxtFJ 00 ,,,,,,,,,, δδ Let define ( ) ( ) T xxx n FFtxtxtFp    =      = ••• • ,,,,: 1  and suppose that [ ]               =                 ∂ ∂ ∂ ∂ =     ∂ ∂ =                 ∂ ∂ ∂ ∂ ≡ • ••• nnnn n n n xxxxxx xxxxxx xxxxxx xxx n T x T xx FFF FFF FFF FFF x x F xx F x F             21 21212 12111 21 ,,, 1 is nonsingular for t ϵ [t0, tf ] (regular problem), then we can solve ) (locally, because of the Implicit Function Theorem) as a function of , by using Legendre’s Dual Transformation. ( )tx • ( ) ( )tptxt ,, Return to Table of Content
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous – 1) 12.1 Legendre’s Dual Transformation Adrien-Marie Legendre 1752-1833 Let consider a function of n variables xi, m variables, and time t:ii xy ≡ ( )mn yyxxtF ,,,,,, 11  and introduce a new set of n variables pi defined by the transformation: ni y F p i i ,,2,1: = ∂ ∂ = We can see that for t ϵ (t0, tf )                             ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ +                             ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ ∂ =             m mnn m n nn n n dx dx dx xy F xy F xy F xy F dy dy dy y F yy F yy F y F dp dp dp          2 1 2 1 2 1 2 11 2 2 1 2 2 1 2 1 2 2 1 2 2 1 We want to replace the variables dyi (i=1,2,…,n) by the new variables dpi (i=1,2,…,n). We can see that the new n variables are independent if the Hessian Matrix ni njji ni njji xx F yy F ,,1 ,,1 2 ,,1 ,,1 2      = = = =         ∂∂ ∂ =         ∂∂ ∂ is nonsingular in the interval t ϵ (t0, tf ). According to the Implicit Function Theorem (Appendix 1) we can obtain a unique function in the interval t ϵ (t0, tf ).( )pxtxy ii ,,: =
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –2) Let define a new function H (Hamiltonian) of the variables t, xi, pi. ( )nm n i ix n i ii ppxxtHxFFypFH i ,,,,,,: 11 11  =+−=+−= ∑∑ == Then: ( ) ∑∑∑∑∑ =====             ∂ ∂ −++ ∂ ∂ − ∂ ∂ −=++ ∂ ∂ − ∂ ∂ − ∂ ∂ −= n i i i iii n j j j n i iiii n i i i n j j j dy y F pdpydx x F dt t F dpydypdy y F dx x F dt t F dH 11111 But because ( )nm ppxxtHH ,,,,,, 11 = ∑ ∑ ∂ ∂ + ∂ ∂ + ∂ ∂ = = = n j n i i i j j dp p H dx x H dt t H dH 1 1 and because all the variations are independent we have: ;,,1;,,1&, mj x F x H ni y F p p H y t F t H jji i i i  = ∂ ∂ −= ∂ ∂ = ∂ ∂ = ∂ ∂ = ∂ ∂ −= ∂ ∂ Now we can define the Dual Legendre’s Transformation from ( ) ( ) ( )yxtFyppxtHtoyxtF n i ii ,,,,,, 1 −= ∑= by using ni p H xy ni x F y F p i ii ii i ,,2,1 ,,2,1    = ∂ ∂ == = ∂ ∂ = ∂ ∂ = The variables t, , and H are called Canonical Variables corresponding to the functional J. x p
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –3) If we apply now the Legendre Transformation to and H we obtainp ( ) ( )yxtFpxtHyp n i ii ,,,, 1 =−∑= The Legendre Transformation is an involution, i.e. a transformation which is its own inverse. ( ) ( ) ( ) ( )∫       −      ++−= •f f t t T x t t T dttxp dt d txtxtFxdpdtHJ 0 0 ,, δδ Let write δJ in terms of the Canonical Variables: From this expression the necessary conditions such that δJ is zero are ( ) 00 =+− t T xdpdtH ( ) ( )      = • txtxtFp dt d x ,, ( ) 0=+− ft T xdpdtH We found before that the necessary conditions such that δJ is zero for those admissible solutions passing through the points and are the Euler-Lagrange Equations: ( )* 0 * 0 * , xtA ( )*** , ff xtB ( ) ( ) ( ) ( ) 0,,,, =      −      •• • txtxtF dt d txtxtF x x niF y F p x ,,2,1:  == ∂ ∂ =Since the two equations are identical. Return to Table of Content
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –4)       ∂ ∂ =⇒= ∂ ∂ == p H xni p H xy i ii  ,,2,1and x H x F Fmj x F x H x jj ∂ ∂ −= ∂ ∂ =⇒= ∂ ∂ −= ∂ ∂ :;,,1 also ( ) ( )      = • txtxtFp dt d x ,,we obtain Canonical Euler-Lagrange Equation or Hamilton’s Equations x H td pd p H td xd ∂ ∂ −= ∂ ∂ = we can write the Euler-Lagrange Equations in the form x H ∂ ∂ −= ( ) ( )      = • • txtxtFp x ,,:using William Rowan Hamilton (1805-1855)
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –5) Canonical Euler-Lagrange Equation or Hamilton’s Equations x H td pd p H td xd ∂ ∂ −= ∂ ∂ = Therefore we transformed the n Second Order Euler-Lagrange Differential Equations in 2 n First Order Hamilton’s Equations. Let find out when this problem is well-posed, i.e.: • a solution exists. • the solution is unique. • the solution depends continuously on the initial values. First we must remember that the Hamilton’s Equation where derived only for regular problems: is nonsingular for t ϵ (t0, tf ).xxF  From the theory of First Order Differential Equations (see Appendix 3) the solution exists if       ∂ ∂       ∂ ∂ x H p H , exists, are continuous in t ϵ (t0, tf ) (except a finite number of points). This implies that is continuous and has continuous partial derivatives.( ) ( )yxtFyppxtH n i ii ,,,, 1 −= ∑= The solution is unique and depends continuously on the initial values if in addition the problem has 2 n defined boundary conditions. Therefore the general solutions of the Hamilton’s Equations are therefore two vector parameters solutions . Those parameters are defined by 2n boundary conditions. ( ) ( )T n T n βββααα ,,,,, 11  == ( ) ( )βαϕ ,,ttx = Return to Table of Content
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –6) 12.2 Transversality Conditions (Canonical Variables ) For other admissible variations we shall need to add the additional necessary conditions, such that δJ is zero, called Transversality Conditions Equations: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) fitxdtxtxtFdttxtxtxtFtxtxtF iiii T x iiiii T x iii ,00,,,,,, ==      +            −      •••• •• ( ) ( )( ) ( ) fitxdpdttptxtH i T iiii ,00,, ==+−or (a) Suppose that the following relation defines the boundary: ( ) ( ) ( ) ( ) ( ) iitiiiii dttdtt dt d txdttx Ψ=Ψ=→Ψ= then the Transversality Conditions Equations are: ( ) ( ) ( ) ( ) ( ) ( ) fitxttxtxtFtxtxtF iiiii T x iii ,00,,,, ==      −Ψ      +      ••• • ( ) ( )( ) ( ) ( ) fittptptxtH i T iiii ,00,, ==Ψ+−or (b) Suppose that ti and are not defined, and is not a function of ti , therefore d ti and are independent differentials and the Transversality Conditions will be: ix ix ixd ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,, 0,,,, =      =      −      • ••• • • iii x iiii T x iii txtxtF txtxtxtFtxtxtF ( ) ( )( ) ( ) 0 0,, = = i iii tp tptxtH or Return to Table of Content
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –8) 12.3 Weierstrass-Erdmann Corner Conditions (Canonical Variables) At the corners c we found: ( ) 0 0000 =      −+               −−      −= + • − • + • − • •• c t T xt T x c t T x t T x txdFFdtxFFxFFJ cccc δ Using the Canonical Variables we obtain: ( ) ( )[ ] ( ) ( )[ ] ( ) 00000 =−++−= +−+− cc T c T ccc txdtptpdttHtHJδ (a) If they are apriori conditions at the corner like: ( ) ( ) ( ) ( ) ( ) cctccccc dttdtt dt d txdttx Ψ=Ψ=→Ψ= then the necessary conditions at the corner are: ( ) ( ) ( ) ( ) ( ) ( )0000 ++−− −Ψ=−Ψ cctc T cctc T tHttptHttp (b) If they are not apriori conditions at the corner; i.e. the function is not apriori defined then dti and independent variables and ( ) ( )cc ttx Ψ= cxd ( ) ( ) ( ) ( )0000 & +−+− == cccc tptptHtH Return to Table of Content
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –9) 12.4 First Integrals of the Euler-Lagrange Equations A First Integral of a system of differential equations is a function which has a constant value along each integral curve of the system. We defined ( ) ( ) ( )yxtFypyxtFyppxtH T n i ii ,,,,,, 1 −=−= ∑= and ( ) ( ) ( ) ( ) td pd p pxtH td xd x pxtH t pxtH td pxtHd TT       ∂ ∂ +      ∂ ∂ + ∂ ∂ = ,,,,,,,, Using the Canonical Euler-Lagrange Equations x H td pd p H td xd ∂ ∂ −= ∂ ∂ = we obtain from ( ) ( ) ( )yxtFypyxtFyppxtH T n i ii ,,,,,, 1 −=−= ∑= t H x H p H p H x H t H td Hd TT ∂ ∂ = ∂ ∂       ∂ ∂ − ∂ ∂       ∂ ∂ + ∂ ∂ = If does not depend on t explicitly, H doesn’t depend on t explicitly and is is constant over the optimal path, i.e. is a First Integral of the Euler-Lagrange Equations. ( )yxtF ,, ( )pxH , 0= ∂ ∂ = t H td Hd
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –10) First Integrals of the Euler-Lagrange Equations (continue – 1) Consider an arbitrary function ( )pxt ,,Φ=Φ and compute ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x pxtH p pxt p pxtH x pxt t pxt td pd p pxt td xd x pxt t pxt td pxtd TT TT ∂ ∂       ∂ Φ∂ − ∂ ∂       ∂ Φ∂ + ∂ Φ∂ =       ∂ Φ∂ +      ∂ Φ∂ + ∂ Φ∂ = Φ ,,,,,,,,,, ,,,,,,,, Define Poisson Bracket [ ] ( ) ( ) ( ) ( ) x pxtH p pxt p pxtH x pxt H TT ∂ ∂       ∂ Φ∂ − ∂ ∂       ∂ Φ∂ =Φ ,,,,,,,, :, Siméon Denis Poisson 1781-1840 From which [ ]H ttd d ,Φ+ ∂ Φ∂ = Φ is constant over the optimal path, i.e. is a First Integral of the Euler-Lagrange Equations iff 1.F and Φ do not depend on t explicitly. 2.[Φ,H] = 0. ( )px,Φ Return to Table of Content
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –11) 12.5 Equivalence Between Euler-Lagrange and Hamilton Functionals The Euler-Lagrange Functional [ ] ( )∫= ft t dtxxtFxJ 0 ,,  is optimized by the solution of the Euler-Lagrange Equations: ( ) ( ) ( ) ( ) 0,,,, =      −      •• • txtxtF dt d txtxtF x x We set ( ) ( )      = • • txtxtFp x ,,: and the Hamiltonian ( ) ( ) ( ) ( ) xppxtHxxtFxxtFxppxtH TT  +−=⇒−= ,,,,,,:,, We define the Hamilton Functional [ ] ( )[ ]∫ +−= ft t T dtxppxtHpxJ 0 ,,:,  Since: ( ) ( )xxtFxppxtH T  ,,,, =+− ( )[ ] ( )∫∫ =+− ff t t t t T dtxxtFdtxppxtH 00 ,,,,  William Rowan Hamilton (1805-1855)
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –12) Equivalence Between Euler-Lagrange and Hamilton Functionals (continue – 1) Hamilton Functional [ ] ( )[ ]∫ +−= ft t T dtxppxtHpxJ 0 ,,:,  William Rowan Hamilton (1805-1855) Let find the Euler-Lagrange Equations for the Hamilton Functional ( )[ ] ( )[ ] ( )[ ] ( )[ ] 0,,,, 0,,,, =       +− ∂ ∂ −+− ∂ ∂ =       +− ∂ ∂ −+− ∂ ∂ xppxtH ptd d xppxtH p xppxtH xtd d xppxtH x TT TT       or 0 0 =+ ∂ ∂ − =− ∂ ∂ − td xd p H td pd x H We recovered the Canonical Euler-Lagrange (Hamilton) Equations (William R. Hamilton 1835) Return to Table of Content
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –13) 12.6 Equivalent Functionals Two functionals are said to be equivalent if they have the same extremal trajectories. Suppose we have an arbitrary continuous and differentiable function: ( )xtS , Define ( ) ( ) x x S t S xtS td d xxt T        ∂ ∂ + ∂ ∂ ==Ψ ,:,, Let compute ( ) x x S xt S xxt x  2 22 ,, ∂ ∂ + ∂∂ ∂ =Ψ ∂ ∂ ( ) x x S tx S xtd d x S xxt x     2 22 ,, ∂ ∂ + ∂∂ ∂ =      ∂ Ψ∂ ⇒ ∂ ∂ =Ψ ∂ ∂ Since tx S xt S ∂∂ ∂ = ∂∂ ∂ 22 0=      ∂ Ψ∂ − ∂ Ψ∂ xtd d x  Similar to Euler-Lagrange (E.-L.) Equations The functionals [ ] ( )∫= ft t dtxxtFxJ 0 ,,  0 .. =      ∂ ∂ − ∂ ∂ ⇒ − x F td d x FLE  [ ] ( ) ( )[ ]∫ Ψ−= ft t dtxxtxxtFxJ 0 ,,,, ~  0 .. =      ∂ ∂ − ∂ ∂ =      ∂ Ψ∂ + ∂ Ψ∂ −      ∂ ∂ − ∂ ∂ ⇒ − x F td d x F xtd d xx F td d x FLE  and have the same Euler-Lagrange Equations.
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –14) Equivalent Functionals (continue – 1) Two functionals are said to be equivalent if they have the same extremal trajectories. We can see that [ ] ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( )[ ] [ ] ( )[ ] ( )[ ]00 00 ,, ,,,, , ,,,,,, ~ 0 00 txtStxtSxJ txtStxtSdtxxtF dt td xtdS xxtFdtxxtxxtFxJ ff ff t t t t t t f ff +−= +−=       −=Ψ−= ∫ ∫∫   The functionals and are called equivalent functionals.( )xJ ( )xJ ~ Return to Table of Content
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –15) 12.7 Canonical Transformations The functional [ ] ( )[ ]∫ +−= ft t T dtxppxtHpxJ 0 ,,:,  is optimized by the trajectories derived from the Canonical Euler-Lagrange (Hamilton) Equations x H td pd p H td xd ∂ ∂ −= ∂ ∂ = Let perform a change of variables, from to according topx, px ~,~ ( ) ( )pxpp pxxx ,~~ ,~~ = = Since                   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =      pd xd p p x p p x x x pd xd ~~ ~~ ~ ~ this is possible iff ( ) ( ) 0 , ~,~ :~~ ~~ det ≠ ∂ ∂ =             ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ px px p p x p p x x x
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –16) Canonical Transformations (continue – 1) We look for transformations under which the Canonical Euler-Lagrange (Hamilton) Equations preserve their form. They are called Canonical Transformations. x H td pd p H td xd ~ ~~ ~ ~~ ∂ ∂ −= ∂ ∂ = and optimize the functional [ ] ( )[ ]∫ +−= ft t T dtxppxtHpxJ 0 ~~~,~, ~ :~,~  The two functional are equivalent if ( ) ( ) ( )xtS td d td xd ppxtH td xd ppxtH TT , ~ ~~,~, ~ ,, −+−=+− From which ( ) ( ) ( )( )pxxtSdxdptdpxtHxdptdpxtH TT ~,~,~~~,~, ~ ,, −+−=+− or ( )( ) ( ) ( )[ ] tdpxtHpxtHxdpxdppxxtSd TT ,, ~~,~,~~~,~, −+−=
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –17) Canonical Transformations (continue – 2) ( )( ) ( ) ( )[ ] tdpxtHpxtHxdpxdppxxtSd TT ,, ~~,~,~~~,~, −+−= From xd x x p x xd p x pdpd p x xd x x xd ~ ~~~ ~~ ~ ~ ~ 11 ∂ ∂       ∂ ∂ −      ∂ ∂ =⇒ ∂ ∂ + ∂ ∂ = −− we can use instead of to obtainxx ~, px ~,~ ( )( ) ( ) ( ) ( )[ ] tdpxtHpxtHxdpxdp xd x S xd x S td t S xxtSdpxxtSd TT TT ,, ~~,~,~~ ~ ~ ~,,~,~, −+−=       ∂ ∂ +      ∂ ∂ + ∂ ∂ == Finally we obtain: ( ) ( ) t S pxtHpxtH x S p x S p ∂ ∂ =− ∂ ∂ −= ∂ ∂ = ,, ~~,~,~ ~ Return to Table of Content
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –18) 12.8 Caratheodory's Lemma Consider the problem of minimizing the functional [ ] ( ) ( )∫       = ⋅fxt xt dttxtxtFxtJ , , 00 ,,, defined in a simple connected domain Ω in the plane. Given an initial point suppose that one and only one extremal of exists between the initial point and every point . ( )xt, ( ) Ω∈00 , xt [ ]xtJ , ( ) Ω∈xt, Assume that is continuous for all and all admissible . Define      ⋅ xxtF ,, ( ) Ω∈xt, x ( ) ( ) ( ) ( )∫       = ⋅ Ω∈ fxt xt xt dttxtxtFxtS , , , 00 ,,min:, called the geodesic distance (Hamilton called it optical distance) or the Hamilton's characteristic function. Since we have one and only one extremal connecting with along a curve Γ Ω. is a single-valued function for all .ϵ ( ) Ω∈00 , xt ( ) Ω∈xt, ( )xtS , ( ) Ω∈xt, Field of Extremals Starting from ( ) Ω∈00 , xt
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –19) Caratheodory's Lemma (continue – 1) Field of Extremals Starting from ( ) Ω∈00 , xt Suppose that we calculate the functional along a neighbor curve Γε Ω that connects with .ϵ [ ]xtJ , ( )00 , xt ( )xt, By definition of ( )xtS , [ ] ( ) ( ) ( ) ( ) ( )∫ Γ ⋅ Γ Ω∈∀≥      −      =− fxt xt xtdtxtS td d txtxtFxtSxtJ , , 00 ,0,,,,, ε ε where we use the fact that ( ) ( ) ( ) ( )xtwithxtconnectingdtxtS td d xtS fxt xt ,,,, 00 , , 00 ε ε Γ∀      = ∫ Γ Along , if is continuous and differentiable relative to t and we haveΩ∈Γε ( )xtS , x ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )εεεεεεε ,,,,,,,,,,,, txtxtStxtStx t txtS x txtS t txtS td d T xt T += ∂ ∂       ∂ ∂ + ∂ ∂ = we obtain: [ ] ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )∫ Γ ⋅ Γ Ω∈∀≥      −−      =− fxt xt T xt xtdttxtxtStxtStxtxtFxtSxtJ , , 00 ,0,,,,,,,,,,, ε ε εεεεε  Since this is true for all and curve is the only optimal curve, the last equation is equivalent to: ( ) Ω∈xt, Ω∈Γ ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )εεεεεεε ε ,&,,0,,,,,,,,, txtxttxtxtStxtStxtxtF T xt  Γ≠Γ∈∀>−−      ⋅ ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) Γ∈=−−      ⋅ txtfortxtxtStxtStxtxtF T xt ,0,,,,  and
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –20) Caratheodory's Lemma (continue – 2) Field of Extremals Starting from ( ) Ω∈00 , xt Since at any point the change in the curve direction is defined by its slope we can write ( ) Ω∈xt, ( )xtX , Carathéodory's Lemma If the Hamiltonian's characteristic function is defined on an admissible set of simple connected region of terminations Ω and if S is continuous and differentiable on it's arguments, then every element of an optimal trajectory that lies entirely in Ω is characterized by ( )xtS , ( )xxt ,, ( ) ( ) ( ) ( ) ( )[ ] 0,,,,min,,,, =−−=−−      ⋅ XxtSxtSXxtFxxtSxtSxxtF T xt X T xt   Constantin Carathéodory (1873-1950) R.E Bellman arrived to the Carathéodory's Lemma in a different way which will be described elsewhere. Return to Table of Content
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –21) 12.9 Hamilton-Jacobi Equations From the Carathéodory's Lemma we can derive the following Theorem (a) If is continuous for all , is not constraint and if is continuous and differentiable on it's arguments, then for every element of an optimal trajectory that lies entirely in Ω, except for the corners of , the following two conditions have to hold       ⋅ xxtF ,, ( ) Ω∈xt, x ( )xtS , ( )xxt ,, x ( ) xxxtFxxtFxtS T xt        −      = ⋅⋅ ,,,,, ( )       = ⋅ xxtFxtS xx ,,,  (b) Here is the uniquely determined slope of the optimal trajectory at the point and is viewed as a function of t and . x ( ) Ω∈xt, x First Proof of (a) Using the Carathéodory's Lemma, let define: ( ) ( ) ( ) 0,,,,:,, ≥−−=      ⋅ XxtSxtSXxtFxxtE T xt  we see that if is not constraint, from the ordinary differential calculus, the necessary conditions for the minimum are, that on the optimal trajectory . x ( )xxt ,,
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –22) Hamilton-Jacobi Equations (continue – 1) First Proof of (a) (continue – 1) ( ) ( ) ( )[ ] ( ) ( ) 0,,,,,,, =−=−− ∂ ∂ = ∂ ∂ xtSxxtFxxtSxtSxxtF xx E xx T xt    and this gives ( )       = ⋅ xxtFxtS xx ,,,  ( ) ( ) ( )[ ] ( ) 0,,,,,,22 2 ≥=−− ∂ ∂ = ∂ ∂ xxtFxxtSxtSxxtF xx E xx T xt    This is the Legendre's Necessary Condition. ( )       = ⋅ xxtFxtS xx ,,, If we substitute in Carathéodory's Lemma ( ) ( ) ( ) ( ) ( )[ ] 0,,,,min,,,, =−−=−−      ⋅ XxtSxtSXxtFxxtSxtSxxtF T xt X T xt   we obtain for the optimal trajectory ( )xxt ,, ( ) ( ) ( ) 0,,,,,,,,, =      −−      =−−      ⋅⋅⋅ xxxtFxtSxxtFxxtSxtSxxtF T xt T xt   If we substitute and in we obtain ( ) xxxtFxxtFxtS T xt        −      = ⋅⋅ ,,,,, ( )       = ⋅ xxtFxtS xx ,,,  ( ) ( ) ( ) 0,,,,:,, ≥−−=      ⋅ XxtSxtSXxtFxxtE T xt  ( ) ( ) ( ) ( ) 0,,,,,,,, ≥−−−=      ⋅ xXxxtFxxtFXxtFxxtE T x   Weierstrass' Condition E is called Weierstrass' Excess Function. End of Proof (a)
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –23) Hamilton-Jacobi Equations (continue – 2) Second Proof of (a) (Geometrical) Suppose that is a point on the optimal trajectory .( )( ) Ω∈ii txt , Let plot ( ) ( )      = • txtxtF ,,η ( ) ( ) xxtSxtS T xtS ,, +=η • = xξas a function of and the hyper-plane From Carathéodory's Lemma those two functions intersect at and the hyper-plane must stay on one side of , therefore it is tangent to it. ( )( ) Ω∈ii txt , ( ) ( )      = • txtxtF ,,η
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –24) Hamilton-Jacobi Equations (continue – 3) Second Proof of (a) (Geometrical) (continue – 1) On the other side the hyper-plane tangent at is given by( ) ( ) ( )      == •• iiiiii txtxtFtx ,,,ηξ ( ) ( ) ( ) ( ) ( )      +      −      = ••• iiiiiii T xT txtxtFtxtxtxtF ,,,, ξη  Since ηT ≡ ηS we must have ( ) xxxtFxxtFxtS T xt        −      = ⋅⋅ ,,,,, ( )       = ⋅ xxtFxtS xx ,,,  We can see from Figure that E is given by the difference between and the tangent to hyper-plane. We can see that the condition for minimality is that the tangent hyper-plane remains bellow the surface . ( ) ( )       == • XtxtxtF ,,η ( ) ( )      = • txtxtF ,,η End of Geometrical Proof of (a) Geometric Representation of , and Weierstrass Condition( ) ( )      = • txtxtF ,,η ( ) ( ) xxtSxtS xtS ,, +=η
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –25) Hamilton-Jacobi Equations (continue – 4) Proof of (b) Let use the Canonical Forms. Start with definition ( )xxtF x F p x    ,,: = ∂ ∂ = If the problem is regular, i.e. is nonsingular, we proved, that according to the Implicit Function Theorem we obtain a unique function in the interval t ϵ (t0, tf ). ( )xxtF xx  ,, ( )pxtxx ,,:  = End of Proof of (b) (b) Here is the uniquely determined slope of the optimal trajectory at the point and is viewed as a function of t and . x ( ) Ω∈xt, x Theorem (continue) Note ( )       = ⋅ xxtFxtS xx ,,,  ( )xxtF x F p x    ,,: = ∂ ∂ =Using and ( ) ( ) ( )xtppxtSxxtF x F p xx ,,,,: =⇒== ∂ ∂ =    End Note
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –26) Hamilton-Jacobi Equations (continue – 5) We defined, also, the function H (Hamiltonian) of the variables t, px, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )xtHpxtHxxxtFxxtFxxxtpxxtFH xtpppxtxx T x T , ~ ,,,,,,,,,,: ,,,: == ==+−=+−=    If we compare this expression with , we obtain:( ) xxxtFxxtFxtS T xt        −      = ⋅⋅ ,,,,, ( ) ( )pxtHxtSt ,,, −= If we compare with , we obtain:( )xxtF x F p x    ,,: = ∂ ∂ = ( )       = ⋅ xxtFxtS xx ,,,  ( ) pxtSx =, Therefore the Carathéodory's Lemma equation ( ) ( ) ( ) ( ) ( )[ ] 0,,,,min,,,, =−−=−−      ⋅ XxtSxtSXxtFxxtSxtSxxtF T xt X xt   can be rewritten as ( ) ( ) 0,,, =+ xt SxtHxtS Hamilton-Jacobi Equation William Rowan Hamilton (1805-1855) Carl Gustav Jacob Jacobi 1804-1851 The Hamilton-Jacobi Equation is a Partial Differential Equation in which is in general nonlinear.( )xtS , Return to Table of Content
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –27) Jacobi’s Theorem Carl Gustav Jacob Jacobi 1804-1851 Let be a general solution of the Hamilton-Jacobi equation:( )α,, xtS ( ) ( ) 0,,,,,, =      ∂ ∂ + ∂ ∂ αα xt x S xtHxt t S depending on the parameters ( )n T ααα ,,1 = Assume also that ( ) 0,,detdet 2 ≠      ∂∂ ∂ = α α α xt x S S x Let n arbitrary constants.( )n T βββ ,,1 = The two-parameter family of solutions of the Hamilton Equations ( ) ( )βαβα ,,,,, tpptxx == x H td pd p H td xd ∂ ∂ −= ∂ ∂ = are obtained from ( ) βα α = ∂ ∂ ,, xt S together with ( )α,, xt x S p ∂ ∂ =
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –28) Proof of Jacobi’s Theorem Carl Gustav Jacob Jacobi 1804-1851 Since det Sαx ≠ 0 using the Implicit Function Theorem (see Appendix 1) we can use to uniquely find as a function of( ) βα α = ∂ ∂ ,, xt S x ( )βα ,,t ( ) ( )βαβα α α ,,,, Im 0det txxxt S Theorem Function plicit S x =⇒= ∂ ∂ ≠ Substitute this back in and differentiate with respect to t( ) βαα =,, xtS ( )( ) 0,,,, 22 = ∂∂ ∂ + ∂∂ ∂ =      ∂ ∂ td xd x S t S txt S td d αα αβα α Now, take the partial differential of the Hamilton-Jacobi equation with respect to α ( ) ( ) 0,,,,,, 22 = ∂ ∂ ∂∂ ∂ + ∂∂ ∂ =      ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ xS H x S t S xt x S xtHxt t S αα α α α α If we use in this equation the fact that Sαt = Stα and , we obtainxSp = ( ) ( ) 0,,,,,, 22 = ∂ ∂ ∂∂ ∂ + ∂∂ ∂ =      ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ p H x S t S xt x S xtHxt t S αα α α α α therefore 0 2 =      ∂ ∂ − ∂∂ ∂ p H td xd x S α + - Since det Sαx ≠ 0 the previous equation is satisfied only if 0= ∂ ∂ − p H td xd
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –29) Proof of Jacobi’s Theorem (continue – 1) Carl Gustav Jacob Jacobi 1804-1851 ( )α,, xt x S p ∂ ∂ =Let differentiate with respect to t ( ) p H xx S xt S td xd xx S xt S xt x S td d td pd ∂ ∂ ∂∂ ∂ + ∂∂ ∂ = ∂∂ ∂ + ∂∂ ∂ = ∂ ∂ = 2222 ,, α Now, take the partial differential of the Hamilton-Jacobi equation with respect to x ( ) ( ) 0,,,,,, 22 = ∂ ∂ ∂∂ ∂ + ∂ ∂ + ∂∂ ∂ =      ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ xS H xx S x H tx S xt x S xtH x xt t S x αα If we use in this equation the fact that Sαt = Stα and , we obtainxSp = x H p H xx S xt S ∂ ∂ −= ∂ ∂ ∂∂ ∂ + ∂∂ ∂ 22 x H td pd ∂ ∂ −= We obtain q.e.d. Return to Table of Content
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –30) Example: Geometrical Optics and Fermat’s Principle ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += For the Geometrical Optics we obtained: Define ( ) [ ] ( ) [ ] 2/122 2/122 1 ,, : 1 ,, : zy zzyxn z F p zy yzyxn y F p z y       ++ = ∂ ∂ = ++ = ∂ ∂ = Adding the square of those two equations gives ( )( ) ( )2222222 1 zynzypp zy  +=+++ from which ( ) ( )222 2 22 1 zy ppn n zy +− =++  Substitute this equation in that of F ( ) ( )222 2 ,,,, zy zy ppn n ppzyxF +− = ( ) ( )222 222 zy z zy y ppn p z ppn p y +− = +− =   solve for zy ,
SOLO Calculus of Variations Canonical Form of Euler-Lagrange Equations (continuous –31) Example: Geometrical Optics and Fermat’s Principle (continue – 1) Define the Hamiltonian ( ) ( ) ( ) ( ) ( ) ( ) ( )222 222 2 222 2 222 2 ,, ,,,,:,,,, zy zy z zy y zy zyzyzy ppzyxn ppn p ppn p ppn n zpypppzyxFppzyxH +−−= +− + +− + +− −=++−=  The canonical equations are ( ) ( )222 222 zy z z zy y y ppn p p H xd zd z ppn p p H xd yd y +− = ∂ ∂ == +− = ∂ ∂ ==   ( ) ( )222 222 zy z zy y ppn z n n z H xd pd ppn y n n y H xd pd +− ∂ ∂ −= ∂ ∂ −= +− ∂ ∂ −= ∂ ∂ −= We recover the previous equations. We can also see that if n is constant, than H is not an explicit function of x, y, z and is also constant since from previous equations both py and pz are constant. Return to Table of Content
136 [1] C. Carathéodory, “Calculus of Variations and Partial Differential Equations of the First Order”, Part I and Part II, Holden-Day Inc, 1965, English translation from German 1935 References SOLO Calculus of Variations [2]O. Bolza, “Lectures on the Calculus of Variations”, Dover Publications, New York, 1961, Republication of a work published by Univ. of Chicago 1904 [3] G.A. Bliss, “Lectures on the Calculus of Variations”, Univ. of Chicago Press, Chicago, 1946 [4] W.S. Kimball, “Calculus of Variations, by Parallel Displacement”, Butterworths Scientific Publications, 1952 [5] L.E. Elsgolc, , “Calculus of Variations”, Pergamon Press, Addison-Wesley, 1962 [6] I.M. Gelfand, S.V. Fomin, “Calculus of Variations”, Prentice-Hall, 1963 [7] G. Leitmann, “Calculus of Variations and Optimal Control, An Introduction”, Plenum Press, 1981 [8] H. Sagan, “Introduction to Calculus of Variations”, Dover Publication, New York, 1969 [9] D. Lovelock, H. Rund, “Tensors Differential Forms, and Variational Principles”, Dover Publication, New York, 1975, 1989 [10] J.L. Troutman, “Variational Calculus with Elementary Convexity”, Springer-Verlag, 1983 [11] R. Weinstock, “Calculus of Variations with Applications to Physics and Engineering”, Dover Publication, New York, 1952, 1974
137 SOLO References Calculus of Variations
138 SOLO References (continue – 1) Return to Table of Content Calculus of Variations
February 20, 2015 139 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA
SOLO Appendix: Useful Mathematical Theorems Calculus of Variations The following mathematical theorems are useful in the Calculus of Variations: •Implicit Function Theorem •Heine-Borel Theorem •Ordinary Differential Equations Theorems •Euler-Lagrange Ordinary Differential Equations Theorems •Partial Differential Equations of the First Order Theorems
SOLO Appendix 1: Implicit Functions Theorem Calculus of Variations Let continuous functions on a domain D of the parameters:( ) ( ) 0,,:, 1 == T nffuxf  ( ) n T n Rxxx ∈= ,,: 1  ( ) m T m Ruuu ∈= ,,: 1  having continuous first partial derivatives                 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ =                 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ = m nn m u n nn n x u f u f u f u f u f f x f x f x f x f x f f ,, ,, : ,, ,, : 1 1 1 1 1 1 1 1       Consider an interior point of the domain of definition of for which( )00 ,uxP ( )uxf , ( ) ( ) ( ) 0, 00 =uxf and the following Jacobian is nonzero: ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) 0 ,, ,, 00 00 00 ,1 1 1 1 , , ≠ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ = uxn nn n ux uxx x f x f x f x f x f f   
SOLO Appendix 1: Implicit Functions Theorem (continue – 1) Calculus of Variations Then there exists a certain neighborhood of this point a unique system of continuous functions that satisfies the conditions: ( ) ( ) ( ) { }δδ ≤−= 00 : uuuuN ( )ux ϕ= ( ) ( ) ( )00 ux ϕ=a ( )( ) ( ) ( )0 0:, uNuuuf δϕ ∈∀=b u∂ ∂ϕc exists in the same neighborhood, are continuous and are found by solving: ( )( ) ( ) ( ) 0 ,, : , = ∂ ∂ + ∂ ∂ ∂ ∂ = u uxf ux uxf ud uufd ϕϕ If are of class C(p) in than is also of class C(p) .u( )uxf , ( )uϕ Proof Existenc e ( ) ( ) 0,:, 1 2 ≥= ∑= n i i uxfuxFDefine the scalar and the neighborhoods: ( ) ( ) ( ) { } ( ) ( ) ( ) { }    ≤−= ≤−= δ ρ δ ρ 00 00 : : uuuuN xxxxN ( )0 u ( )0 x ( ) δ+0 u( ) δ−0 u ( ) ρ+0 x ( ) ρ−0 x ( ) σ−0 x ( ) σ+0 x u ( ) 0, =uxf D
SOLO Appendix 1: Implicit Functions Theorem (continue – 2) Calculus of Variations Proof of Existence (continue – 1) ( )0 u ( )0 x ( ) δ+0 u( ) δ−0 u ( ) ρ+0 x ( ) ρ−0 x ( ) σ−0 x ( ) σ+0 x u ( ) 0, =uxf D Let be the set of boundary points of defined as( ) ( )0 xNρ∂ ( ) ( )0 xNρ ( ) ( ) ( ) { }ρρ =−=∂ 00 : xxxxN ( ) ( ) ( ) 0, 00 =uxfAccording to we have ( ) ( ) ( ) ( ) ( ) ( ) 0,:, 1 00200 == ∑= n i i uxfuxF Let choose such thatu ( ) ( )0 uNu δ∈ Since is continuous and compact (bounded and closed) in and for and chosen . ( )uxf , ( ) ( )0 xNρ ( ) ( )0 uNδ ( ) ( )0 xNx ρ∂∈ ( ) ( )0 uNu δ∈ ( ) ( ) 0,, 1 2 >= ∑= n i i uxfuxF and attains its minimum value m on ( ) ( )0 xNρ∂ ( )( ) ( )( ) ( )uxFm uNu xNx ,inf 0 0 δ ρ ∈ ∂∈ = Since we can find a σ < ρ such that and( ) ( ) ( ) 0, 00 =uxF ( ) ( ) ( ) ( )00 xNxN ρσ ⊂ ( )( )( )( ) ( ) ( ) ( ) ( )0 2 , 2 ,inf 0 0 xNxfor m uxF m uxF uNu xNx σ δ σ ∂∈>→= ∈ ∂∈
SOLO Appendix 1: Implicit Functions Theorem (continue – 3) Calculus of Variations Proof of Existence (continue – 2) ( )0 u ( )0 x ( ) δ+0 u( ) δ−0 u ( ) ρ+0 x ( ) ρ−0 x ( ) σ−0 x ( ) σ+0 x u ( ) 0, =uxf D Also since we can diminish σ < ρ such that( ) 0, 00 =uxF ( ) ( ) ( )0 2 , xNinsidexfor m uxF σ< Since on the boundary of and , at some point inside , the scalar attains its minimum inside and ( ) 2 , m uxF > ( ) ( )0 xNσ ( ) 2 , m uxF < ( ) ( )0 xNσ ( )uxF , ( ) ( )0 xNσ ( ) ( ) ( ) ( ) ( ) 0 , , , ,, 1 11 1 =         ∂ ∂ ⋅= ∂ ∂ ⋅= ∑ ∑∑∑ = == = n j j n i j i i n i n j j j i i dx x uxf uxfdx x uxf uxfuxFd This must hold for each , therefore ( ) ( )0 xNxd j σ∈ ( ) ( ) nj x uxf uxf n i j i i ,,10 , , 1 =∀= ∂ ∂ ⋅∑= ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0, , , , ,, ,, 1 1 1 1 1 =      ∂ ∂ =                           ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ uxf x uxf uxf uxf x uxf x uxf x uxf x uxf n n nn n     or ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) 0 ,, ,, 00 00 00 ,1 1 1 1 , , ≠ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ = uxn nn n ux uxx x f x f x f x f x f f    Since it follows that the previous equality is possible only if: ( ) 0, =uxf We proved that for every , exist at least one such that . ( ) ( )0 uNu δ∈ ( ) ( )0 xNx σ∈ ( ) 0, =uxf
SOLO Appendix 1: Implicit Functions Theorem (continue – 4) Calculus of Variations Proof of Uniqueness ( )0 u ( )0 x ( ) δ+0 u( ) δ−0 u ( ) ρ+0 x ( ) ρ−0 x ( ) σ−0 x ( ) σ+0 x u ( ) 0, =uxf D Suppose that for a given we have two values such that . ( ) ( )0 uNu δ∈ ( ) ( ) ( ) ( )021 , xNxx σ∈ ( ) ( ) ( ) ( ) 0,, 21 == uxfuxf We can write this equation as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )uxxxfuxxxfuxfuxf niniii ,,,,,,,,,, 11 2 1 1 22 2 2 1 12  −=− ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )uxxxxfuxxxxf uxxxxfuxxxxf uxxxxfuxxxxf uxxxxfuxxxxf nini nini nini nini ,,,,,,,,,, ,,,,,,,,,, ,,,,,,,,,, ,,,,,,,,,, 11 3 1 2 1 1 21 3 1 2 1 1 11 3 1 2 1 1 22 3 1 2 1 1 22 3 1 2 1 1 22 3 2 2 1 1 22 3 2 2 1 1 22 3 2 2 2 1      −+ −+ −+ −= ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )12 1 0 1221211 1 211 1 221 1 ,,,,, ,,,,,,,,,, jjijjjnjjj j i njinji xxBxxduxxxxx x f uxxxfuxxxf −=−−+ ∂ ∂ = − ∫ θθ  We can write where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ −+ ∂ ∂ = +− 1 0 22 1 1211 1 1 1 21 ,,,,,,,:,, θθ duxxxxxxx x f uxxB njjjjj j i ij  We can see that ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ux x f duxxxxx x f uxxB j i njjj j i ij ,,,,,,,,,, 1 1 0 22 1 11 1 1 1 11 ∂ ∂ =∫ ∂ ∂ = +− θ
SOLO Appendix 1: Implicit Functions Theorem (continue – 5) Calculus of Variations Proof of Uniqueness (continue – 1) ( )0 u ( )0 x ( ) δ+0 u( ) δ−0 u ( ) ρ+0 x ( ) ρ−0 x ( ) σ−0 x ( ) σ+0 x u ( ) 0, =uxf D Therefore we can write ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,, ,,,,,,,,,, 1 1221 11 2 1 1 22 2 2 1 12 =−= −=− ∑= n j jjij niniii xxuxxB uxxxfuxxxfuxfuxf  Those equations hold for i= 1,2,…,j, therefore we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) 0,, 1221 12 1 2 2 2 1 1 2 1 21 22221 11211 =−=               − − −             xxuxxB xx xx xx BBB BBB BBB ij nn nnnn n n      Since we have ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )( ) ( ) ( )( ) 0 ,, ,, ,, 00 00 00 ,1 1 1 1 , 000 , ≠ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ == uxn nn n ux ijuxx x f x f x f x f x f uxxBf    and is continuous in , if we choose σ small enough we can assure that and the equation is satisfied only if . ( )uxfx , ( )ux, ( ) ( ) ( )[ ] 0,det 21 ≠xxBij ( ) ( ) ( )[ ] ( ) ( ) ( ) 0, 1221 =− xxxxBij ( ) ( )12 xx = This proves that for every , exist at least one (σ small enough) such that we can write . ( ) ( )0 uNu δ∈ ( ) ( )0 xNx σ∈ ( ) 0, =uxf ( )ux ϕ=
SOLO Appendix 1: Implicit Functions Theorem (continue – 6) Calculus of Variations Proof of Continuity ( )0 u ( )0 x ( ) δ+0 u( ) δ−0 u ( ) ρ+0 x ( ) ρ−0 x ( ) σ−0 x ( ) σ+0 x u ( ) 0, =uxf D By taking the derivative of with respect to , we obtain( )( )uuf ,ϕ u ( )( ) ( ) ( ) 0 ,,, = ∂ ∂ + ∂ ∂ ∂ ∂ = u uxf ux uxf ud uufd ϕϕ from which we can see that exists in the same neighborhood, and are continuous because and exist and are continuous. u∂ ∂ϕ ( ) x uxf ∂ ∂ , ( ) u uxf ∂ ∂ , If are of class C(p) in than is also of class C(p) .u( )uxf , ( )uϕ q.e.d.
SOLO Appendix 1: Implicit Functions Theorem (continue – 7) Calculus of Variations Extension of the Implicit Functions Theorem to all Domain of Definition of ( ) ( ) 0,,:, 1 == T nffuxf  We found the unique function that satisfies the conditions:( )ux ϕ= ( ) ( ) ( )00 ux ϕ= ( )( ) ( ) ( )0 0, uNuuuf δϕ ∈∀= ( )uxf ,We want to extend this result to the domain D where the functions are defined provided that ( ) ( ){ }0,&,0 ,, ,, 1 1 1 1 =∈∀≠ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ = uxfDux x f x f x f x f x f f n nn n x    a) b) D is compact (closed and bounded) For this purpose we use the Heine-Borel Theorem ( see Appendix 2 for proof): A compact domain S can be covered by a given finite number of open covering sub-domains.
SOLO Appendix 1: Implicit Functions Theorem (continue – 8) Calculus of Variations Extension of the Implicit Functions Theorem to all Domain of Definition of (continue – 1) ( ) ( ) 0,,:, 1 == T nffuxf  We are using the following procedure: Choose( ) ( ) ( )12 uNu δ∈ Find the unique ( ) ( ) ( ) ( ) ( ) ( )01 1 1 1 xNux σ ϕ ∈= 2) Define a neighborhood ( ) ( )1 uNδ 1) Choose Find the unique ( ) ( ) ( ) ( ) ( ) ( )12 2 2 2 xNux σ ϕ ∈= Choose( ) ( ) ( )1− ∈ NN uNu δ N) Define a neighborhood ( ) ( )N uNδ Find the unique ( ) ( ) ( ) ( ) ( ) ( )1 2 − ∈= NNN xNux N σ ϕ ( )0 u ( )0 x ( )1 u ( ) 0, =uxf D ( )1 x According to Heine-Borel Theorem the compact domain D for which is covered by a finite number N of sets. ( ) 0, =uxf Return to Table of Content
150 Heinrich Eduard Heine ( 1821 - 1881) Félix Édouard Justin Émile Borel (1871 –1956) Appendix 2: Heine–Borel Theorem SOLO Calculus of Variations A compact domain S can be covered by a given finite number of open covering sub-domains. Proof of Heine–Borel Theorem Both S and the sub-domains Ti are given beforehand, since it is no hard to pick out a single open interval which completely covers a bounded set S. Let S be contained in the interval -N ≤ x ≤ N (S is bounded). Now divide this closed interval into two equal intervals (1) -N ≤ x ≤ 0 (2) 0 ≤ x ≤ N Any element x of S will belong to either (1) or (2). If the theorem is false, it will not be possible to cover the points of S in both (1) and (2), by a finite number of sub-domains of T, so the points of S in either (1) or (2) require an infinite covering. Assume that the elements of S in (1) still require an infinite covering, We subdivide this interval into two equal parts and repeat the above argument. In this way we construct a sequence of sets such that each Si is closed and bounded and such that the diameters of the  ⊃⊃⊃⊃⊃ i SSSS 321 0lim →∞→ ii S
151 Heinrich Eduard Heine ( 1821 - 1881) Félix Édouard Justin Émile Borel (1871 –1956) Appendix 2: Heine–Borel Theorem (continue – 1) SOLO Calculus of Variations A compact domain S can be covered by a given finite number of open covering sub-domains. Proof of Heine–Borel Theorem (continue – 1) Because the sub-domains are nested there exists a unique point P which is contained in each Si. Since P is in Si, one of the open intervals of T, say Tp, will cover P. This Tp has a finite nonzero diameter so that eventually one of the Si will be contained in Tp, since . But by assumption all the elements of this Si require an infinite number of the sub-domains in T to cover them. This is a direct contradiction to the fact that a single Tp covers them. Hence our original assumption is wrong, and the theorem is proved. 0lim → ∞→ i i S Return to Table of Content
152 Appendix 3: Ordinary Differential Equations (ODE) SOLO Calculus of Variations is a normal system of ordinary differential equations. Well-Posed Problems A differential equation problem is well-posed if: •A solution exists. •The solution is unique. •The solution depends continuously on the initial values. The well-posedness requires proving theorems of existence (there is a solution), uniqueness (there is only one solution), and continuity (the solution depends continuously on the initial value). ( ) givenConditionsInitialn nitxxxf td xd nii i ,,1,,,,,1  ==
153 Appendix 3: Ordinary Differential Equations (ODE) (continue – 1) SOLO Calculus of Variations ( ) givenConditionsInitialn nitxxxf td xd nii i ,,1,,,,,1  == is a normal system of ordinary differential equations. Definitions: • Solution of an ODE means an explicit solution xi = φi (t) defined in a region R. • A neighborhood of a point is defined as a sphere contained this point as is center, satisfying (r some positive constant) ( )00 ,tx ( ) ( ) 22 0 2 0 rttxx <−+− • A point is an interior point of R if it contains a neighborhood that is wholly contained in R. It is an exterior point of R if it contains a neighborhood that doesn’t contain any point of R. ( )00 ,tx • A point is a boundary point of R if every neighborhood has both interior and exterior points in R. ( )00 ,tx ( ) ( )txxxtx ni ,,,,,:, 1 =• Neighborhood Interior Point Boundary Point
154 Appendix 3: Ordinary Differential Equations (ODE) (continue – 2 SOLO Calculus of Variations ( ) givenConditionsInitialn txf td xd ,= is a normal system of ordinary differential equations. Definitions continue – 1): Interior Point Boundary Point • Limit of a Sequence is equivalent to for each ε >0 exist an integer N such that . ( ) ( )00 ,, txtx i ii ∞→ ⇒ ( ) ( ) Nittxx ii >∀<−+− 22 0 2 0 ε • Limit of a Function in a region R. We say that if for every sequence in R such that converges to the same limit A. ( )txf , ( ) ( ) ( ) Atxf txtx = → ,lim 00 ,, ( ) ( )00 ,, txtx i ii ∞→ ⇒ ( )txf , • A Function is Continuous at a point if( )txf , ( )00 ,tx ( ) ( ) ( ) ( )00 ,, ,,lim 00 txftxf txtx = → • Uniform Convergence A sequence Uniform Converge to a limit if for each positive ε there is an integer N, independent on , such that ( )ii tx , ( )tx, x ( ) ( ) Nitxtxi >∀<− ε
155 Appendix 3: Ordinary Differential Equations (ODE) (continue – 3) SOLO Calculus of Variations Derivation of a Solution of the Ordinary Differential Equations. ( ) givenConditionsInitialntxf td xd ,= Theorem I (Existence) (Cauchy- Peano)If the functions are continuous in a closed and bounded region R of n+1 dimensional space , then through each interior point of the region there exists at least one continuously derivable curve which is defined in an interval |t-t0| < a ( )txf , ( )tx, ( )00 ,tx ( )txx = Giuseppe Peano 1858 - 1932 Augustin-Louis Cauchy 1789 –1857 Pro ofSince R is closed and bounded, the functions are uniformly bounded in R and there exists a positive number M such that ( )txf , ( ) ( ) RtxniMtxfi ∈∀=< ,,,1,  ( ) ( ) ( ) ( ) ( ) ( )1111 121112 010001 , , , −−−− −+= −+= −+= NNNNNN tttxfxx tttxfxx tttxfxx  A solution can be build by dividing the interval |t-t0| < a in small intervals t0 < t1 <… <tj<… < tN < t0+a & |tj+1-tj| <δ. Start from and perform( )00 ,tx Interior Point
156 Appendix 3: Ordinary Differential Equations (ODE) (continue – 4) SOLO Calculus of Variations Derivation of a Solution of the Ordinary Differential Equations (continue – 1). Proof (continue -1)This construction defines a polygon, that for small enough, approximate a solution ( )txx = ( ) ( )∫+= t dxfxtx 0 0 , ττ We can see that ( ) ( ) ( ) ( ) ( ) ( ) ( )001211 01211012110 ttMttMttMttM xxxxxxxxxxxxxx jjjjj jjjjjjjjj −=−++−+−≤ −++−+−≤−++−+−=− −−− −−−−−−   Therefore the solution is bounded in the region ( )00 ttMxx −≤− We can obtain also this result from ( ) ( ) ( ) ( )0 000 0 ,, ttMdMdxfdxfxtx ttt −=≤≤=− ∫∫∫ τττττ Interior Point Slope +M Slope -M The Theorem that proves only “Existence”, not “Uniqueness”, was first discovered by Giuseppe Peano in 1890. This solution is due to Augustin Cauchy and the polygon is called “Cauchy Polygon”.
157 Appendix 3: Ordinary Differential Equations (ODE) (continue – 5) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Rudolf Lipschitz (1832 – 1903) ( ) givenConditionsInitialn txf td xd ,= To obtain Uniqueness of a Solution of the Ordinary Differential Equations we need in addition to the condition the condition ( ) ( ) RtxniMtxfi ∈∀=< ,,,1,  ( ) ( ) ( ) ( ) Rtxtxconstknixxktxftxf iiii ∈∀==−<− ,~&,.,,,1~,~,  Lipschit z Conditio nUnder suitable hypothesis the Mean Value Theorem, gives ( ) ( ) ( ) ( ) xxxx x tf txftxf i ii ~~, ,~, ≥≥− ∂ ∂ =− η η This means that we can replace the Lipschitz Condition with the more restrictive condition ( ) ( ) Rtxconstknik x txf ii i ∈∀==≤ ∂ ∂ ,.,,,1 , 
158 Appendix 3: Ordinary Differential Equations (ODE) (continue – 6) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. ( )txf td xd ,= Theorem (Existence and Uniqueness) (Picard–Lindelöf Theorem) In a Bounded region R , let be continuous and satisfying( )txf , ( ) ( ) RtxniMtxf ii ∈∀=< ,,,1,  as well as ( ) ( ) ( ) ( ) Rtxtxconstknixxktxftxf iiii ∈∀==−<− ,~&,.,,,1~,~,  or . ( ) ( ) Rtxconstknik x txf ii i ∈∀==≤ ∂ ∂ ,.,,,1 ,  The ODE has one, and only one, solution containing the internal point . The solution lies in the shadow region (defined by ) and can be extended to the right and the left of until it meets the boundary of R. ( )txx = ( )00 ,tx Interior Point Slope +M Slope -M ( ) ii Mtxf <, 0x Lipschit z Conditio n Those conditions are “sufficient” but not “necessary” for existence and uniqueness of solutions. There are cases when those conditions are not satisfied and a unique solution exists.
159 Appendix 3: Ordinary Differential Equations (ODE) (continue – 7) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (Picard–Lindelöf Theorem) We introduce the Picard Successive Approximations, called also the Picard Iterations, after the French Mathematician Charles Picard: Slope +M Slope -M Shaded Region Charles Émile Picard 1856 - 1941 ( ) ( ) ( )txf td xd x txf td xd x txf td xd x n n n ,:' ,:' ,:' 1 1 2 2 0 1 1 −== == ==  Ernst Leonard Lindelöf 1870 - 1946 Let show first that all those curves are defined for a ≤ t ≤ b and lie in the Shaded Region defined by . ( )txfx ii ,' 1−= 00 ttxx −≤− Assume that the graph is defined for a ≤ t ≤ b and lies in the Shaded Region, then for a ≤ t ≤ b ; hence , and ( )txx n= ( ) mtxf n ≤, ( ) mtxfx nn ≤=+ ,' 1 ( ) ( ) 0110 00 '' ttmdttxdttxxx t t n t t n −≤≤=− ∫∫ ++ Therefore lies in the Shadow Region.( )txx n=
160 Appendix 3: Ordinary Differential Equations (ODE) (continue – 8) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (continue – 1) Slope +M Slope -M Shaded Region Let estimate now the difference between two successive approximations. Define: ( ) ( ) ( ) ( ) ( ) ( ) 0&: 0 01 0 001 =−=−= −−  txtxtwtxtxtw nnnnnn ( ) ( )txfxandtxfx nnnn ,',' 11 −+ ==We have Subtract and take the norm ( ) ( ) 111 ,,'' −−+ −≤−=− nn Lipschitz nnnn xxktxftxfxx or ( ) ( ) ( )twktwtw td d n Lipschitz nn ≤= ++ 11 ' ( ) ( ) ( ) ( ) 00000011 00 , ttmtdmtdtxftxtxtw t t t t −=≤=−= ∫∫Let compute: ( ) ( ) ( )twktw td d twk nnn ≤≤− +1 ( ) ( ) ( ) 00121 1 ttmktwktw td d twk n −=≤≤−⇒ = Integrating from t0 to t (since the integrand doesn’t change sign the inequalities are preserved after integration): ( ) ( ) ( ) 22 2 0 0000221 2 0 0 1 0 tt mkttmktwtwtwk tt mk t t n − =−≤−≤−= − −⇒ ∫ =
161 Appendix 3: Ordinary Differential Equations (ODE) (continue – 9) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (continue – 2) Slope +M Slope -M Shaded Region ( ) ( ) ( )twktw td d twk nnn ≤≤− +1Start from: ( ) ( ) 22 2 0 0 0 022 2 0 0 1 tt kmtwtw tt km n − ≤−≤ − −⇒ =  ( ) ( ) !3!3 3 02 0 0 033 3 02 0 2 tt kmtwtw tt km n − ≤−≤ − −⇒ =  ( ) ( ) !! 01 0 0 0 01 0 1 n tt kmtwtw n tt km n n nn n n n − ≤−≤ − −⇒ −− −  ( ) ( ) ( ) 22 2 02 0232 2 02 0 2 tt kmtwktw td d twk tt km n − =≤≤−= − −⇒ = Integration Induction
162 Appendix 3: Ordinary Differential Equations (ODE) (continue – 10) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (continue – 2) Slope +M Slope -M Shaded Region We obtained: ( ) ( ) !! 01 0 0 0 01 0 1 n tt kmtwtw n tt km n n nn n n n − ≤−≤ − −⇒ −− −  Therefore: ( ) ( ) ! 00 n ttk k m tw n n − ≤ Let compute: ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ] 11012110 wwwtxtxtxtxtxtxtxtx nnnnnnn +++=−++−+−=− −−−−  ( ) ( ) ( ) ( )1 ! 00 1 00 110 −≤ − ≤+++≤− − = − ∑ ttk n i i nnn e k m i ttk k m wwwtxtx  ( ) ( ) ( )1lim 00 0 −≤− − ∞→ ttk n n e k m txtxLet take the limit n→∞ of the previous expression: Therefore the limit : ( ) ( )txtxn n = ∞→ lim exists as an Uniform Limit on the interval a ≤ t ≤ b. Using again Lipschitz Condition: ( ) ( ) ( ) ( ) btatxtxktxftxf nn ≤≤−≤− ,, we obtain: ( ) ( ) btatxftxf n n ≤≤= ∞→ ,,lim q.e.d. Existence
163 Appendix 3: Ordinary Differential Equations (ODE) (continue – 11) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (continue – 3) Slope +M Slope -M Shaded RegionUniqueness: Assume the existence of another solution, such that: ( ) ( ) ( )00 ~,,~ ~ txtxtxf td xd == ( ) ( ) ( )001 ,, txtxtxf td xd nn n == −Use the sequence: ( ) ( ) ( ) 000 00 ,~~ ttmtdmtdtxftxtx t t t t −=≤=− ∫∫Compute: Subtracting those equations and taking the norm, we obtain: ( ) ( ) 11 ~,,~ ~ −− −≤−=− n Lipschitz n n xxktxftxf td xd td xd 00 1 1 ~ ~ ttkmxxk td xd td xdn −≤−≤−⇒ =
164 Appendix 3: Ordinary Differential Equations (ODE) (continue – 12) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (continue – 4) Slope +M Slope -M Shaded Region Uniqueness (continue – 1): 0 1 0 1 ~ ttkm td xd td xd ttkm n −≤−≤−−⇒ = ( ) ( ) 2 ~ 2 2 0 1 2 0 1 tt kmtxtx tt km n − ≤−≤ − −⇒ = 11 ~ ~ ~ −− −≤−≤−− n n n xxk td xd td xd xxk Integration We obtained: 2 ~ ~ ~ 2 2 02 1 2 1 2 02 2 tt kmxxk td xd td xd xxk tt km n − ≤−≤−≤−−≤ − −⇒ = Integration !3 ~~~ !3 3 02 121 3 02 2 tt kmxxkxxxxk tt km n − ≤−≤−≤−−≤ − −⇒ = ( ) ( ) ( ) ( ) ( ) ( )!1 ~ !1 1 0 1 0 + − ≤−≤ + − −⇒ ++ n ttk k m txtx n ttk k m n n n n Induction
165 Appendix 3: Ordinary Differential Equations (ODE) (continue – 13) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (continue – 5) Slope +M Slope -M Shaded Region Uniqueness (continue – 1): We obtained: ( ) ( ) ( ) ( )!1 ~ 1 0 + − ≤− + n ttk k m txtx n n We see that: ( ) ( ) 0~ ∞→ ⇒− n n txtx Therefore the limit : ( ) ( )txtxn n ~lim = ∞→ is Unique and exists as an Uniform Limit on the interval a ≤ t ≤ b. q.e.d.
166 Appendix 3: Ordinary Differential Equations (ODE) (continue – 14) SOLO Calculus of Variations Continuous Dependence of Solution of the Ordinary Differential Equations. Thomas Hakon Grönwall (1877 – 1932) We want to show that the Solution of the Ordinary Differential Equations depends continuously on the Initial Values. For this let state the: Grönwall Inequality Let u and v be continuous function satisfying u (x) > 0 and v (x) ≥ 0 on [a,b]. Let c ≥ 0 be a constant. If ( ) ( ) ( ) bxatdtvtucxv x a ≤≤+≤ ∫ , then ( ) ( ){ } bxatdtucxv x a ≤≤≤ ∫ ,exp Prof of Grönwall Inequality : First assume c > 0 and define ( ) ( ) ( ) bxatdtvtucxV x a ≤≤+= ∫ ,: Then V (x) ≥ v (x), and since u and v are nonnegative, V (x) ≥ V (a) = c on [a,b]. Moreover, V’(x) = u (x) v (x0 ≤ u (x) V (x). Dividing by V (x), we get
167 Appendix 3: Ordinary Differential Equations (ODE) (continue – 15) SOLO Calculus of Variations Continuous Dependence of Solution of the Ordinary Differential Equations. Thomas Hakon Grönwall (1877 – 1932) Prof of Grönwall Inequality (continue – 1): If we take c → 0+, we get v (x) ≤ 0, which is the same result obtained from Grönwall Inequality with c = 0. ( ) ( ) ( ) bxa xV xV xu ≤≤≥ , ' Integrate both sides of this equation, from a to x ( ) ( ) ( ) ( ) ( )( )cxVsVsd sV sV sdsu x a x a x a /lnln ' ==≥ ∫∫ Since logarithmic and exponential function are increasing function with their argument, we can resolve and preserve the inequality ( ) ( ){ } bxasdsucxV x a ≤≤≤ ∫ ,exp Since V (x) ≥ v (x) we proved that .( ) ( ){ } 0&,exp >≤≤≤ ∫ cbxatdtucxv x a q.e.d. Grönwall Inequality
168 Appendix 3: Ordinary Differential Equations (ODE) (continue – 16) SOLO Calculus of Variations Continuous Dependence of Solution of the Ordinary Differential Equations. Let use Grönwall Inequality to prove the following Continuous Dependence of ODE on Initial Value ( ) ( ) hk exxtxtx 0000 ~,,~ −≤−φφ ( )tx ,~ 0φ where k is any positive constant such that for all . Moreover as approaches , the solution approaches uniformly in . kxf ≤∂∂ / ( ) 0, Rtx ⊂ 0x0 ~x ( )tx ,0φ htt ≤− 0 Let continuous functions on an open rectangle containing the point . Assume that for all sufficiently close to , the solution of the ODE exists on the interval and the graph lies within a closed region . Then, for ( ) xfandtxf ∂∂ /, ( ){ }dxcbtatxR <<<<= ,:, ( )00 ,tx 0 ~x 0x ( )tx ,~ 0φ htt ≤− 0RR ⊂0 htt ≤− 0
169 Appendix 3: Ordinary Differential Equations (ODE) (continue – 17) SOLO Calculus of Variations Continuous Dependence of Solution of the Ordinary Differential Equations. Proof of Continuous Dependence of ODE on Initial Value: The Solution of ODE satisfies the Integral Equation:( )tx ,0φ ( ) ( )( ) httsdssxfxtx t t ≤−+= ∫ 0000 ,,,, 0 φφ Similarly the Solution of ODE satisfies the Integral Equation:( )tx ,~ 0φ ( ) ( )( ) httsdssxfxtx t t ≤−+= ∫ 0000 ,,,~~,~ 0 φφ Subtracting the second equation from the first gives: ( ) ( ) ( )( ) ( )( )[ ]∫ −+−=− t t sdssxfssxfxxtxtx 0 ,,~,,~,~, 00000 φφφφ Assume that t > t0 and tacking the norm of both sides ( ) ( ) ( )( ) ( )( )∫ −+−≤− t t sdssxfssxfxxtxtx 0 ,,~,,~,~, 00000 φφφφ kxf ≤∂∂ /Since or satisfies the Lipschitz Condition( )txf , ( )( ) ( )( ) ( ) ( )sxsxkssxfssxf ,~,,,~,, 0000 φφφφ −≤− ( ) ( ) ( ) ( )∫ −+−≤− t t sdsxsxkxxtxtx 0 ,~,~,~, 00000 φφφφWe obtain:
170 Appendix 3: Ordinary Differential Equations (ODE) (continue – 18) SOLO Calculus of Variations Continuous Dependence of Solution of the Ordinary Differential Equations. Proof of Continuous Dependence of ODE on Initial Value (continue – 1): ( ) ( ) ( ) ( )∫ −+−≤− t t sdsxsxkxxtxtx 0 ,~,~,~, 00000 φφφφWe obtained: Use Grönwall Inequality: ( ) ( ) ( ) ( ) ( ) 0&0,0 , ≥≤≤≥> ≤≤+≤ ∫ cbxaonxvxu bxatdtvtucxv x a ( ) ( ){ } bxatdtucxv x a ≤≤≤ ∫ ,exp ( ) ( ) ( ) ( ) 0~:&0,~,:,0: 000 ≥−=≥−=>= xxctxtxtvktu φφwith: ( ) ( ) { } ( ) hk htt ttkt t exxexxsdkxxtxtx ~~exp~,~, 00000 0 0 0 −≤−=−≤− ≤− − ∫φφ For t < t0, we can use t0 – t instead of t. ( )tx ,~ 0φWe see that as approaches , the solution approaches uniformly in . 0 ~x 0x ( )tx ,0φ htt ≤− 0 q.e.d.
171 Appendix 3: Ordinary Differential Equations (ODE) (continue – 19) SOLO Calculus of Variations Let use Grönwall Inequality to prove the following Continuous Dependence on of Solutions of ODE ( ) ( ) ( ) RtxtxftxF ⊂∀≤− ,,,, ε Continuous Dependence on of Solution of the Ordinary Differential Equations.( )txf , ( )txf , Let continuous functions on an open rectangle containing the point . Let be continuous in R and assume that ( ) xfandtxf ∂∂ /, ( ){ }dxcbtatxR <<<<= ,:, ( )00 ,tx ( )txF , Let be the solution to the initial value problem:( )tx,φ ( ) ( ) 00,, xtxtxf td xd == Let be the solution to the initial value problem:( )tx,ψ ( ) ( ) 00,, xtxtxF td xd == RR ⊂0 Assume both solutions exist on [t0 – h, t0 +h] and their graphs lie in a closed region . Then for |t-t0| ≤ h, ( ) ( )txandtx ,, ψφ ( ) ( ) hk ehtxtx εψφ ≤− ,, where k is any positive constant such that for all . Moreover as approaches uniformly on R, that is, as ε→0+, the solution approaches uniformly in . kxf ≤∂∂ / ( ) 0, Rtx ⊂ fF htt ≤− 0 ( )tx,ψ ( )tx,φ
172 Appendix 3: Ordinary Differential Equations (ODE) (continue – 20) SOLO Calculus of Variations Continuous Dependence on of Solution of the Ordinary Differential Equations.( )txf , Proof of Continuous Dependence on of Solutions of ODE( )txf , ( ) ( )( ) httsdssxfxtx t t ≤−+= ∫ 00 ,,,, 0 φφ The integral representation of , is( ) ( )txandtx ,, ψφ ( ) ( )( ) httsdssxFxtx t t ≤−+= ∫ 00 ,,,, 0 ψψ Subtracting those equations gives ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )[ ] ( )( ) ( )( )[ ] httsdssxFssxfsdssxfssxf sdssxFsdssxftxtx t t t t t t t t ≤−−+−= −=− ∫∫ ∫∫ 0 00 00 ,,,,,,,, ,,,,,, ψψψφ ψφψφ Applying the norm and using the Triangle Inequality, we obtain ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) httsdssxFssxfsdssxfssxftxtx t t t t ≤−−+−≤− ∫∫ 0 00 ,,,,,,,,,, ψψψφψφ ( )( ) ( )( ) ( ) ( )sxsxkssxfssxf kxf Lipschitz or ,,,,,, / ψφψφ −≤− ≤∂∂ use and ( )( ) ( )( ) εψψ ≤− ssxFssxf ,,,, ( ) ( ) ( ) ( ) ( ) ( ) htthsdsxsxksdsdsxsxktxtx t t t t t t ≤−+−≤+−≤− ∫∫∫ 0 000 ,,,,,, εψφεψφψφ
173 Appendix 3: Ordinary Differential Equations (ODE) (continue – 21) SOLO Calculus of Variations Return to Table of Content We obtained: Use Grönwall Inequality: ( ) ( ) ( ) ( ) ( ) 0&0,0 , ≥≤≤≥> ≤≤+≤ ∫ cbxaonxvxu bxatdtvtucxv x a ( ) ( ){ } bxatdtucxv x a ≤≤≤ ∫ ,exp ( ) ( ) ( ) ( ) hctxtxtvktu εψφ =≥−=>= :&,0,,:,0:with: ( ) ( ) { } ( ) hk htt ttkt t ehehsdkhtxtx εεεψφ ≤− − ≤=≤− ∫ 0 0 0 exp,, Continuous Dependence on of Solution of the Ordinary Differential Equations.( )txf , Proof of Continuous Dependence on of Solutions of ODE (continue – 1)( )txf , ( ) ( ) ( ) ( ) httsdsxsxkhtxtx t t ≤−−+≤− ∫ 0 0 ,,,, ψφεψφ In particular as approaches uniformly on R, that is, as ε→0+, the solution approaches uniformly in . fF htt ≤− 0 ( )tx,ψ ( )tx,φ q.e.d.

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Calculus of variations

  • 1.
    1 Calculus of Variations SOLOHERMELIN "weak" neighbor ( )( )000 , txtA ( )( )fff txtB , ( )txx t "strong" neighbor ( )ε,tx http://www.solohermelin.com
  • 2.
    2 Table of Content Calculusof VariationsSOLO . Introduction 1. General Formulation of the Simplest Problem of Calculus of Variations 2. Solution Method 2.1 Neighborhoods and Variations 3. Variations of the Functional J 4. Necessary Conditions for Extremum 4.4 Special Cases 4.5Examples 5. Boundary Conditions 6. Corner Conditions 7 Sufficient Conditions and Additional Necessary Conditions for a Weak Extremum 4.1 The First Fundamental Lemma of the Calculus of Variations 4.2The Euler-Lagrange Equation 4.3 The Second Fundamental Lemma of the Calculus of Variations 8 Legendre’s Necessary Conditions for a Weak Minimum (Maximum)
  • 3.
    Table of Content(continue – 1) Calculus of VariationsSOLO 9. Jacobi’s Differential Equation (1837) and Conjugate Points 9.1 Conjugate Points 9.2 Fields Definition 10. Hilbert’s Invariant Integral 11. The Weierstrass Necessary Condition for a Strong Minimum (Maximum) Summary 12. Canonical Form of Euler-Lagrange Equations 12.1 Legendre’s Dual Transformation 12.2 Transversality Conditions (Canonical Variables ) 12.3 Weierstrass-Erdmann Corner Conditions (Canonical Variables) 12.4 First Integrals of the Euler-Lagrange Equations 12.5 Equivalence Between Euler-Lagrange and Hamilton Functionals 12.6 Equivalent Functionals 12.7 Canonical Transformations 12.8 Caratheodory's Lemma 12.9 Hamilton-Jacobi Equations Jacobi’s Theorem
  • 4.
    Table of Content(continue – 2) Calculus of VariationsSOLO References Appendix 1: Implicit Functions Theorem Appendix: Useful Mathematical Theorems Appendix 2: Heine–Borel Theorem Appendix 3: Ordinary Differential Equations
  • 5.
    5 HISTORY OF CALCULUSOF VARIATIONSSOLO “When the Tyrian princess Dido landed on the Mediterranean sea she was welcomed by a local chieftain. He offered her all the land that she could enclose between the shoreline and a rope of knotted cowhide. While the legend does not tell us, we may assume that Princess Dido arrived at the correct solution by stretching the rope into the shape of a circular arc and thereby maximized the area of the land upon which she was to found Carthage. This story of the founding of Carthage is apocryphal. Nonetheless it is probably the first account of a problem of the kind that inspired an entire mathematical discipline, the calculus of variations and its extensions such as the theory of optimal control.” (George Leitmann “The Calculus of Variations and Optimal Control – An Introduction” Plenum Press, 1981) Dido Maximum Area Problem
  • 6.
    6 ( ) ∫∫− == a a xy dxyAdJ maxmaxmax Given a rope of length P connected to each end of straight line of length 2 a < P find the shape of the rope necessary to enclose the maximum area between the rope and the straight line. The problem can be formulated as: Dido Maximum Area Problem HISTORY OF CALCULUS OF VARIATIONS ( ) ( ) ( ) ∫∫∫∫∫ −−− =+=      +=+== a a a a a a dxdxdx xd yd ydxdsdP θθ sectan11 2 2 22 subject o the isoperimetric constraint: where: θtan= xd yd SOLO Return to Table of Content Rope of length P ( )xθ x y a+a− y dx
  • 7.
    7 1. General Formulationof the Simplest Problem of Calculus of Variations Given: (1) A Functional (function of functions) J [x (t)] Calculus of VariationsSOLO ( )[ ] ( ) ( ) ( ) ( )( ) ( ) ( )∫∫       == ⋅ff t t t t nn dttxtxtFdttxtxtxtxtFtxJ 00 ,,,,,,,, 11  ( ) ( ) ( )( )T n txtxtx ,,: 1 = ( ) ( ) ( )( ) ( ) ( ) T n T n tx dt d tx dt d txtxtx       == ⋅ ,,,,: 11  where: ( ) ( )      ⋅ txtxtF ,, ( ) ( )txtxt ⋅ ,, (2) shall be continuous and admit continuous partial derivatives of the first, second and third order in a domain which contains all points . .
  • 8.
    8 General Formulation ofthe Simplest Problem of Calculus of Variations Calculus of VariationsSOLO . ( ) ( ) ( ) ( ) ( )( )T n tftftftftx ,,, 21 ==(1) The vector of functions where t0 ≤ t ≤tf, fi (t)i=1,n being single valued of t that minimizes (maximizes) the functional J in a weak neighborhood. (2) fi (t)i=1,n are continuous and consist of a finite number of arcs of continuously turning tangent, not parallel to the x axis; i.e. fi (t)€ D (1) (3) passes through two points (constant vectors), defined or not.( )tx ( )tx xt,(4) lies in a given region of the space. corner points( )( )000 , txtA ( )( )fff txtB , ( )txx t Find: Figure: A Possible Solution for the Problem of Calculus of Variations
  • 9.
    9 General Formulation ofthe Simplest Problem of Calculus of Variations Calculus of VariationsSOLO Examples of Calculus of Variations Problems 1. Brachistochrone Problem A particle slides on a frictionless wire between two fixed points A(0,0) and B (xfc, yfc) in a constant gravity field g. The curve such that the particle takes the least time to go from A to B is called brachistochrone (βραχιστόσ Greek for “shortest“, χρόνοσ greek for “time). The brachistochrone problem was posed by John Bernoulli in 1696, and played an important part in the development of calculus of variations. The problem was solved by Johann Bernoulli, Jacob Bernoulli, Isaac Newton, Gottfried Leibniz and Guillaume de L’Hôpital. Let choose a system of coordinates with the origin at point A (0,0) and the y axis in the constant g direction x y V ( )tγ ( )fcfc yxB ,fcx fcy N  ( )0,0A
  • 10.
    10 General Formulation ofthe Simplest Problem of Calculus of Variations Calculus of VariationsSOLO Examples of Calculus of Variations Problems 1. Brachistochrone Problem Since the motion of the particle is in a frictionless fixed gravitational field the total energy is conserved ( ) ygVyVygVV 2 2 1 2 1 2 0 22 0 +=→−= x y V ( )tγ ( )fcfc yxB ,fcx fcy N  ( )0,0A Second way to get this relation is: ( ) ygVVdygdVV sd yd ggV sd Vd td sd sd Vd td Vd =−→=→==== 2 0 2 2 1 sinγ where V0 is the velocity of the particle at point A and ( ) ( )22 ydxdsd += td xd xd yd td yd td xd td sd V 222 1       +=      +      == We have xd ygV xd yd xd V xd yd td 2 11 2 0 22 +       + =       + = The cost function is ∫∫       = +       + = cfcf xx xd xd yd yxFxd ygV xd yd J 00 2 0 2 ,, 2 1
  • 11.
    11 HISTORY OF CALCULUSOF VARIATIONS The brachistochrone problem In 1696 proposed the Brachistochrone (“shortest time”) Problem: Given two points A and B in the vertical plane, what is the curve traced by a point acted only by gravity, which starts at A and reaches B in the shortest time. Johann Bernoulli 1667-1748 SOLO
  • 12.
    12 The brachistochrone problem JacobBernoulli (1654-1705) Gottfried Wilhelm von Leibniz (1646-1716) Isaac Newton (1643-1727) The solutions of Leibniz, Johann Bernoulli, Jacob Bernoulli and Newton were published on May 1697 publication of Acta Eruditorum. L’Hôpital solution was published only in 1988. Guillaume François Antoine de L’Hôpital (1661-1704) SOLO HISTORY OF CALCULUS OF VARIATIONS
  • 13.
    13 General Formulation ofthe Simplest Problem of Calculus of Variations Calculus of VariationsSOLO Examples of Calculus of Variations Problems 2. Problem of Minimum Surface of Revolution Given two points A (a,ya) and B (b, yb) a≠b in the plane. Find the curve that joints these two points with a continuous derivative, in such a way that the surface generated by the rotation of this curve about the x axis has the smallest possible area. x y ( )bybB , ( )ayaA , ( ) ( )22 ydxdsd += y Minimum Surface of Revolution The surface generated by the rotation of y (x) curve about the x – axis can be calculated using ( ) ( ) xd xd yd yydxdysdydS 2 22 1222       +=+== πππ Therefore ( )∫       +== b a xd xd yd xySJ 2 12: π 2 1,,       +=      xd yd y xd yd yxF We can see that F is not an explicit function of x.
  • 14.
    14 General Formulation ofthe Simplest Problem of Calculus of Variations Calculus of VariationsSOLO Examples of Calculus of Variations Problems 3. Geometrical Optics and Fermat Principle The Principle of Fermat (principle of the shortest optical path) asserts that the optical length of an actual ray between any two points is shorter than the optical ray of any other curve that joints these two points and which is in a certain neighborhood of it. An other formulation of the Fermat’s Principle requires only Stationarity (instead of minimal length). ∫ 2 1 P P dsn An other form of the Fermat’s Principle is: Principle of Least Time The path following by a ray in going from one point in space to another is the path that makes the time of transit of the associated wave stationary (usually a minimum).
  • 15.
    15 SOLO We have: constS = constdSS=+ sˆ ∫ 2 1 P P dsn 1 P 2 P ( ) ( ) ( )∫∫∫∫ =      +      +=== 2 1 2 1 2 1 ,,,, 1 1,, 1 ,, 1 0 22 00 P P P P P P xdzyzyxF c xd xd zd xd yd zyxn c dszyxn c tdJ  The stationarity conditions of the Optical Path using the Calculus of Variations ( ) ( ) ( ) xd xd zd xd yd zdydxdds 22 222 1       +      +=++= Define: xd zd z xd yd y ==  &: ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += General Formulation of the Simplest Problem of Calculus of Variations Calculus of Variations Examples of Calculus of Variations Problems 3. Geometrical Optics and Fermat Principle Paths of Rays Between Two Points
  • 16.
    16 SOLO General Formulation ofthe Simplest Problem of Calculus of Variations Calculus of Variations Examples of Calculus of Variations Problems 4. Hamilton Principle for Conservative Systems The motion of a conservative system, from time t0 to tf is such that the integral ( )∫= ft t dtqqLJ 0 ,  has a stationary value ( δJ = 0), where ( ) ( ) ( )qVqqTqqL −=  ,:, qq , ( )qqT , ( )qV δ is the Lagrangian of the system are the generalized coordinate vector of the system and its derivatives kinetic energy of the system potential energy of the system the variation that will be defined in the next section. Since the system is conservative, the external forces acting on the system are given by ( ) ( )qVqQ ∇= For a non-conservative system the Extended Hamilton Principle is ( ) ( ) 0, 00 =+ ∫∫ ff t t t t dtqqQdtqqT δδ  The Hamilton Principle doesn’t require the minimization but only stationarity (vanishing of the first variation δJ = 0).
  • 17.
    17 SOLO General Formulation ofthe Simplest Problem of Calculus of Variations Calculus of Variations Examples of Calculus of Variations Problems 5. Geodesics Suppose we have a surface specified by two parameters u and v and the vector .( )vur ,  The shortest path lying on the surface and connecting to points of the surface is called a geodesic. A B ( )vur ,  vdrv  udru  rd  The Shortest Path on a Surface The arc length differential is td td vd v r td ud u r td vd v r td ud u r td td rd td rd td td rd ds 2/12/1             ∂ ∂ + ∂ ∂ ⋅      ∂ ∂ + ∂ ∂ =      ⋅==  td td vd v r v r td vd td ud v r u r td ud u r u r 2/122 2                     ∂ ∂ ⋅ ∂ ∂ +                  ∂ ∂ ⋅ ∂ ∂ +            ∂ ∂ ⋅ ∂ ∂ = 
  • 18.
    18 SOLO General Formulation ofthe Simplest Problem of Calculus of Variations Calculus of Variations Examples of Calculus of Variations Problems 5. Geodesics (continue) A B ( )vur ,  vdrv  udru  rd  The Shortest Path on a Surface The length of the path between the two points A and B is ∫               +            +      == B A r r td td vd G td vd td ud F td ud ESJ   2/122 2:       ∂ ∂ ⋅ ∂ ∂ = u r u r E  :       ∂ ∂ ⋅ ∂ ∂ = v r u r F  :       ∂ ∂ ⋅ ∂ ∂ = v r v r G  : where Return to Table of Content
  • 19.
    19 SOLO Calculus of Variations 2.Solution Method ( )tx ( )ε,tx To find a candidate for the minimizing (maximizing) trajectory , construct variations (neighbors) of this trajectory and find the conditions under which those variations increase (decrease) the value of the functional J [x (t)]. The results of this method are known as the Calculus of Variations. "weak" neighbor ( )( )000 , txtA ( )( )fff txtB , ( )txx t "strong" neighbor ( )ε,tx Return to Table of Content
  • 20.
    20 SOLO Calculus of Variations 2.1Neighborhoods and Variations "weak" neighbor ( )( )000 , txtA ( )( )fff txtB , ( )txx t "strong" neighbor ( )ε,tx ( )tx( )ε,txLet define a function of the closeness of order k to Weak Neighborhood is a “weak” neighborhood of order k if:( )ε,tx ( ) ( ) ( ) ( ) ( ) ( )         ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ = → → → tx t tx t tx t tx t txtx k k k k ε ε ε ε ε ε ,lim ,lim ,lim 0 0 0  ( ) ( )( ) ( )( )00000 ,,, txtAtxt ∈εεε ( ) ( )( ) ( )( )fffff txtBtxt ,,, ∈εεε Strong Neighborhood If we only have (only for k = 0) then is called a “strong” neighborhood. If k > 0 then it is a “weak” neighborhood. ( ) ( )txtx = → ε ε ,lim 0 ( )ε,tx ( ) ( ) ( ) ( ) ( ) ( )32 0 2 2 0 , 2 1, ,:, εε ε ε ε ε ε εε εε Ο/+ ∂ ∂ + ∂ ∂ =−=∆ == d tx d tx txtxtxLet compute: ( ) 0lim 2 3 0 → Ο/ → ε ε ε w
  • 21.
    21 SOLO Calculus of Variations Firstand Second Variations ( )ε,tx ( )txFirst Variation of ( ) ( ) ε ε ε δ ε d tx tx 0 , : =∂ ∂ = i.e. the differential of as a function of ε ( )txThe First Variation of is defined as ( )txSecond Variation of ( )txThe Second Variation of is defined as ( ) ( ) 2 0 2 2 2 , : ε ε ε δ ε d tx tx = ∂ ∂ = ( )( )fff txtB , x t ( )2,εtx ( )1,εtx ( )tx ft ( )1εft ( )2εft At the boundaries t0 and tf are functions of ε (see Figure)
  • 22.
    22 SOLO Calculus of Variations Firstand Second Variations at the Boundary Therefore at the boundaries we have ( )( ) fiitx ,0 , = εε ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )xdxdxd d tx d tx txtxtx ii tt ii ii 32 32 0 2 2 0 2 1 , 2 1, ,:, Ο/++= =Ο/+ ∂ ∂ + ∂ ∂ =−=∆ == εε ε εε ε ε εε εεεε εε ε ε ε d x xd i i t t 0: = ∂ ∂ = 2 0 2 2 2 0 2 :&: ε ε ε ε εε d x xdd x xd i i i i t t t t == ∂ ∂ = ∂ ∂ = where: ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( )      ∂ ∂ +      ∂ ∂ +      ∂ ∂ = =      = εε εεε ε ε εε ε ε εε ε εε εε εε ε ,,, ,,2 2 i i i i i ii tx d d d dt d d tx td dt tx td d tx d d d d tx d d ( )       ∂ ∂ + ∂∂ ∂ + ∂ ∂ +      ∂∂ ∂ + ∂ ∂ = 2 22 2 22 2 2 εεεε ε εεε x d dt t x d td t x d dt t x d dt t x iiii 2 2 2 222 2 2 2 εεεεε ∂ ∂ + ∂ ∂ + ∂∂ ∂ +      ∂ ∂ = x d td t x d dt t x d dt t x iii ( )( ) ( )( ) ( ) ( )( )εε εε ε εεεε ε ,,, i i ii tx d dt tx t tx d d ∂ ∂ + ∂ ∂ = We have: ( )( )fff txtB , x t ( )fxd 3 Ο/ ( )ε,tx ( )tx ff dtt +( )0=εft fxδ fxd fx∆ fxd 2 ff dtx • fdt Variations at the Boundary tf
  • 23.
    23 SOLO Calculus of Variations Firstand Second Variations at the Boundary ( )( )fff txtB , x t ( )fxd 3 Ο/ ( )ε,tx ( )tx ff dtt +( )0=εft fxδ fxd fx∆ fxd 2 ff dtx • fdt Variations at the Boundary tf ( )( ) ( ) ( )( ) εεε ε ε ε ε εε εεε dtxd d dt tx t xd i i ii 000 ,, === ∂ ∂ +      ∂ ∂ = ( )( ) ( )( )εεεε ε ,, 0 i tt i txx dt xd tx t ii •• = === ∂ ∂ ( ) i i dtd d dt = = ε ε ε ε 0 ( ) ( )( ) εεε ε δ ε dtxtx ii 0 ,: =∂ ∂ = ( ) ( ) fitxdttxxd iii ,0=+= • δ But Therefore we obtain:
  • 24.
    ( ) 2 0 2 2 2 :ε ε ε ε d d td td i i = =and define: 24 SOLO Calculus of Variations First and Second Variations at the Boundary ( )( )fff txtB , x t ( )fxd 3 Ο/ ( )ε,tx ( )tx ff dtt +( )0=εft fxδ fxd fx∆ fxd 2 ff dtx • fdt Variations at the Boundary tf ( ) ( ) ( ) ( ) ε ε ε ε ε ε ε ε εε εεεε d d dt d t tx d d dt t tx xd iiii i 00 2 00 2 2 2 , 2 , ====       ∂ ∂ ∂ ∂ +      ∂ ∂ = ( ) ( ) ( ) 2 0 2 2 2 0 2 2 0 ,, ε ε ε ε ε εε εεε d tx d d td t tx iii === ∂ ∂ + ∂ ∂ + ( ) ( ) ( )ii i txtx dt d t tx •• = == ∂ ∂ 2 2 0 2 2 , ε ε ( ) ( )i i txd t tx • = =      ∂ ∂ ∂ ∂ δε ε ε ε 0 , ( ) ( ) 2 0 2 2 2 , : ε ε ε δ ε d tx tx i i = ∂ ∂ = ( )( ) ( ) ( ) ( ) fitxtdtxdttxdttxxd iiiiiiiii ,02 2222 =+++= •••• δδ Also we have: But: Therefore Return to Table of Content
  • 25.
    25 SOLO Calculus of Variations 3.Variations of the Functional J The value of the functional J in the neighborhood of is( )ε,tx ( )tx ( ) ( ) ( ) ( ) ( ) ∫       = • ε ε εεε ft t dttxtxtFJ 0 ,,,, ( ) ( )εε ,:, tx t tx ∂ ∂ = • where We can write: ( ) ( ) ( ) ( )JJJd d Jd d d dJ JJJ 3232 0 2 2 0 2 1 2 1 0: δδδεε ε ε ε εε εε Ο/++=Ο/++= =−=∆ == where ε ε δ ε d d dJ J 0 : = = the first variation of J 2 0 2 2 2 : ε ε δ ε d d Jd J = = the second variation of J ( ) 0lim 2 3 0 → Ο/ → ε ε ε
  • 26.
    26 SOLO Calculus of Variations FirstVariation of the Functional J ( ) ( ) ( ) ( ) ( )               = ∫ • ε ε εε εε ε ft t dttxtxtF d d d dJ 0 ,,,, ( ) ( ) ( ) ( ) ( ) ( ) ε ε εε ε ε εε d dt txtxtF d dt txtxtF f fff 0 000 ,,,,,,,,       −      = •• ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫       ∂ ∂       + ∂ ∂       + ••• • ε ε ε ε εεε ε εε ft t x x dttxtxtxtFtxtxtxtF 0 ,,,,,,,,,, T n nn x x F x F x F x F x F x F F x x x xxtF       ∂ ∂ ∂ ∂ ∂ ∂ =                       ∂ ∂ ∂ ∂ ∂ ∂ =                       ∂ ∂ ∂ ∂ ∂ ∂ =      • ,,,:,, 21 2 1 2 1   T n x x F x F x F xxtF       ∂ ∂ ∂ ∂ ∂ ∂ =      • •    ,,,:,, 21 and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫             +      +       −      == ••• •• = • ft t T x T x ffff dttxtxtxtFtxtxtxtF dttxtxtFdttxtxtF d dJ J 0 ,,,, ,,,, 0000 0 δδ ε ε δ ε
  • 27.
    27 SOLO Calculus of Variations SecondVariation of the Functional J ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               ∂ ∂       + ∂ ∂       +             +      −             +      =    = ∫ ••• •• •• ε ε ε ε εεε ε εε ε ε ε ε εε ε ε εε ε ε ε ε εε ε ε εε εεεε ft t T x T x f fff f fff dttxtxtxtFtxtxtxtF d d d dt d d txtxtF d dt txtxtF d d d dt d d txtxtF d dt txtxtF d d d dJ d d d Jd 0 ,,,,,,,,,, ,,,,,,,, ,,,,,,,, 0 000 0 000 2 2  In this equation we have: ( ) ( ) fi x d dt t x F x d dt t x F d dt FtxtxtF d d it iT x iT x i tiii ,0,,,, =                 ∂ ∂ + ∂ ∂ +      ∂ ∂ + ∂ ∂ +=      •• • εεεεε εε ε  t F Ft ∂ ∂ =: ( ) ( ) 2 2 ε ε ε ε ε d td d dt d d ii =      ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       ∫       ∂ ∂       + ∂ ∂       •••ε ε ε ε εεε ε εε ε ft t T x T x dttxtxtxtFtxtxtxtF d d 0 ,,,,,,,,,,  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ε ε ε ε εεε ε εε d dt txtxtxtFtxtxtxtF f ffff T xffff T x       ∂ ∂       + ∂ ∂       = ••• ,,,,,,,,,,  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ε ε ε ε εεε ε εε d dt txtxtxtFtxtxtxtF T x T x 0 00000000 ,,,,,,,,,,       ∂ ∂       + ∂ ∂       − •••  ( ) ( ) td x F xx F xx F x F xx F xx F xx T xx T T x t t xx T xx T T x f              ∂ ∂         ∂ ∂ +      ∂ ∂         ∂ ∂ + ∂ ∂ +             ∂ ∂       ∂ ∂ +      ∂ ∂       ∂ ∂ + ∂ ∂ + ••••• ••••• ∫ εεεεεεεεεε ε ε 2 2 2 2 0 and
  • 28.
    28 SOLO Calculus of Variations SecondVariation of the Functional J (continue – 1) [ ] [ ]               =                 ∂ ∂ ∂ ∂ = ∂ ∂ =               ∂ ∂ ∂ ∂ = nnnn n n n xxxxxx xxxxxx xxxxxx xxx T x T xx FFF FFF FFF FFF x x F xx F x F      21 21212 12111 21 ,,,: 1 1 [ ] [ ]               =                 ∂ ∂ ∂ ∂ = ∂ ∂ =               ∂ ∂ ∂ ∂ = ••• nnnn n n n xxxxxx xxxxxx xxxxxx xxx T x T xx FFF FFF FFF FFF x x F x x F x F            21 21212 12111 21 ,,,: 1 1 [ ]               =                 ∂ ∂ ∂ ∂ =     ∂ ∂ =                 ∂ ∂ ∂ ∂ = •• • nnnn n n n xxxxxx xxxxxx xxxxxx xxx T x T xx FFF FFF FFF FFF x x F xx F x F          21 21212 12111 21 ,,,: 1 1 [ ]               =                 ∂ ∂ ∂ ∂ =     ∂ ∂ =                 ∂ ∂ ∂ ∂ = • ••• nnnn n n n xxxxxx xxxxxx xxxxxx xxx T x T xx FFF FFF FFF FFF x x F xx F x F             21 21212 12111 21 ,,,: 1 1
  • 29.
    29 SOLO Calculus of Variations SecondVariation of the Functional J (continue – 2) xx T xxxx ijji xx FFF x F xx F x F jiij =→=      ∂ ∂ ∂ ∂ =         ∂ ∂ ∂ ∂ =: xxxx T xxxx ijji xx FFFF x F xx F x F jiij   ≠=→=      ∂ ∂ ∂ ∂ =         ∂ ∂ ∂ ∂ =: xx T xxxx ijji xx FFF x F xx F x F jiij   =→=      ∂ ∂ ∂ ∂ =         ∂ ∂ ∂ ∂ =: Let integrate by parts the term ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∂ ∂       − ∂ ∂ =∫ ∂ ∂ ••• • ε ε ε ε ε ε εεε f f f t t T x t t T x t t T x dt x F dt dx Fdt x F 0 0 0 2 2 2 2 2 2 By using all those developments we get: 2 2 2 2 εεεεεεεε d td F d dtx d dt t x F x d dt t x F d dt F d Jd f t f t fT x fT x f t f f +                 ∂ ∂ + ∂ ∂ +      ∂ ∂ + ∂ ∂ += •• • 2 0 2 0000 0 0 εεεεεεε d td F d dtx d dt t x F x d dt t x F d dt F t t T x T xt +                 ∂ ∂ + ∂ ∂ +      ∂ ∂ + ∂ ∂ +− •• • ( ) ( )εε εεεεεεεε ff f t T x t T x t T x T x f t T x T x x F x F d dtx F x F d dtx F x F 0 0 2 2 2 2 0 ∂ ∂ − ∂ ∂ +         ∂ ∂ + ∂ ∂ −         ∂ ∂ + ∂ ∂ + •••• •• ( ) ( ) dt x F xx F xx F x F xx F xx F dt d F xx T xx T T x t t xx T xx TT x x f             ∂ ∂         ∂ ∂ +      ∂ ∂         ∂ ∂ + ∂ ∂ +             ∂ ∂       ∂ ∂ +      ∂ ∂       ∂ ∂ + ∂ ∂       −+ ••••• •••••• ∫ εεεεεεεεεε ε ε 2 2 2 2 0
  • 30.
    30 SOLO Calculus of Variations SecondVariation of the Functional J (continue – 3) ft fffT x ffT x f t d td F x d dtx d dt t x F d dtx d dt t x F d dt F d Jd         +         ∂ ∂ + ∂ ∂ +      ∂ ∂ +      ∂ ∂ + ∂ ∂ +      = •• • 2 2 2 2 22 2 2 22 εεεεεεεεεε 0 2 0 2 2 2 0 2 000 2 0 22 t T x T xt d td F x d dtx d dt t x F d dtx d dt t x F d dt F         +         ∂ ∂ + ∂ ∂ +      ∂ ∂ +      ∂ ∂ + ∂ ∂ +      − •• • εεεεεεεεε ( ) ( ) dt x F xx F xx F xx F xx F dt d F xx T xx T t t xx T xx TT x x f              ∂ ∂         ∂ ∂ +      ∂ ∂         ∂ ∂ +             ∂ ∂       ∂ ∂ +      ∂ ∂       ∂ ∂ + ∂ ∂       −+ •••• ••••• ∫ εεεεεεεεε ε ε0 2 2 ( ) ( ) ft fff T x ff T xft tdFxdtxdtxFdtxdtxFdtFd d Jd J       +      +++      ++== •••• = • 22222 0 2 2 2 22 δδδε ε δ ε ( ) ( ) ft fff T x ff T xft tdFxdtxdtxFdtxdtxFdtF       +      +++      ++− •••• • 2222 22 δδδ ( ) ( ) ( ) ( ) ( ) ( ) ∫                     +      +      ++      −+ •••• ••••• ε ε δδδδδδδδδ ft t xx T xx T xx T xx T T x x dtxFxxFxxFxxFxxF dt d F 0 2 Therefore
  • 31.
    31 SOLO Calculus of Variations SecondVariation of the Functional J (continue – 4) ( ) ( ) fitxdttxxd iii ,0=+= • δ But we found that: ( ) ( ) ( ) ( ) ( ) fitxtdtxdttxdttxxd iiiiiiiii ,02 2222 =+++= •••• δδ ( ) −      +      −+      −+= •• • ft ff T x ff T xft tdFtdxxdFdtdtxxdFdtFJ 22222 2δ ( ) −      +      −+      −+− •• • 0 0 2 0 22 00 2 0 2 t T x T xt tdFtdxxdFdtdtxxdFdtF ( ) ( ) ( ) ( ) ( ) ( ) ∫                     +      +      ++      −+ •••• ••••• ε ε δδδδδδδδδ ft t xx T xx T xx T xx T T x x dtxFxxFxxFxxFxxF dt d F 0 2 Hence: and the final result is: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫                     +      +      ++      −+ +            −+++      −−             −+++      −= •••• •• •• ••••• •• •• ε ε δδδδδδδδδ δ f f t t xx T xx T xx T xx T T x x t T x T x T x T xt t f T x f T x ff T xf T xt dtxFxxFxxFxxFxxF dt d F tdxFFxdFdtxdFdtxFF tdxFFxdFdtxdFdtxFFJ 0 0 2 0 2 0 2 00 2 0 2222 2 2
  • 32.
    32 SOLO Calculus of Variations 4.Necessary Conditions for Extremum We found that: ( ) ( ) ( ) ( )JJJd d Jd d d dJ JJJ 3232 0 2 2 0 2 1 2 1 0: δδδεε ε ε ε εε εε Ο/++=Ο/++==−=∆ == For a Minimum Solution of the Functional J we must have: ΔJ ≥ 0 for any small dε (see Figure) ( )( )ε,txJ 0=ε ε Minimum of J as function of ε For a Maximum Solution of the Functional J we must have: ΔJ ≤ 0 for any small dε To prevent that the sign of dε to change the same of ΔJ the Necessary Condition for Extremum is 00 0 == = Jord d dJ δε ε ε This condition must be fulfilled for any admissible variation.
  • 33.
    33 SOLO Calculus of Variations NecessaryConditions for Extremum (continue – 1) Suppose that is an extremal solution with the fixed end points and .( )tx* ( )0 * 0 * , xtA ( )0 * 0 * , xtB Let choose first all the variation that passes through those points (see Figure ). ( )( )000 * , txtA ( )( )fff txtB ,* ( )tx*x t ( )ε,1 tx ( )ε,2 tx Variations Passing through Fixed End Points 0&0 0 22 0 ==== tdtddtdt ff ( ) ( ) ( ) ( ) 0&0 0 22 0 ==== txtxtxtx ff δδδδ ( ) ( ) ( ) ( ) 0&0 0 22 0 ==== txdtxdtxdtxd ff Therefore: ( ) ( ) ( ) ( ) ( ) ( ) 0,,,, 0 =            +      = ∫ ••• • ft t T x T x dttxtxtxtFtxtxtxtFJ δδδ where: ( ) ( ) εε ε δ ε dtxtx 0 , =∂ ∂ = ( ) ( ) εε ε δ ε dtxtx 0 , = •• ∂ ∂ =
  • 34.
    34 SOLO Calculus of Variations NecessaryConditions for Extremum (continue – 2) Transformation of the First Variation δ J by integration by parts (a) First way: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫       −      =∫       •••• ••• f f f t t T x t t T x t t T x dttxtxtxtF dt d txtxtxtFdttxtxtxtF 0 0 0 ,,,,,, δδδ Because we have( ) ( ) 00 == txtx f δδ ( ) ( ) ( ) ( ) ( ) 0,,,, 0 =            −      = ∫ •• • ft t T x x dttxtxtxtF dt d txtxtFJ δδ δ J must be zero for all admissible variations , where and dt0 = dtf = 0. ( )txδ ( ) ( ) 00 == txtx f δδ Note: •••••••       +      +      =      • xxxtGxxxtGxxtGxxtG dt d T x T xt ,,,,,,,, therefore integration by parts assumes however, that not only , but also exists and is continuous in (t0, tf ). • x •• x End Note Return to Table of Content
  • 35.
    35 SOLO Calculus of Variations NecessaryConditions for Extremum (continue – 3) 4.1 The First Fundamental Lemma of the Calculus of Variations (Du Bois-Reymond-1879) If M(t) is a continuous function of t in (t0, tf ) and if ( ) ( ) 0 0 =∫ ft t tdtxtM δ for all functions that vanish at t0 and tf and which admit a continuous derivative in (t0, tf ), then ( )txδ ( ) fttttM ≤≤= 00 Paul David Gustav Du Bois-Reymond (1831-1889) Proof: Suppose M (t) ≠ 0, say greater than zero at a point t1 on the interval (t0, tf ). Because M(t) is continuous exists a neighbor of t1 say (t1-ζ, t1+ζ) in which we chose ( ) ( ) ( ) ( )   +−∈−−+− +−∉ = ζζζζ ζζ δ 11 2 1 2 1 11 , ,0 ttttttt ttt x kk ( )tM tft0t 1t ζ+1tζ−1t xδ admits a continuous derivative in (t0, tf ) and vanishes at t0 and t1 and nevertheless makes ( ) ( ) 0 0 >∫ ft t tdtxtM δ contrary to the hypothesis; therefore M (t) ≠ 0 is impossible for al t0≤ t ≤tf. q.e.d. Return to Table of Content
  • 36.
    36 SOLO Calculus of Variations NecessaryConditions for Extremum (continue – 4) 4.2 The Euler-Lagrange Equation The Necessary Condition for an extremal is δ J = 0, where ( ) ( ) ( ) ( ) ( ) 0,,,, 0 =            −      = ∫ •• • ft t T x x dttxtxtxtF dt d txtxtFJ δδ For all variations satisfying and dt0 = dtf = 0.( )txδ ( ) ( ) 00 == txtx f δδ ( ) ( ) ( ) ( )( ) f T n ttttxtxtxtx ≤≤= 021 ,,, δδδδ  By choosing for i=1,…,n δ xi(t) ≠ 0 and δ xj(t) = 0 for all j ≠ i and using the First Fundamental Lemma, we can see that δ J= 0 for all admissible variations if and only if( )txδ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )           =      −      =      −      =      −      •• •• •• • • • 0,,,, 0,,,, 0,,,, 2 2 1 1 txtxtF dt d txtxtF txtxtF dt d txtxtF txtxtF dt d txtxtF n n x x x x x x 
  • 37.
    37 SOLO Calculus of Variations NecessaryConditions for Extremum (continue – 5) The Euler-Lagrange Equation (continue – 1) As a matrix equation ( ) ( ) ( ) ( ) 0,,,, =      −      •• • txtxtF dt d txtxtF x x Euler-Lagrange Equation It was discovered by Euler in 1744. Later in 1760 Lagrange discussed this equation and introduced the notation δ and the notion of Variation. Leonhard Euler (1707-1783) Joseph-Louis Lagrange (1736-1813) By developing this equation we get: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,,,,,,,, =      −      +      +      ••••••• •••• txtxtFtxtxtFtxtxtxtFtxtxtxtF x txxxxx This is a Nonhomogeneous, Second Order, Differential Equation. ( ) ( )      • •• txtxtF xx ,,If is nonsingular on t0 ≤ t ≤tf, then ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )             −      +                  −= •••• − ••• •••• txtxtFtxtxtFtxtxtxtFtxtxtFtx x txxxxx ,,,,,,,, 1 The existence of is achieved if the matrix has an inverse for all t in (t0, tf ). If this condition is satisfied we have a Regular Problem. The problem is well defined if 2n boundary conditions are defined (see Appendix 3 ). ( )tx •• ( ) ( )      • •• txtxtF xx ,,
  • 38.
    38 SOLO Calculus of Variations NecessaryConditions for Extremum (continue – 6) The Euler-Lagrange Equation (continue – 2) Leonhard Euler (1707-1783) Joseph-Louis Lagrange (1736-1813) Therefore the general solutions of the Euler-Lagrange Equations are therefore two vector parameters solutions( ) ( )T n T n βββααα ,,,,, 11  == ( ) ( )βαϕ ,,ttx = and those parameters are defined by the 2n boundary conditions.
  • 39.
    39 SOLO Calculus of Variations NecessaryConditions for Extremum (continue – 7) (b) Second way: Du Bois-Reymond and Hilbert integrated the first, instead of the second, term of δ J by parts ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫ ∫ •••• •••               −      +            =             +      = • • f f f f f t t Tt t x x t t t t T x t t T x T x dttxdtxxtFxxtFtxdtxxtF dttxtxtxtFtxtxtxtFJ 0 0 0 0 0 ,,,,,, ,,,, δδ δδδ Paul David Gustav Du Bois-Reymond (1831-1889) David Hilbert (1862 – 1943) Because , we have:( ) ( ) 00 == txtx f δδ ( ) 0,,,, 0 0 =               −      = ∫ ∫ ••• • f ft t Tt t x x dttxdtxxtFxxtFJ δδ δ J must be zero for all admissible variations , such that .( )txδ ( ) ( ) 00 == txtx f δδ Return to Table of Content
  • 40.
    40 SOLO Calculus of Variations NecessaryConditions for Extremum (continue – 8) 4.3 The Second Fundamental Lemma of the Calculus of Variations (Du Bois-Reymond-1879) Paul David Gustav Du Bois-Reymond (1831-1889) Proof: q.e.d. If N(t) is a continuous function of t in (t0, tf ) and if for all functions of classes C(1) that vanish at t0 and tf then N (t) must be constant in (t0, tf ). ( )txδ ( ) ( ) 0 0 =∫ •ft t dttxtN δ Let subtract from the previous equation the identity: ( ) ( ) ( )[ ] 00 0 =−=∫ • txtxCdttxC f t t f δδδ where C is a constant. ( )[ ] ( ) 0 0 =∫ − •ft t dttxCtN δ From all the possible variations let choose the following particular variation: ( ) ( )[ ] 0>−= • εεδ CtNtx For this variation we must have: ( )[ ] ( ) ( )[ ] 0 00 2 =∫ −=∫ − • ff t t t t dtCtNdttxCtN δ This is possible only if N (t) = C. Therefore N (t) = C is a necessary condition. The sufficiency condition is proven by substituting N (t) = C in the original equation.
  • 41.
    41 SOLO Calculus of Variations NecessaryConditions for Extremum (continue – 9) The Second Fundamental Lemma of the Calculus of Variations (Du Bois-Reymond-1879) (continue – 1) Let apply the Second Fundamental Lemma of the Calculus of Variations to the equation: ( ) 0,,,, 0 0 =               −      = ∫ ∫ ••• • f ft t Tt t x x dttxdtxxtFxxtFJ δδ ( ) ( ) ( ) ( ) f T n ttttxtxtxtx ≤≤      = •••• 021 ,,, δδδδ where We obtain the following form of the Euler-Lagrange Equation: ∫       +=      •• • ft t x x dtxxtFCxxtF 0 ,,,, From this equation we can see that every solution of our problem with continuous first derivative – not only those admitting a second derivative – must satisfy the Euler-Lagrange Equation; i.e. the existence of is not necessary.( )tx •• Return to Table of Content
  • 42.
    42 SOLO Calculus of Variations 4.4Special Cases F doesn’t depend explicitly on the free variable t ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) xxxF td d xxF xxxF td d xxxFxxxFxxxFxxxFxxF td d T xx T x T x T x T x T x           −=       −−+=− ,, ,,,,,, For an extremal the Euler-Lagrange equation applies, and we have ( ) ( ) ( ) ( ) 0,, =      −      •• • txtxF dt d txtxF x x ( ) ( )[ ] 0,, =− xxxFxxF td d T x   Therefore ( ) ( ) constCxxxFxxF T x ==−   ,, Let perform the following:
  • 43.
    43 SOLO Calculus of Variations SpecialCases (continue – 1) F is not an explicit function of x In this case the Euler-Lagrange equation is: ( ) 0, =      • • txtF dt d x that can be integrated to give ( ) constCtxtF x ==      • • , F is not an explicit function of x In this case the Euler-Lagrange equation is: ( )( ) 0, =txtFx ( )( ) ( ) ( ) 0,..,0,det =∀≠ xtFtsxttxtF xxIf we can find that satisfies this equation.( )txx = According to Implicit Function Theorem this solution is unique..
  • 44.
    44 SOLO Calculus of Variations SpecialCases (continue – 2) F is an exact differential ( ) ( )( ) ( ) ( ) ( ) xxtVxtVxtV td d txtxtF T xt  ,,,,, +=≡ If this is true than ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )00 , , , , ,,,,, 000000 xtVxtVdtxtV td d dttxtxtF ff xtP xtP xtP xtP ffffff −=∫=∫  therefore the functional is independent on the integration path. Let find what conditions F must satisfy in order to be an exact differential. Let compute ( ) ( )( ) ( ) ( ) xxtVxtVtxtxtF xxxtx  ,,,, += ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) xxtVxtVtxtxtF td d xtVtxtxtF T txtxxxx   ,,,,,,, +=→= From those relations we can see that the condition that F is an exact differential if and only if the Euler-Lagrange equation is an identity. ( ) ( )( ) ( ) ( )( )txtxtFtxtxtF td d xx  ,,,, ≡ Return to Table of Content
  • 45.
    45 SOLO Calculus of Variations 4.5Example 1: Brachistocrone A particle slides on a frictionless wire between two fixed points A(0,0) and B (xfc, yfc) in a constant gravity field g. The curve such that the particle takes the least time to go from A to B is called brachistochrone. x y V ( )tγ ( )fcfc yxB ,fcx fcy N  ( )0,0A ∫∫       = +       + = cfcf xx xd xd yd yxFxd ygV xd yd J 00 2 0 2 ,, 2 1 We derived the cost function: xd yd y ygV y xd yd yxF = + + =      : 2 1 :,, 2 0 2   wher e F doesn’t depend explicitly on the free variable x, therefore if we replace and we use the result obtained for F not depending explicitly on x, we obtain ( ) ( )xtyx ,, → ( ) ( ) const ygVy y ygV y yyFyyyF y == ++ − + + =− α 212 1 ,, 2 0 2 2 2 0 2     const ygVy == ++ α 21 1 2 0 2  or
  • 46.
    46 SOLO Calculus of Variations Example1: Brachistocrone (continue – 1) x y V ( )tγ ( )fcfc yxB ,fcx fcy N  ( )0,0A const ygVy == ++ α 21 1 2 0 2  Let define a parameter τ such that τcos 1 1 2 =       + xd yd and const ygV ygV xd yd == + = +      + α τ 2 cos 21 1 2 02 0 2 From which ( ) ( )ττ αα τ 2cos12cos1 4 1 2 cos 2 22 22 0 +=+==+ r ggg V y Tacking the derivative of this equation with respect to τ we obtain τ τ 2sin2r d yd −=
  • 47.
    47 SOLO Calculus of Variations Example1: Brachistocrone (continue – 2) τ ττ τ ττ 2 22 2 2 cos /1 1 =       +            =       + d yd d xd d xd d xd d yd ( ) ( )ττ ττ τ τττ τ τττ ττ 2sin2 2cos12cos4 cossin16sin cos2sin4cos 0 2 4222 2 2222 22 +±=→ +±=±=→ =      → +      =      → rxx rr d xd r d xd r d xd d xd Let change variables to 2τ = θ – π, to get ( ) ( )θ θθ cos1 2 sin 2 0 0 −=+ −+= r g V y rxx θsinr θcosr θr x y 0x 0V g V 2 2 0 r r A B ),( yx θ We obtain the equation of a cycloid generated by a circle of radius r rolling upon the horizontal line and starting at the point g V y 2 2 0 −=         −− g V x 2 , 2 0 0
  • 48.
    48 HISTORY OF CALCULUSOF VARIATIONS The brachistochrone problem ( ) ( )     −−= −+= g V ry rxx 2 cos1 sin 2 0 0 θ θθ Cycloid Equation ∫∫∫∫       = +       + === cfcfcf xxxt xd xd yd yxFxd ygV xd yd V sd tdJ 00 2 0 2 00 ,, 2 1 Minimization Problem Solution of the Brachistochrone Problem: SOLO Johann Bernoulli 1667-1748
  • 49.
    49 SOLO Calculus of Variations Example2: Minimum Surface of Revolution x y ( )bybB , ( )ayaA , ( ) ( )22 ydxdsd += y ( )∫       += b a xd xd yd xyJ 2 12π For this problem we derived the cost function: Given two points A (a,ya) and B (b, yb) a≠b in the plane. Find the curve that joints these two points with a continuous derivative, in such a way that the surface generated by the rotation of this curve about the x axis has the smallest possible area. We have ( ) ( ) ( ) ( ) xd yd xyxyxyyyxF =+= :12:,, 2  π F doesn’t depend explicitly on the free variable x, therefore we can apply the results for this special case, with ( ) ( )xtyx ,, → ( ) ( ) C y y yyyyyFyyyF y ππ 2 1 12,, 2 2 2 =         + −+=−     2 1 yCy +=or Separating variables, we obtain C xd C y C yd = −      1 2 1 2 −      = C y y
  • 50.
    50 SOLO Calculus of Variations Example2: Minimum Surface of Revolution (continue – 1) x y ( )bybB , ( )ayaA , ( ) ( )22 ydxdsd += y C xd C y C yd = −      1 2 Integration of this equation, gives         −      +=− 1ln 2 1 C y C y CCx from which 1exp 2 1 −      +=      − C y C y C Cx take the square 1exp211212122exp 1 222 1 −      − =−         −      +=−      +−      =      − C Cx C y C y C y C y C y C y C y C Cx From this equation we can compute 2 expexp 11       − −+      − = C Cx C Cx C y ( )       − = C Cx Cxy 1 coshor The solution is a curve called a catenary (catena = chain in Latin) and the surface of revolution which is generated is called a catenoid of revolution.
  • 51.
    51 SOLO Example 3: GeometricalOptics and Fermat Principle We have: constS = constdSS =+ sˆ ∫ 2 1 P P dsn 1 P 2 P ( ) ( ) ( )∫∫∫∫ =      +      +=== 2 1 2 1 2 1 ,,,, 1 1,, 1 ,, 1 0 22 00 P P P P P P xdzyzyxF c xd xd zd xd yd zyxn c dszyxn c tdJ  Let find the stationarity conditions of the Optical Path using the Calculus of Variations ( ) ( ) ( ) xd xd zd xd yd zdydxdds 22 222 1       +      +=++= Define: xd zd z xd yd y ==  &: ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += Calculus of Variations
  • 52.
    52 SOLO Necessary Conditions forStationarity (Euler-Lagrange Equations) ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += 0= ∂ ∂ −      ∂ ∂ y F y F dx d  ( ) [ ] 2/122 1 ,, zy yzyxn y F    ++ = ∂ ∂ [ ] ( ) y zyxn zy y F ∂ ∂ ++= ∂ ∂ ,, 1 2/122  ( ) [ ] [ ] 01 1 ,, 2/122 2/122 = ∂ ∂ ++−         ++ y n zy zy yzyxn xd d    0= ∂ ∂ −      ∂ ∂ z F z F dx d  [ ] [ ] 0 11 2/1222/122 = ∂ ∂ −         ++++ y n zy yn xdzy d    Calculus of Variations Example 3: Geometrical Optics and Fermat Principle (continue – 1)
  • 53.
    53 SOLO Necessary Conditions forStationarity (continue - 1) We have [ ] 0 1 2/122 = ∂ ∂ −         ++ y n zy yn sd d   y n sd yd n sd d ∂ ∂ =      In the same way [ ] 0 1 2/122 = ∂ ∂ −         ++ z n zy zn sd d   z n sd zd n sd d ∂ ∂ =      Calculus of Variations Example 3: Geometrical Optics and Fermat Principle (continue –2)
  • 54.
    54 SOLO Necessary Conditions forStationarity (continue - 2) Using ( ) ( ) ( ) xd xd zd xd yd zdydxdds 22 222 1       +      +=++= we obtain 1 222 =      +      +      sd zd sd yd sd xd Differentiate this equation with respect to s and multiply by n      sd d 0=            +            +            sd zd sd d n sd zd sd yd sd d n sd yd sd xd sd d n sd xd sd nd sd zd sd nd sd yd sd nd sd xd sd nd =      +      +      222 sd nd and sd nd sd zd n sd d sd zd sd yd n sd d sd yd sd xd n sd d sd xd =            +            +            add those two equations Calculus of Variations Example 3: Geometrical Optics and Fermat Principle (continue – 3)
  • 55.
    55 SOLO Necessary Conditions forStationarity (continue - 3) sd nd sd zd n sd d sd zd sd yd n sd d sd yd sd xd n sd d sd xd =            +            +            Multiply this by and use the fact that to obtain xd sd cd ad cd bd bd ad = xd nd sd zd n sd d xd zd sd yd n sd d xd yd sd xd n sd d =            +            +      Substitute and in this equation to obtain y n sd yd n sd d ∂ ∂ =      z n sd zd n sd d ∂ ∂ =      xd zd z n xd yd y n xd nd sd xd n sd d ∂ ∂ − ∂ ∂ −=      Since n is a function of x, y, z x n xd zd z n xd yd y n xd nd zd z n yd y n xd x n nd ∂ ∂ = ∂ ∂ − ∂ ∂ −→ ∂ ∂ + ∂ ∂ + ∂ ∂ = and the previous equation becomes x n sd xd n sd d ∂ ∂ =      Calculus of Variations Example 3: Geometrical Optics and Fermat Principle (continue – 4)
  • 56.
    56 SOLO Necessary Conditions forStationarity (continue - 4) We obtained the Euler-Lagrange Equations: x n sd xd n sd d ∂ ∂ =      y n sd yd n sd d ∂ ∂ =      z n sd zd n sd d ∂ ∂ =      k sd zd j sd yd i sd xd sd rd kzjyixr ˆˆˆ ˆˆˆ ++= ++=   Define the unit vectors in the x, y, z directionskji ˆ,ˆ,ˆ The Euler-Lagrange Equations can be written as: n sd rd n sd d ∇=       This is the Eikonal Equation from Geometrical Optics. Calculus of Variations Example 3: Geometrical Optics and Fermat Principle (continue – 5) Return to Table of Content
  • 57.
    57 SOLO Calculus of Variations 5.Boundary Conditions Until now we considered only the variations passing through the end points and . But those are not all the admissible variations. If or are not specified then if we consider all admissible variations (see Figure), then δ J will be given by: ( )* 0 * 0 * , xtA ( )*** , ff xtB ( )00 , xtA ( )ff xtB , ( )txδ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫             +      +      −      = ••••• • ft t T x xffff dttxtxtxtFtxtxtxtFdttxtxtFdttxtxtFJ 0 ,,,,,,,, 0000 δδδ ( )( )000 , txtA ( )( )fff txtB , ( )tx*x t ( )ε,1 tx ( )ε,2 tx Variations that Satisfy the Boundary Conditions Integrating by parts the second term of the integral as before we have: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫             −      +             +      −            +      = •• •••• • •• f f t t T x x t T x t T x dttxtxtxtF dt d txtxtF xtxtxtFdttxtxtFxtxtxtFdttxtxtFJ 0 0 ,,,, ,,,,,,,, δ δδδ
  • 58.
    58 SOLO Calculus of Variations BoundaryConditions (continue – 1) ( )( )000 , txtA ( )( )fff txtB , ( )tx*x t ( )ε,1 tx ( )ε,2 tx Variations that Satisfy the Boundary Conditions ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫             −      +             +      −            +      = •• •••• • •• f f t t T x x t T x t T x dttxtxtxtF dt d txtxtF xtxtxtFdttxtxtFxtxtxtFdttxtxtFJ 0 0 ,,,, ,,,,,,,, δ δδδ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫             −      +             +            −      −             +            −      = •• •••• •••• • •• •• f f t t T x x t T x T x t T x T x dttxtxtxtF dt d txtxtF xdtxtxtFdtxtxtxtFtxtxtF xdtxtxtFdtxtxtxtFtxtxtFJ 0 0 ,,,, ,,,,,, ,,,,,, δ δ But ( ) ( ) ( ) iiiii dttxtxdtx • −= δ Therefore:
  • 59.
    59 SOLO Calculus of Variations BoundaryConditions (continue – 2) We found before that the necessary conditions such that δ J is zero for those admissible solutions passing through the points and are the Euler-Lagrange Equation:( )* 0 * 0 * , xtA ( )*** , ff xtB ( ) ( ) ( ) ( ) 0,,,, =      −      •• • txtxtF dt d txtxtF x x For other admissible variations we shall need to add the additional necessary conditions, such that δ J is zero, called Transversality Conditions Equations: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) fitxdtxtxtFdttxtxtxtFtxtxtF iiii x iiiii T x iii ,00,,,,,, ==      +            −      •••• •• (a) Suppose that the following relation defines the boundary: ( ) ( ) ( ) ( ) ( ) fidttdtt dt d txdttx iitiiiii ,0=Ψ=Ψ=→Ψ= then the Transversality Conditions Equations are: ( ) ( ) ( ) ( ) ( ) ( ) fitxttxtxtFtxtxtF iitiii T x iii ,00,,,, ==      −Ψ      +      ••• •
  • 60.
    60 SOLO Calculus of Variations BoundaryConditions (continue – 3) Geometric Interpretation of the Transversality Conditions Let plot as a function of . The hyper-plane tangent at( ) ( )      = • txtxtF ,,η • = xξ ( ) ( ) ( )      == •• iiiiii txtxtFtx ,,,ηξ is given by ( ) ( ) ( ) ( ) ( )      +      −      = ••• iiiiiii T x txtxtFtxtxtxtF ,,,, ξη  ( ) ( ) ( ) ( ) ( ) ( ) fitxttxtxtFtxtxtF iitiii T x iii ,00,,,, ==      −Ψ      +      ••• • We can see that for η = 0 the last equation is identical to the Transversality Conditions Equation. Geometric Representation of the Transversality Conditions ( ) ( ) ( ) ( ) ( ) fidttdtt dt d txdttx iitiiiii ,0=Ψ=Ψ=→Ψ=
  • 61.
    61 SOLO Calculus of Variations BoundaryConditions (continue – 4) Transversality Conditions Suppose that ti and are not defined and is not a function of ti, then dti and are independent differentials and therefore both coefficients of dti and must be zero. ix ix ixd ixd ( ) ( ) ( ) ( ) ( ) 0,,,, =      −      ••• • iiii T x iii txtxtxtFtxtxtF ( ) ( ) 0,, =      • • iii x txtxtF Or, by using the second equation to simplify the first we get: ( ) ( ) ( ) ( ) fi txtxtF txtxtF iii x iii ,0 0,, 0,, =        =      =      • • • Those Equations are called Natural Boundary Conditions because they arise naturally when the original problem doesn’t specify boundary conditions.
  • 62.
    62 SOLO Calculus of Variations BoundaryConditions (continue – 5) Example: Transversality Conditions for Geometrical Optics and Fermat’s Principle ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += For the Geometrical Optics we obtained: Assume that the initial and final boundaries are defined by the surfaces A (x0, y0, z0) and B (xf, yf, zf) respectively. The transversality conditions at the boundaries i=0,f are defined by ( ) ( ) ( )[ ] ( ) ( ) 0,,,,,,,, ,,,,,,,,,,,, =++ −− iziy izy dzzyzyxFdyzyzyxF dxzyzyxFzzyzyxFyzyzyxF     ( ) [ ] [ ] [ ] [ ] ( ) sd xd zyxn zy n zy zn z zy yn yzynFzFyzyzyxF zy ,, 1 11 1,,,, 2/122 2/1222/122 2/122 = ++ = ++ − ++ −++=−−         ( ) [ ] ( ) ( ) [ ] ( ) sd zd zyxn zy zzyxn z F F sd yd zyxn zy yzyxn y F F z y ,, 1 ,, ,, 1 ,, 2/122 2/122 = ++ = ∂ ∂ = = ++ = ∂ ∂ =         For are tangent to the boundary surfaces A (x0, y0, z0) and B (xf, yf, zf).fird i ,0=  From Transversality Conditions we can see that the rays are normal (transversal) to the boundary surfaces (see Figure). Transversality Conditions Return to Table of Content
  • 63.
    63 SOLO Calculus of Variations 6.Corner Conditions In the development of the Euler-Lagrange Equation we assumed that not only is continuous, but also . However, there are a number of problems, for which this assumption is not true, for example problems of reflection or refraction. ( )tx ( )tx • We define such problems as follows: Find the curve that passes through the boundary points (given or not) and and extremizes the functional . This curve should reach the point after having been reflected by a given function (see Figure). ( )tx ( )00 , xtA ( )ff xtB , ( )[ ] ( ) ( )∫       = •ft t dttxtxtFtxJ 0 ,, ( )ff xtB , ( )tx Ψ= corner point ( )( )000 , txtA ( )( )fff txtB , ( )tx*x tct ( )tΨ The Corner Point of the Trajectories
  • 64.
    64 SOLO Calculus of Variations CornerConditions (continue – 1) corner point ( )( )000 , txtA ( )( )fff txtB , ( )tx*x tct ( )tΨ The Corner Point of the Trajectories Solution: Let define by tc the unknown time when the extremal is reflected. Then we can express the functional J in the form: ( )tx • ( )[ ] ( ) ( ) ( ) ( )∫∫       +      = •• f c c t t t t dttxtxtFdttxtxtFtxJ ,,,, 0 We suppose that is continuous in each of the intervals (t0, tc-), (tc+, tf). Then for both intervals we have: ( )tx • (1) The Euler-Lagrange Equation is: ( ) ( ) ( ) ( ) cf x x ttttttxtxtF dt d txtxtF ≠≤≤=      −      •• • 00,,,, ( )tx •• if exists, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ + • − •       +=            +=      •• •• f c c t t x x t t x x dttxtxtFCtxtxtF dttxtxtFCtxtxtF 0 0 0 ,,,, ,,,, ( )tx •• if doesn’t exist
  • 65.
    65 SOLO Calculus of Variations CornerConditions (continue – 2) corner point ( )( )000 , txtA ( )( )fff txtB , ( )tx*x tct ( )tΨ The Corner Point of the Trajectories Solution (continue – 1): (2) The Transversality Conditions at the initial (0) and final (f) points are: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) fitxdtxtxtFdttxtxtxtFtxtxtF iiii T x iiiii T x iii ,00,,,,,, ==      +            −      •••• •• Then: ( ) ( ) 00000 0000 =         +      −−         +      −= ++ • −− • + • + • − • − • c t T x c t T x c t T x c t T x txdFdtxFFtxdFdtxFFJ cccc δ But tc- = tc+ = tc and , thereforeccc xdxdxd == +− 00 ( ) 0 0000 =      −+               −−      −= + • − • + • − • •• c t T xt T x c t T x t T x txdFFdtxFFxFFJ cccc δ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,,,, ,,,, ,,,, 00 000 000 =            −      +          +      −          −      + • − • + • + • + • − • − • − • •• • • cccc x ccc x ccccc x ccc cccc x ccc txdtxtxtFtxtxtF dttxtxtxtFtxtxtF txtxtxtFtxtxtF The necessary conditions for extremal at the corners are: Those are the Weierstrass-Erdmann Corner Conditions. Those equations were developed independently by Weierstrass and Erdmann in 1877. Karl Theodor Wilhelm Weierstrass 1815-1897
  • 66.
    66 SOLO Calculus of Variations CornerConditions (continue – 3) corner point ( )( )000 , txtA ( )( )fff txtB , ( )tx*x tct ( )tΨ The Corner Point of the Trajectories Solution (continue – 2): (a) If they are a priori conditions at the corner like: ( ) ( ) ( ) ( ) ( ) cctccccc dttdtt dt d txdttx Ψ=Ψ=→Ψ= then the necessary conditions at the corner are: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )      −Ψ      +            −Ψ      +      + • + • + • − • − • − • • • 000 000 ,,,, ,,,, cctccc T x ccc cctccc T x ccc txttxtxtFtxtxtF txttxtxtFtxtxtF
  • 67.
    67 SOLO Calculus of Variations CornerConditions (continue – 4) Solution (continue – 3): (b) If they are not a priory conditions at the corner; i.e. the function is not a priori defined then dtc and are independent variables and ( ) ( )cc ttx Ψ= cxd ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )      =            −      =       −      + • − • + • + • + • − • − • − • •• • • 00 000 000 ,,,, ,,,, ,,,, ccc x ccc x cccc T x ccc cccc T x ccc txtxtFtxtxtF txtxtxtFtxtxtF txtxtxtFtxtxtF corner point ( )( )000 , txtA ( )( )fff txtB , ( )tx*x tct ( )tΨ The Corner Point of the Trajectories
  • 68.
    68 SOLO Calculus of Variations CornerConditions (continue – 5) Geometric Interpretation of the Corner Conditions Since the Corner Conditions where derived from the Transversality Conditions, we have a similar geometrical interpretation. Let plot as a function of .( ) ( )      = • txtxtF ,,η • = xξ Since the hyper-plane tangent at is given by( ) ( )+− = cc txtx ( ) ( ) ( )      == − • −− • − cccccc txtxtFtx ,,,ηξ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )− • − • − • − • − • − • − •       −      +      =       +      −      = cccc T xcccccc T x ccccccc T x txtxtxtFtxtxtFtxtxtF txtxtFtxtxtxtF ,,,,,, ,,,,   ξ ξη The hyper-plane tangent at is given by( ) ( ) ( )      == + • ++ • + cccccc txtxtFtx ,,,ηξ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )+ • + • + • + • + • + • + •       −       +      =       +      −      = cccc T x cccccc T x ccccccc T x txtxtxtF txtxtFtxtxtF txtxtFtxtxtxtF ,, ,,,, ,,,,    ξ ξη But according to the Corner Conditions the two tangent hyper-planes are the same (see Figure )
  • 69.
    SOLO Calculus of Variations CornerConditions (continue – 6) Example: Corner Conditions for Geometrical Optics and Fermat’s Principle ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += For the Geometrical Optics we obtained: Let examine the following two cases: 1. The optical path passes between two regions with different refractive indexes n1 to n2 (see Figure) In region (1) we have: In region (2) we have: ( ) ( ) 22 11 1,,,,,, zyzyxnzyzyxF  ++= ( ) ( ) 22 22 1,,,,,, zyzyxnzyzyxF  ++= ( ) ( ) ( )[ ]{ ( ) ( ) ( )[ ]} ( ) ( )[ ] ( ) ( )[ ] 0,,,,,,,, ,,,,,,,, ,,,,,,,,,,,, ,,,,,,,,,,,, 222111 222111 22222222222 11111111111 =−+ −+ −−− −− dzzyzyxFzyzyxF dyzyzyxFzyzyxF dxzyzyxFzzyzyxFyzyzyxF zyzyxFzzyzyxFyzyzyxF zz yy zy zy         The Weierstrass-Erdmann necessary condition at the boundary between the two regions is where dx, dy, dz are on the boundary between the two regions.
  • 70.
    SOLO Calculus of Variations CornerConditions (continue – 7) Example: Corner Conditions for Geometrical Optics and Fermat’s Principle (continue – 1) ( ) [ ] [ ] [ ] [ ] ( ) sd xd zyxn zy n zy zn z zy yn yzynFzFyzyzyxF zy ,, 1 11 1,,,, 2/122 2/1222/122 2/122 = ++ = ++ − ++ −++=−−         ( ) [ ] ( ) ( ) [ ] ( ) sd zd zyxn zy zzyxn z F F sd yd zyxn zy yzyxn y F F z y ,, 1 ,, ,, 1 ,, 2/122 2/122 = ++ = ∂ ∂ = = ++ = ∂ ∂ =         ( ) ( ) 0 21 21 =⋅        − rd sd rd n sd rd n rayray   where is on the boundary between the two regions andrd  ( ) ( ) sd rd s sd rd s rayray 2 :ˆ, 1 :ˆ 21  == are the unit vectors in the direction of propagation of the rays.
  • 71.
    SOLO Calculus of Variations CornerConditions (continue – 8) Example: Corner Conditions for Geometrical Optics and Fermat’s Principle (continue – 2) ( ) 0ˆˆ 2211 =⋅− rdsnsn  2211 ˆˆ snsn −Therefore is normal to .rd  Since can be in any direction on the boundary between the two regions (see Figure ) is parallel to the unit vector normal to the boundary surface, and we have rd  2211 ˆˆ snsn − 21 ˆ −n ( ) 0ˆˆˆ 221121 =−×− snsnn This the Snell’s Law of Geometrical Optics
  • 72.
    SOLO Calculus of Variations CornerConditions (continue – 9) Example: Corner Conditions for Geometrical Optics and Fermat’s Principle (continue – 3) 2. The optical path is reflected at the boundary. ( ) ( ) ( ) 0ˆˆ 21 21 =⋅−=⋅        − rdssrd sd rd sd rd rayray   n1 = n2 , we obtain i.e. is normal to , i.e. to the boundary where the reflection occurs. Also we can write 21 ˆˆ ss − rd  ( ) 0ˆˆˆ 2121 =−×− ssn ( ) ( ) ( ) 0ˆˆ 21 221121 =⋅−=⋅      − rdsnsnrd sd rd n sd rd n rayray   In this case, if we substitute in the equation Return to Table of Content
  • 73.
    SOLO Calculus of Variations 7Sufficient Conditions and Additional Necessary Conditions for a Weak Extremum We found that: ( ) ( ) ( ) ( )JJJd d Jd d d dJ JJJ 3232 0 2 2 0 2 1 2 1 0: δδδεε ε ε ε εε εε Ο/++=Ο/++==−=∆ == The Necessary Condition that Δ J ≥ 0 or ≤ 0 for all small dε is 00 0 == = Jor d dJ δ ε ε For Sufficient Conditions for a “Weak” Local Extremum we must add the following: ( ) ( )0000 2 0 2 2 ≤≥≤≥ = Jor d Jd δ ε ε for a minimum (maximum) solution. The expression for δ2 J is ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫                     +      +      ++      −+             −+++      −−             −+++      −= •••• •• •• ••••• •• •• ε ε δδδδδδδδδ δ f f t t xx T xx T xx T xx T T x x t T x T x T x T xt t f T x f T x ff T xf T xt dtxFxxFxxFxxFxxF dt d F tdxFFxdFdtxdFdtxFF tdxFFxdFdtxdFdtxFFJ 0 0 2 0 2 0 2 00 2 0 2222 2 2
  • 74.
    SOLO Calculus of Variations SufficientConditions and Additional Necessary Conditions for a Weak Extremum (continue -1) Suppose first that the end points are fixed; i.e.: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 00 00 00 2 0 2 0 2 0 2 0 2 0 2 0 ==== ==== ==== ff ff ff txdtxdtxdtxd txtxtxtx tdtddtdt δδδδ In this case: ( ) ( ) ( ) ( ) ( ) ( ) ∫                     +      +      ++      −= •••• ••••• ε ε δδδδδδδδδδ ft t xx T xx T xx T xx T T x x dtxFxxFxxFxxFxxF dt d FJ 0 22 For an extremal solution the Euler-Lagrange Equation holds, therefore0=− • x x F dt d F ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫                               = ∫                     +      +      += • • •••• ••• • •••• ε ε ε ε δ δ δδ δδδδδδδδδ f f t t xxxx xx xxT T t t xx T xx T xx T xx T dt x x FF FF xx dtxFxxFxxFxxFxJ 0 0 2 We have the following properties of the derivatives T xxxx T xxxx T xxxx FFFFFF  ===
  • 75.
    SOLO Calculus of Variations SufficientConditions and Additional Necessary Conditions for a Weak Extremum (continue -2) Let define: TT xxxx T xxxx TT xxxx RFFRFFQPFFP ======== •• :,:,: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ++= +++=                             = • •== == ε ε ε ε ε ε δδδδδδ δδδδδδδδ δ δ δδδ f f f ii ii t t TTT t t TTTTT t t T T T xdxd tddt dtxPxxQxxPx dtxPxxQxxQxxPx dt x x RQ QP xxJ 0 0 0 2 2 2 00 00 2   Therefore: Return to Table of Content
  • 76.
    SOLO Calculus of Variations 8Legendre’s Necessary Conditions for a Weak Minimum (Maximum) – 1786 Adrien-Marie Legendre 1752-1833 Proof: Let start from the necessary condition of an Minimal Optimal Trajectory, that ( ) ( ) ( ) 02 0 2 2 00 00 2 ≥++== ∫ == == ε ε δδδδδδδ f ii ii t t TTT xdxd tddt dtxPxxQxxPxJ  (The same reasoning applies for a Maximal Optimal trajectory where it is required that δ2 J ≤ 0) Suppose that R is not positive definite and we have a constant vector such that at some point t = τ, on our curve. Since R (t) was assumed continuous, this inequality will hold over some sufficiently small interval [τ-h, τ+h] . We now define the function so that it vanishes outside and at the end points of the interval, it has all the necessary derivatives; is sufficiently small in absolute value in the interval, but performs fairly rapid oscillations. v 0<vRvT ( )txδ ( )          +<<      − − ≤<−      − + = elsewhere ht h t h th h t h tx 0 1 1 ττν τ ε ττν τ ε δ ( )          +<<− ≤<− =⇒ elsewhere ht h th h tx 0 ττν ε ττν ε δ  The matrix must be Positive (Negative) Definite along a Minimal (Maximal) Optimal Trajectory xxFR =
  • 77.
    SOLO Calculus of Variations Legendre’sNecessary Conditions for a Weak Minimum (Maximum) – 1786 (continue – 1) Adrien-Marie Legendre 1752-1833 Proof (continue – 1): ixδ t 1h−τ 2h−τ 2h+τ1h+τ hε          +<<      − − <<−      − + = elsewhere ht h t h th h t h x 0 1 1 ττν τ ε ττν τ ε δ          +<<− <<− = elsewhere ht h th h x 0 ττν ε ττν ε δ  t 1h−τ 2h−τ 2h+τ1h+τ h ε ixδ Since P (t) and Q (t) are continuous matrix functions in the interval t € [0, tf] we can find two positive numbers M1 and M2 such that: ( ) ( ) [ ]f TT ttMvtQvMvtPv ,0, 21 ∈∀<<
  • 78.
    SOLO Calculus of Variations Legendre’sNecessary Conditions for a Weak Minimum (Maximum) – 1786 (continue – 2) Proof (continue – 2): ixδ t 1h−τ 2h−τ 2h+τ1h+τ hε          +<<      − − <<−      − + = elsewhere ht h t h th h t h x 0 1 1 ττν τ ε ττν τ ε δ          +<<− <<− = elsewhere ht h th h x 0 ττν ε ττν ε δ  t 1h−τ 2h−τ 2h+τ1h+τ h ε ixδ ( ) ( ) [ ]f TT ttMvtQvMvtPv ,0, 21 ∈∀<< 2 2 0 1 0 1 2 2 0 0 22 211 11 00 Mhdt h t dt h t M dtQv h t dtQv h t dtxQxdtxQx h h h h TT t t T t t T ff ε ττ ε ν τ εν τ εδδδδ τ τ τ τ ≤             ∫       − −+∫       − +≤ ∫ ∫       − −+      − +=∫≤∫ + < − < − +   1 22 0 1 2 0 1 2 1 2 0 0 2 2 2 2 211 11 00 Mhdt h t dt h t Mh dtPv h t hdtPv h t hdtxPxdtxPx h h h h TT t t T t t T ff ε ττ ε ν τ εν τ εδδδδ τ τ τ τ ≤             ∫       − −+∫       − +≤ ∫ ∫       − −+      − +=∫≤∫ + < − < − +  We have ( ) ( ) ( ) ( )[ ]{ } ( )[ ] 10212 21 2 22 0 ≤≤−+= +−+−=∫=∫ + − λλτ ε ντλτλ ε ν ε δδ τ τ someforhRv h hhhRv h dttRv h dtxRx T T h h T t t T f 
  • 79.
    SOLO Calculus of Variations Legendre’sNecessary Conditions for a Weak Minimum (Maximum) – 1786 (continue – 3) ixδ t 1h−τ 2h−τ 2h+τ1h+τ hε          +<<      − − <<−      − + = elsewhere ht h t h th h t h x 0 1 1 ττν τ ε ττν τ ε δ          +<<− <<− = elsewhere ht h th h x 0 ττν ε ττν ε δ  t 1h−τ 2h−τ 2h+τ1h+τ h ε ixδ ( ) ( ) ( ) ( )[ ]{ } ( )[ ] 10212 21 2 22 0 ≤≤−+= +−+−=∫=∫ + − λλτ ε ντλτλ ε ν ε δδ τ τ someforhRv h hhhRv h dttRv h dtxRx T T h h T t t T f  Since by assumption and R (t) is a continuous matrix functions in the interval t € [0, tf] we can find a small h1 such that for all h ≤ h1 we have ( ) 0<vRvT τ ( )[ ] 021 2 <−≤−+ µνλτ hR Therefore 1 22 2 0 hhdtxRx ft t T ≤∀−≤∫ µεδδ  Using the previous results we can write ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 21 22 00 00 2 22 22 0000 2 2 hhMhMh dtxPxdtxQxdtxPxdtxPxxQxxPxJ ffff ii ii t t T t t T t t T t t TTT xdxd tddt ≤∀−+≤ ++≤++= ∫∫∫∫ == == µε δδδδδδδδδδδδδ ε ε ε ε ε ε ε ε  Since we can find a small h2 ≤ h1 such that for all h ≤ h1 ( ) 02 0 2 21 2 =−+ =h MhMh µ ( ) 2 2 21 222 022 hhMhMhJ ≤∀<−+≤ µεδ δ2 J turn out negative, which contradicts the before mentioned necessary condition for minimum; i.e. δ2 J ≥ 0. Therefore must be Positive Definite along the trajectory to have a minimum.xxFR = q.e.d.
  • 80.
    SOLO Calculus of Variations Legendre’sNecessary Conditions for a Weak Minimum (Maximum) – 1786 (continue – 4) Example: Geometrical Optics and Fermat’s Principle ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += For the Geometrical Optics we obtained: ( ) [ ] [ ] [ ] ( ) [ ] 2/322 2 2/322 2 2/1222/1222 2 1 1 111 ,, zy zn zy yn zy n zy yzyxn yy F        ++ + = ++ − ++ =         ++∂ ∂ = ∂ ∂ ( ) [ ] [ ] 2/3222/122 2 11 ,, zy zyn zy zzyxn yzy F      ++ −=         ++∂ ∂ = ∂∂ ∂ ( ) [ ] ( ) [ ] 2/322 2 2/1222 2 1 1 1 ,, zy yn zy zzyxn zz F      ++ + =         ++∂ ∂ = ∂ ∂ From those equations we obtain: ( ) [ ] ( ) ( )        +− −+ ++ = 2 2 2/322 '' 1'' 1 1 ,, yyx zyz zy zyxn F XX    Let use Sylvester Theorem to check the positiveness of ''XXF [ ] ( ) ( ) [ ] ( )( )[ ] [ ] 0 1 11 11'' 1 det 1 det 2/122 2222 2/3222 2 2/322'' > ++ =−++ ++ =         +− −+ ++ = zy n zyyz zy n yyx zyz zy n F XX      1 ( ) 01 2 >+ z2 We can see that according to Sylvester Theorem is Positive Definite.''XXF James Joseph Sylvester 1814-1897 Return to Table of Content
  • 81.
    SOLO Calculus of Variations 9.Jacobi’s Differential Equation (1837) and Conjugate Points Let start from the necessary condition of a Minimal Optimal Trajectory, that ( ) ( ) ( ) 02 0 2 2 00 00 2 ≥∫ ++= == == ε ε δδδδδδδ fii ii t t TTT xdxd tddt dtxRxxQxxPxJ  Define ( ) xRxxQxxPxxx TTT  δδδδδδδδ ++=Ω 2:, Therefore ( ) ( ) ( ) ( ) ∫Ω= == == ε ε δδδδ f ii ii t t xdxd tddt dtxxxJ 0 2 2 , 00 00 2  We have ( ) ( ) xRxQ x xx xQxP x xx T      δδ δ δδ δδ δ δδ 22 , 22 , += ∂ Ω∂ += ∂ Ω∂ We can see that ( ) ( ) ( ) x x xx x x xx xx TT     δ δ δδ δ δ δδ δδ       ∂ Ω∂ +      ∂ Ω∂ =Ω ,, ,2 Since ( ) ( ) ( ) ( ) ( ) ∫       ∂ Ω∂ −      ∂ Ω∂ =∫       ∂ Ω∂ = = f f f f t t Tt t T tx tx t t T T dtx x xx td d x x xx dtx x xx 0 0 0 0 ,,, 0 0 0 δ δ δδ δ δ δδ δ δ δδ δ δ           we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫       +−+= ∫       ∂ Ω∂ − ∂ Ω∂ = ∫               ∂ Ω∂ +      ∂ Ω∂ =∫ Ω= == == f f ffii ii t t TTTTT t t T t t TT t t xdxd tddt dtxRxQx dt d QxPx dtx x xx dt d x xx dtx x xx x x xx dtxxxJ 0 0 00 2 2 ,, 2 1 ,, 2 1 , 00 00 2 δδδδδ δ δ δδ δ δδ δ δ δδ δ δ δδ δδδδ ε ε       
  • 82.
    SOLO Calculus of Variations Jacobi’sDifferential Equation (1837) and Conjugate Points (continue – 1) we have ( ) ( ) ( )∫     +−+= ft t TTTTT dtxRxQx dt d QxPxxJ 0 2 δδδδδδδ  Gilbert Ames Bliss (1876 –1951) Gilbert A. Bliss (1876-1951) suggested to show that the minimum of δ2 J for all possible is non-negative. If this is true thanxδ ( ) ( ) 0min 22 ≥> xJxJ x δδδδ δ We obtain the following Secondary (Accessory) Variational Problem: ( ) ( ) ( ) ( ) ∫ Ω= == == ε εδδ δδδδ fii ii t tx xdxd tddtx dtxxxJ 0 2 2 ,minmin 00 00 2  The necessary conditions for a minimum are satisfaction of 1.Euler-Lagrange Equations and 2.Transversality 3.Weierstrass-Erdmann Corner Conditions Euler-Lagrange Equation for the Secondary Variational Problem: ( ) ( ) 0 ,, = ∂ Ω∂ − ∂ Ω∂ x xx dt d x xx   δ δδ δ δδ ( ) ( ) 0**** =+−+ xQxPxRxQ dt d T  δδδδor
  • 83.
    SOLO Calculus of Variations Jacobi’sDifferential Equation (1837) and Conjugate Points (continue – 2) ( ) ( ) 0**** =+−+ xQxPxRxQ dt d T  δδδδ Euler-Lagrange Equation for the Secondary Variational Problem: We can see that the extremal makes .*xδ ( ) 0* 00 00 2 2 2 == == = ii ii xdxd tddt xJ δδ Assume that det R ≠ 0 in t € [0,tf], i.e. R is non-singular and has an inverse, in this interval, then 0*** 11 =      −+      −++ −− xPQ dt d RxQQR dt d Rx TT δδδ  Jacobi’s Differential Equation Carl Gustav Jacob Jacobi 1804-1851 This is a Second Order Vectorial Homogeneous Linear Differential Equation with continuous coefficients.
  • 84.
    SOLO Calculus of Variations Jacobi’sDifferential Equation (1837) and Conjugate Points (continue – 3) 0*** 11 =      −+      −++ −− xPQ dt d RxQQR dt d Rx TT δδδ  Carl Gustav Jacob Jacobi 1804-1851 We apply the general existence and uniqueness theorems for linear differential equations and we obtain n solutions, for the initial conditions: U (t) is a nxn matrix and contains the n independent solutions of the Vectorial Homogeneous Linear Differential Equation: 011 2 2 =      −+      −++ −− uP td Qd R td ud QQ td Rd R td ud T T Where is a vector( ) ( ) ( ) ( )            = tu tu tu tu n  2 1 If are the n solutions of the Jacobi’s Vectorial Differential Equation, with initial conditions: ( ) ( ) ( )tututu n,,, 21  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1,,0,0:&0 0,,1,0:&0 0,,0,1:&0 0 2022 10101     === === === nnn etutu etutu etutu then define: ( ) ( ) ( ) ( )[ ]tutututU n,,,: 21 =
  • 85.
    SOLO Calculus of Variations Jacobi’sDifferential Equation (1837) and Conjugate Points (continue – 4) Carl Gustav Jacob Jacobi 1804-1851 Theorem For a weak minimum (maximum) it is necessary that: is Positive (Negative) Definite for t ϵ [0, tf ]( ) xxFtR = ( ) ( ) ( ) ( )[ ]tutututU n,,,: 21 =If is the n solution matrix of the Jacobi’s Homogeneous Linear Differential Equation: 011 2 2 =      −+      −++ −− uP td Qd R td ud QQ td Rd R td ud T T then U (t) must be nonsingular in t ϵ [0, tf ] . Proof Assume that for , then there exists n constants (c1, c2,…,cn) ≠ (0, 0,…,0), such that ( ) ( ) ( ) 0det,0 =→∈ cjcjfcj tUsingularistUtt ( ) ( ) ( ) 02211 =+++ cjnncjcj tuctuctuc  Define ( ) ( ) ( ) ( )     ≤< ≤≤+++ = fcj cjnn ttt ttttuctuctuc tx 0 :* 02211  δ We see that ( ) ( ) 0** 0 == cjtxtx δδ
  • 86.
    SOLO Calculus of Variations Jacobi’sDifferential Equation (1837) and Conjugate Points (continue – 5) Carl Gustav Jacob Jacobi 1804-1851 Proof (continue – 1) Let check the Weierstrass-Erdmann corner conditions at t - tcj ( ) ( ) +− == ∂ Ω∂ = ∂ Ω∂ cjcj tttt x xx x xx     δ δδ δ δδ ,, We have ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0222 , 0 ≠=+= ∂ Ω∂ −−−−−− − cjcjcjcjcjcj t txtRtxtRtxtQ x xx cj    δδδ δ δδ The expression is nonzero since R (tcj) is positive definite and (otherwise is uniquely defined by the terminal conditions ). ( ) 0≠−cjtxδ ( ) ( ) [ ]fttttxtx ,0 0∈∀== δδ ( ) ( ) [ ]fttttxtx ,0 0∈== δδ ( ) ( ) ( ) ( ) ( ) 022 , 00 =+= ∂ Ω∂ +−++ = +     cjcjcjcj tt txtRtxtQ x xx cj δδ δ δδ Therefore ( ) ( ) 0 ,, = ∂ Ω∂ ≠ ∂ Ω∂ +− == cjcj tttt x xx x xx     δ δδ δ δδ The Weierstrass-Erdmann corner conditions at t = tcj are not satisfied, hence is not the minimum of the second variation, therefore exists a variation such that . ( ) 0* 00 00 2 2 2 == == = ii ii xdxd tddt xJ δδ xδ ( ) 02 <xJ δδ q.e.d. Return to Table of Content
  • 87.
    SOLO Calculus of Variations Jacobi’sDifferential Equation (1837) and Conjugate Points (continue – 6) 9.1 Conjugate Points If U (t) is singular in t ϵ [0, tf ] we say that we have a conjugate point. In this case an optimal solution doesn’t exist. The geometric meaning of the conjugate points is as follows: The Second Order Euler-Lagrange Equation has a two-parameter family of solutions. Through any point here passes in general, a one-parameter family of extremals. Let denote this parameter by α and the solutions by . ( )0tx ( )α,tx The solution must satisfy the Euler-Lagrange Equation: ( ) ( ) ( ) ( ) 0,,,,,,,, =      −      •• • αααα txtxtF dt d txtxtF x x Let take the partial derivative with respect to t of previous equation ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )            +      −      +      = •••• αααααααααααα αααα ,,,,,,,,,,,,,,,,,,,,0 txtxtxtFtxtxtxtF dt d txtxtxtFtxtxtxtF xxxxxxxx   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )αααααα αααααααααααα αα αααα ,,,,,,,,,, ,,,,,,,,,,,,,,,,,,,, txtxtxtFtxtxtxtF dt d txtxtxtFtxtxtxtF dt d txtxtxtFtxtxtxtF xxxx xxxxxxxx           −            −       −            −      +      = •• ••••
  • 88.
    SOLO Calculus of Variations Jacobi’sDifferential Equation (1837) and Conjugate Points (continue – 7) Conjugate Points (continue – 1) Rearrange the previous equation ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,,,,,,,,, ,,,,,,,,,,,,, ,,,,, =             −            +             −      +            +       •• ••• • ααααα ααααααα ααα α α α txtxtxtFtxtxtF dt d txtxtxtFtxtxtFtxtxtF dt d txtxtxtF xxxx xxxxxx xx      Using TT xxxx T xxxx TT xxxx RFFRFFQPFFP ======== •• :&:&: we can write ( ) ( ) ( ) 0,,, =      −+      −++ ααα ααα txPQ td d txQQR td d txR TT  which is identical to the Jacobi Equation. Since is a solution of the Jacobi Equation if we have for any tcj ϵ [0, tf ] than we have ( )α,tx ( ) 0, =αα tx ( ) ( ) ( ) ( ) 0, 2211 =+++= cjnncjcjcj tuctuctuctx αα where were defined as the independent solutions of the Jacobi Equation. Therefore U (t) is singular if and according to Theorem the problem doesn’t have a minimum (maximum). ( ) ( ) ( ) ( )[ ]tutututU n,,,: 21 = ( ) [ ]fcjcj tttfortx ,0, 0∈=αα
  • 89.
    SOLO Calculus of Variations Jacobi’sDifferential Equation (1837) and Conjugate Points (continue – 8) Conjugate Points (continue – 2) Let tray to understand the meaning of .( ) [ ]fcjcj tttfort x ,0, 0∈= ∂ ∂ α α Suppose that two close solutions of the family intersect at tcj ϵ [0, tf ].( )α,tx ( ) ( )ααα ,, cjcj txdtx =+ In this case the family of solutions has an envelope (see Figure ) Extremal Trajectories Envelope of Extremal Trajectories G ( )0tx ftconjt 321 0t ( )ftx ( )conjtx Description of Conjugate Points We have ( ) ( ) ( ) [ ]fcj cjcj d cj tttfor d txdtx t x ,0 ,, lim, 0 0 ∈= −+ = ∂ ∂ → α ααα α α α If such a family has an envelope G, then a point of contact of an extremal with the envelope is called a conjugate point to on that extremal. In the Figure point is conjugate to between 0 and tf. ( )0tx ( )conjtx ( )0tx On a minimizing (maximizing) extremal curve connecting point and with nonsingular at each point of it, there can be no point conjugate to , between t0 and tf . ( )00 =tx ( )ftx xxFR = ( )conjtx ( )0tx
  • 90.
    SOLO Calculus of Variations Jacobi’sDifferential Equation (1837) and Conjugate Points (continue – 9) Examples of Conjugate Points: 1. The shortest path between two points A and B on the surface of a sphere is on that one great circle passing trough those two points. If the points are on an opposite diameter there are an infinity of great circles passing through, we don’t have one extremal and those two points are conjugate to each other. A' B' B A Poles as Conjugate Points on a Sphere 2. Rays from a point source refracted by a lens. The refracted rays forms an envelope called caustic. The point P’2 where the reflected ray touches the envelope is called a conjugate point. From the figure we can see that this point is reached by at least two rays with different optical paths. Field of rays passing through a lens and generating a caustic Return to Table of Content
  • 91.
    91 SOLO Calculus of Variations Jacobi’sDifferential Equation (1837) and Conjugate Points (continue – 10) 9.2 Fields Definition Let represent a one parameter family of solutions of the Euler-Lagrange equation in a simply connected region of . This family of solution defines a field if they are not conjugate points in this region. This means that through any point of this region passes one and only one curve of the family. ( )α,tx ( )xt, ( )xt, Field around a solution of Euler- Lagrange equation In Figure we can see a solution of the Euler-Lagrange equation passing trough and . A field of solutions are shown in a simply connected region that contains the solution. The conjugate point is shown outside this region. We say that the solution is Embedded in the Field. ( )0, =αtx ( )00 , xt ( )ff xt , ( )0, =αtx Return to Table of Content
  • 92.
    92 SOLO Calculus of Variations 10.Hilbert’s Invariant Integral Suppose that defines a field of solutions of Euler-Lagrange equation; i.e.( )α,tx ( ) ( ) ( ) ( ) 0,,,,,,,, =      −      •• • αααα txtxtF dt d txtxtF x x Let define any curve C in the field region, that starts at and ends at , and passes through a point with a slope (instead of Field slope ). Trough and passes also the unique extremal solution . ( )00 , xt ( )ff xt , ( )xt, ( )xtX , ( )xtx , ( )00 , xt ( )ff xt , ( )0, =αtx Hilbert’s Integral ( )( ) ( )( ) ( ) ( )( )[ ]∫ −− ft t T xC tdxtXxtxxtxxtFxtxxtF 0 ,,,,,,,,   is invariant on the path C as long as this curve remains in the field of the unique extremal solution. is the field slope and is the path C slope at the point of C.( )xtx , ( )xtX , ( )xt, David Hilbert (1862 – 1943)
  • 93.
    93 SOLO Calculus of Variations Hilbert’sInvariant Integral (continue – 1) Hilbert’s Invariant Integral David Hilbert (1862 – 1943) Proof Since on C we can write C td xd X = ( )( ) ( )( ) ( )[ ] ( )( )∫ +− ft t T x T xC xdxtxxtFtdxtxxtxxtFxtxxtF 0 ,,,,,,,,,,   ( ) ( )( ) ( )( ) ( ) ( ) ( )( )xtxxtFxtN xtxxtxxtFxtxxtFxtM x T x ,,,:, ,,,,,,,:,     = −=Define Rewrite ( ) ( )∫ + ft t T C xdxtNtdxtM 0 ,, This integral is path independent if there exists a function ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )xtVxtN xtVxtM xdxtNdtxtMxdxtVdtxtVxtVd x t TT xt ,, ,, ,,,,, = = +≡+= The following condition must be satisfied ( ) ( ) ( ) ( )xtN t xtM x xtV t xtV x xt ,,,, ∂ ∂ ≡ ∂ ∂ → ∂ ∂ ≡ ∂ ∂
  • 94.
    94 SOLO Calculus of Variations Hilbert’sInvariant Integral (continue – 2) Hilbert’s Invariant Integral David Hilbert (1862 – 1943) Proof (continue – 1) The following condition must be satisfied ( ) ( )xtN t xtM x ,, ∂ ∂ ≡ ∂ ∂ Let check that this is satisfied ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )xtxxtF x xtx xtx x xtx xtxxtFxtxxtxxtF xtxxtF x xtx xtxxtFxtM x x TT xxxx x T x ,,, , , , ,,,,,,, ,,, , ,,,,           ∂ ∂ − ∂ ∂ −− ∂ ∂ += ∂ ∂ ( ) ( )( ) ( )( ) ( ) t xtx xtxxtFxtxxtFxtN t xxtxt ∂ ∂ += ∂ ∂ , ,,,,,,,    Let compute ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( ) 0,,,,,,,,,,, ,, ,,, , , ,,,,,,,,,, , ,,,,,,,, =−++      ∂ ∂ + ∂ ∂ = ∂ ∂ ++− ∂ ∂ += ∂ ∂ − ∂ ∂ xtxxtFxtxxtFxtxxtxxtFxtx x xtx t xtx xtxxtF xtx x xtx xtxxtFxtxxtxxtFxtxxtF t xtx xtxxtFxtxxtFxtM x xtN t xtxxx T xx T xxxxx xxtxt            But ( ) ( ) ( ) ( ) ( )xtx td xtxd xtx x xtx t xtx T , , , ,,     == ∂ ∂ + ∂ ∂ Since satisfies the Euler-Lagrange equation, that is given by( )xtx , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,,,,,,,, =      −      +      +      ••••••• •••• txtxtFtxtxtFtxtxtxtFtxtxtxtF x txxxxx we can see that along C we have ( ) ( ) 0,, = ∂ ∂ − ∂ ∂ xtM x xtN t t q.e.d.
  • 95.
    95 SOLO Calculus of Variations Hilbert’sInvariant Integral (continue – 3) David Hilbert (1862 – 1943) 10.1 Example: Geometrical Optics and Fermat’s Principle ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += For the Geometrical Optics we obtained: The Hilbert’s Invariant Integral is ( ) ( )( ) ( ) ( )[ ] ( ) ( )( ){ ( ) ( ) ( ) ( )[ ] ( ) ( )( )} xdzyxzzyxyzyxFzyxZzyxz zyxzzyxyzyxFzyxYzyxyzyxzzyxyzyxF z zyxP zyxP yC ffff ,,,,,,,,,,,, ,,,,,,,,,,,,,,,,,,,, ,, ,, 0000     −− ∫ −− This is known as Hilbert’s Invariant Integral because it is invariant on the path C as long as this curve remains in the field of the unique extremal solution. ( ) ( ) ( ) ( )zyx x z zyxzzyx x y zyxy ,,,,,,,,, ∂ ∂ = ∂ ∂ =  is the field slope and ( ) ( ) CC x z zyxZ x y zyxY ∂ ∂ = ∂ ∂ = :,,,:,,  is the path C slope at the point (x,y,z) of C we have on path C ( ) ( ) dx x z dxzyxZzddx x y dxzyxYyd C C C C ∂ ∂ == ∂ ∂ == ,,,,, 
  • 96.
    96 SOLO Calculus of Variations Hilbert’sInvariant Integral (continue – 4) David Hilbert (1862 – 1943) Example: Geometrical Optics and Fermat’s Principle (continue – 1) ( ) ( ) ( )[ ]{ ( ) ( ) ( ) ( ) }zdzyzyxFydzyzyxFxdzyzyxFzzyzyxFyzyzyxF zy zyxP zyxP zyC ffff   ,,,,,,,,,,,,,,,,,,,, ,, ,, 0000 −−−−∫ The Hilbert’s Invariant Integral is We can write ( ) [ ] [ ] [ ] [ ] ( ) sd xd zyxn zy n zy zn z zy yn yzynFzFyzyzyxF zy ,, 111 1,,,, 2/1222/1222/122 2/122 = ++ = ++ − ++ −++=−−        ( ) [ ] ( ) ( ) [ ] ( ) sd zd zyxn zy zzyxn z F F sd yd zyxn zy yzyxn y F F z y ,, 1 ,, ,, 1 ,, 2/122 2/122 = ++ = ∂ ∂ = = ++ = ∂ ∂ =         Now we can write the Hilbert’s Invariant Integral as ( ) ( ) ( ) ( ) ∫∫ ⋅=⋅ ffffffff zyxP zyxP zyxP zyxP ray rdsnrd sd rd n ,, ,, ,, ,, 1000010000 ˆ   This is the Lagrange’s Invariant Integral from Geometrical Optics. Joseph-Louis Lagrange (1736-1813) Integration Path through a Ray Bundle Return to Table of Content
  • 97.
    SOLO Calculus of Variations 11.The Weierstrass Necessary Condition for a Strong Minimum (Maximum) –1879 Karl Theodor Wilhelm Weierstrass 1815-1897 11.1 Derivation from Hilbert’s Invariant Integral Along the unique extremal path (denoted as C* - see Figure ), that passes through and we have and the Hilbert Integral becomes ( )00 , xt ( )ff xt , ( ) ( )xtxxtX ,,  = ( )( ) ( )( ) ( ) ( )( )[ ] ( )( ) [ ]xtJtdxtxxtF tdxtXxtxxtxxtFxtxxtF f f t t C t t T xC ,*,,, ,,,,,,,, 0 0 * == −− ∫ ∫    where is the minimum of the functional[ ]xtJ ,* ( )[ ] ( ) ( )( )∫= ft t C dttXtxtFtxJ 0 ,,  Suppose that the extremal is a strong minimum and •C* represents the strong minimum curve •C represents a strong neighbor of C* We can compute ( )[ ] ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( )[ ] ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( )[ ]∫ ∫∫ ∫∫ −−−= −−−= −=∆ f ff ff t t T xC t t T xC t t C t t C t t C tdxtxxtXxtxxtFxtxxtFtXtxtF tdxtXxtxxtxxtFxtxxtFdttXtxtF dttxtxtFdttXtxtFtxJ 0 00 00 ,,,,,,,,,, ,,,,,,,,,, ,,,, *     
  • 98.
    SOLO Calculus of Variations TheWeierstrass Necessary Condition for a Strong Minimum (Maximum) (continue – 1) Karl Theodor Wilhelm Weierstrass 1815-1897 Derivation from Hilbert’s Invariant Integral (continue – 1) Let define the Weierstrass E-function: ( ) ( ) ( ) ( ) ( )xXxxtFxxtFXxtFXxxtE T x   −−−= ,,,,,,:,,, Therefore the strong minimum condition is ( )[ ] ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( )[ ]∫ −−−=∆ ft t T xC tdxtxxtXxtxxtFxtxxtFtXtxtFtxJ 0 ,,,,,,,,,,   ( )[ ] ( ) ( ) ( )( ) 0,,, 0 ≥∫=∆ ft t C dttXtxtxtEtxJ  Weierstrass Necessary Conditions for a Strong Minimum (Maximum) is: ( ) ( )00,,, ≤≥XxxtE  for every admissible set ( )Xxt ,,
  • 99.
    SOLO Calculus of Variations TheWeierstrass Necessary Condition for a Strong Minimum (Maximum) (continue – 2) Karl Theodor Wilhelm Weierstrass 1815-1897 11.2 Weierstrass Derivation Weierstrass, that first derived this necessary condition for a Strong Minimum (Maximum) used the following derivation: 1t δ+1tδε+1t ( )txx = ( ) ( ) ( )[ ]111 txtXtx  −+ δε 1 2 3 0t ft ( ) ( ) [ ] [ ] ( ) ( ) ( ) ( )[ ] [ ] ( ) ( ) ( ) ( )[ ] [ ]       ++∈− − −+ + +∈−−+ +∈ = δδε ε δε δε δ ε 1111 1 11111 110 , 1 , ,, , ttttxtX tt tx ttttxtXtttx ttttttx tx f    Weierstrass Strong Variation Suppose is a candidate trajectory passing trough points 1 and 2, such that it contains no points of discontinuity of and no conjugate points between those points. Let take an arbitrary curve through point 1 such that . Let point 3 a movable point on at t = t1 + εδ . Let connect points 3 with point 2 on The arc 1, 3, 2 constitutes a strong variation (by δ tacking as small as we want). This variation , has a discontinuous derivative at point 1. ( )txx = x ( )tXx = ( ) ( )11 tXtx  ≠ ( )tXx = ( )δ+= 1txx ( )ε,txx =
  • 100.
    SOLO Calculus of Variations TheWeierstrass Necessary Condition for a Strong Minimum (Maximum) (continue – 3) Weierstrass Derivation (continue – 1) 1t δ+1tδε+1t ( )txx = ( ) ( ) ( )[ ]111 txtXtx  −+ δε 1 2 3 0t ft ( ) ( ) [ ] [ ] ( ) ( ) ( ) ( )[ ] [ ] ( ) ( ) ( ) ( )[ ] [ ]       ++∈− − −+ + +∈−−+ +∈ = δδε ε δε δε δ ε 1111 1 11111 110 , 1 , ,, , ttttxtX tt tx ttttxtXtttx ttttttx tx f    Weierstrass Strong Variation ( ) ( ) [ ] [ ] ( ) ( ) ( ) ( )[ ] [ ] ( ) ( ) ( ) ( )[ ] [ ]       ++∈− − −+ + +∈−−+ +∈ = δδε ε δε δε δ ε 1111 1 11111 110 , 1 , ,, , ttttxtX tt tx ttttxtXtttx ttttttx tx f    This variation fails to lie within any weak neighborhood of , no matter how small is δ. ( )txx = Since is a strong minimum, we have:( )[ ]txJ ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ]( ) ( ){ } ( ) ( ) ( ) ( )[ ] ( )∫∫ + + +       −      − − −+−−+= −≤ δ δε δε ε ε εε ε 1 1 1 1 ,, 1 ,,,,,,,, ,0 111111 t t t t dtxxtFtxtXtxtxtFdtxxtFtxtXtxtxtF txJtxJ  Let assume δ→0 ( )[ ] ( )[ ] ( ) ( )( ) ( ){ } ( ) ( ) ( ) ( )[ ] ( ) ε ε ε ε ε ε δε ε δ −       −      − − − +−= − ≤ → 1 ,, 1 ,,, ,,,,, , lim0 111 1 0 xxtFtxtXtxtxtF xxtFtXtxtF txJtxJ  
  • 101.
    SOLO Calculus of Variations TheWeierstrass Necessary Condition for a Strong Minimum (Maximum) (continue – 4) Weierstrass Derivation (continue – 2) 1t δ+1tδε+1t ( )txx = ( ) ( ) ( )[ ]111 txtXtx  −+ δε 1 2 3 0t ft ( ) ( ) [ ] [ ] ( ) ( ) ( ) ( )[ ] [ ] ( ) ( ) ( ) ( )[ ] [ ]       ++∈− − −+ + +∈−−+ +∈ = δδε ε δε δε δ ε 1111 1 11111 110 , 1 , ,, , ttttxtX tt tx ttttxtXtttx ttttttx tx f    Now let take ε→0 ( ) ( )( ) ( ){ } ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )[ ] ( )          XxxtE x T x txtXxxtFxxtFtXtxtF xxtFtxtXxxtFxxtF xxtFtXtxtF ,,, 111 2 11 0 1 ,,,,,, 1 ,,,, 1 ,, lim ,,,,0 −−−= −       −    Ο/+− − − + −≤ → ε ε ε ε ε ε This Inequality is the Weierstrass Necessary Conditions for a Strong Minimum (Maximum) ( ) 0,,, ≥XxxtE or Since the Weierstrass condition directly concerns minimality, rather than stationarity as did Euler-Lagrange condition, it entails no further supporting statements analogous to the Legendre and Jacobi conditions that support the Euler-Lagrange stationary condition. A weak variation is included in the strong variations, therefore a condition that is necessary for a weak local minimum (maximum) is also necessary for a strong local minimum (maximum).
  • 102.
    SOLO Calculus of Variations TheWeierstrass Necessary Condition for a Strong Minimum (Maximum) (continue – 5) 11.3 Geometric Interpretation of Weierstrass Conditions Let plot as a function of (see Figure). The hyper-plane tangent at is given by ( ) ( )      = • txtxtF ,,η • = xξ ( ) ( ) ( )      == •• iiiiii txtxtFtx ,,,ηξ ( ) ( ) ( ) ( ) ( )      +      −      = ••• iiiiiii T x txtxtFtxtxtxtF ,,,, ξη  The E function will be given by the difference between and the tangent hyper-plane. We can see that the condition for minimality is that the tangent hyper-plane remains bellow the surface . ( ) ( )       == • XtxtxtF ,,η ( ) ( )      = • txtxtF ,,η Geometric Representation of Weierstrass Condition
  • 103.
    SOLO Calculus of Variations TheWeierstrass Necessary Condition for a Strong Minimum (Maximum) (continue – 5) 11.4 Example: Geometrical Optics and Fermat’s Principle ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += For the Geometrical Optics we obtained: Weierstrass E Function is defined as ( ) ( ) ( ) ( ) ( ) ( ) ( )zyzyxFzZzyzyxFyYzyzyxFZYzyzyxFZYzyzyxE zy   ,,,,,,,,,,,,,,,,,,:,,,,,, −−−−−= [ ] [ ] ( ) [ ] ( ) [ ] [ ] [ ] ( ) ( )[ ] [ ] [ ] ( ) [ ] [ ] [ ] ( )InequalitySchwarz zyZY zZyY ZYn zZyY zy n ZYn zzZyyYzy zy n ZYn zy zn zZ zy yn yYzynZYn 0 11 1 11 1 1 1 1 1 1 1 '' 1 11 2/1222/122 2/122 2/122 2/122 22 2/122 2/122 2/1222/122 2/1222/122 ≥         ++++ ++ −++= ++ ++ −++= −+−+++ ++ −++= ++ −− ++ −−++−++=               According to Weierstrass Condition if the Jacobi Condition (no conjugate points between and ) is satisfied every extremal is a strong minimum. ( )( )0',',',',',',,, ≥ZYXzyxzyxE Return to Table of Content
  • 104.
    SOLO Calculus of Variations Summary NecessaryConditions for a Weak Relative Minimum (Maximum) Satisfies Boundary Conditions ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) fitxdtxtxtFdttxtxtxtFtxtxtF iiii T x iiiii x iii ,00,,,,,, ==      +            −      •••• •• Satisfies Weierstrass-Erdmann Corner Conditions ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,,,, ,,,,,,,, 00 000000 =            −      +          +         −      −      + • − • + • + • + • − • − • − • •• •• cccc T x ccc T x ccccc T x ccccccc T x ccc txdtxtxtFtxtxtF dttxtxtxtFtxtxtFtxtxtxtFtxtxtF Satisfies the Euler-Lagrange Equation [ ]f x x tttforF dt d F ,0 0∈=− • 1 Satisfies Legendre (Clebsh) Condition2 is Positive (Negative) Definite for( ) xxFtR = [ ]fttt ,0∈ It contains no Conjugate Point for3 [ ]fttt ,0∈ Necessary Conditions for a Strong Relative Minimum (Maximum). Satisfies (1), (2) and (3) and additionally: Weirestrass Necessary Conditions for a Strong Minimum (Maximum) is that: ( ) ( ) ( ) ( ) ( ) ( )00,,,,,,:,,, ≤≥−−−= xXxxtFxxtFXxtFXxxtE T x   4 for every admissible set ( )Xxt ,, Return to Table of Content
  • 105.
    SOLO Calculus of Variations 12.Canonical Form of Euler-Lagrange Equations We found that the first variation of the cost function J is given by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫             −      +             +            −      = •••••• ••• ff t t T x x t t T x T x dttxtxtxtF dt d txtxtFxdtxtxtFdtxtxtxtFtxtxtFJ 00 ,,,,,,,,,, δδ Let define ( ) ( ) T xxx n FFtxtxtFp    =      = ••• • ,,,,: 1  and suppose that [ ]               =                 ∂ ∂ ∂ ∂ =     ∂ ∂ =                 ∂ ∂ ∂ ∂ ≡ • ••• nnnn n n n xxxxxx xxxxxx xxxxxx xxx n T x T xx FFF FFF FFF FFF x x F xx F x F             21 21212 12111 21 ,,, 1 is nonsingular for t ϵ [t0, tf ] (regular problem), then we can solve ) (locally, because of the Implicit Function Theorem) as a function of , by using Legendre’s Dual Transformation. ( )tx • ( ) ( )tptxt ,, Return to Table of Content
  • 106.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous – 1) 12.1 Legendre’s Dual Transformation Adrien-Marie Legendre 1752-1833 Let consider a function of n variables xi, m variables, and time t:ii xy ≡ ( )mn yyxxtF ,,,,,, 11  and introduce a new set of n variables pi defined by the transformation: ni y F p i i ,,2,1: = ∂ ∂ = We can see that for t ϵ (t0, tf )                             ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ +                             ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ ∂ =             m mnn m n nn n n dx dx dx xy F xy F xy F xy F dy dy dy y F yy F yy F y F dp dp dp          2 1 2 1 2 1 2 11 2 2 1 2 2 1 2 1 2 2 1 2 2 1 We want to replace the variables dyi (i=1,2,…,n) by the new variables dpi (i=1,2,…,n). We can see that the new n variables are independent if the Hessian Matrix ni njji ni njji xx F yy F ,,1 ,,1 2 ,,1 ,,1 2      = = = =         ∂∂ ∂ =         ∂∂ ∂ is nonsingular in the interval t ϵ (t0, tf ). According to the Implicit Function Theorem (Appendix 1) we can obtain a unique function in the interval t ϵ (t0, tf ).( )pxtxy ii ,,: =
  • 107.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –2) Let define a new function H (Hamiltonian) of the variables t, xi, pi. ( )nm n i ix n i ii ppxxtHxFFypFH i ,,,,,,: 11 11  =+−=+−= ∑∑ == Then: ( ) ∑∑∑∑∑ =====             ∂ ∂ −++ ∂ ∂ − ∂ ∂ −=++ ∂ ∂ − ∂ ∂ − ∂ ∂ −= n i i i iii n j j j n i iiii n i i i n j j j dy y F pdpydx x F dt t F dpydypdy y F dx x F dt t F dH 11111 But because ( )nm ppxxtHH ,,,,,, 11 = ∑ ∑ ∂ ∂ + ∂ ∂ + ∂ ∂ = = = n j n i i i j j dp p H dx x H dt t H dH 1 1 and because all the variations are independent we have: ;,,1;,,1&, mj x F x H ni y F p p H y t F t H jji i i i  = ∂ ∂ −= ∂ ∂ = ∂ ∂ = ∂ ∂ = ∂ ∂ −= ∂ ∂ Now we can define the Dual Legendre’s Transformation from ( ) ( ) ( )yxtFyppxtHtoyxtF n i ii ,,,,,, 1 −= ∑= by using ni p H xy ni x F y F p i ii ii i ,,2,1 ,,2,1    = ∂ ∂ == = ∂ ∂ = ∂ ∂ = The variables t, , and H are called Canonical Variables corresponding to the functional J. x p
  • 108.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –3) If we apply now the Legendre Transformation to and H we obtainp ( ) ( )yxtFpxtHyp n i ii ,,,, 1 =−∑= The Legendre Transformation is an involution, i.e. a transformation which is its own inverse. ( ) ( ) ( ) ( )∫       −      ++−= •f f t t T x t t T dttxp dt d txtxtFxdpdtHJ 0 0 ,, δδ Let write δJ in terms of the Canonical Variables: From this expression the necessary conditions such that δJ is zero are ( ) 00 =+− t T xdpdtH ( ) ( )      = • txtxtFp dt d x ,, ( ) 0=+− ft T xdpdtH We found before that the necessary conditions such that δJ is zero for those admissible solutions passing through the points and are the Euler-Lagrange Equations: ( )* 0 * 0 * , xtA ( )*** , ff xtB ( ) ( ) ( ) ( ) 0,,,, =      −      •• • txtxtF dt d txtxtF x x niF y F p x ,,2,1:  == ∂ ∂ =Since the two equations are identical. Return to Table of Content
  • 109.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –4)       ∂ ∂ =⇒= ∂ ∂ == p H xni p H xy i ii  ,,2,1and x H x F Fmj x F x H x jj ∂ ∂ −= ∂ ∂ =⇒= ∂ ∂ −= ∂ ∂ :;,,1 also ( ) ( )      = • txtxtFp dt d x ,,we obtain Canonical Euler-Lagrange Equation or Hamilton’s Equations x H td pd p H td xd ∂ ∂ −= ∂ ∂ = we can write the Euler-Lagrange Equations in the form x H ∂ ∂ −= ( ) ( )      = • • txtxtFp x ,,:using William Rowan Hamilton (1805-1855)
  • 110.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –5) Canonical Euler-Lagrange Equation or Hamilton’s Equations x H td pd p H td xd ∂ ∂ −= ∂ ∂ = Therefore we transformed the n Second Order Euler-Lagrange Differential Equations in 2 n First Order Hamilton’s Equations. Let find out when this problem is well-posed, i.e.: • a solution exists. • the solution is unique. • the solution depends continuously on the initial values. First we must remember that the Hamilton’s Equation where derived only for regular problems: is nonsingular for t ϵ (t0, tf ).xxF  From the theory of First Order Differential Equations (see Appendix 3) the solution exists if       ∂ ∂       ∂ ∂ x H p H , exists, are continuous in t ϵ (t0, tf ) (except a finite number of points). This implies that is continuous and has continuous partial derivatives.( ) ( )yxtFyppxtH n i ii ,,,, 1 −= ∑= The solution is unique and depends continuously on the initial values if in addition the problem has 2 n defined boundary conditions. Therefore the general solutions of the Hamilton’s Equations are therefore two vector parameters solutions . Those parameters are defined by 2n boundary conditions. ( ) ( )T n T n βββααα ,,,,, 11  == ( ) ( )βαϕ ,,ttx = Return to Table of Content
  • 111.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –6) 12.2 Transversality Conditions (Canonical Variables ) For other admissible variations we shall need to add the additional necessary conditions, such that δJ is zero, called Transversality Conditions Equations: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) fitxdtxtxtFdttxtxtxtFtxtxtF iiii T x iiiii T x iii ,00,,,,,, ==      +            −      •••• •• ( ) ( )( ) ( ) fitxdpdttptxtH i T iiii ,00,, ==+−or (a) Suppose that the following relation defines the boundary: ( ) ( ) ( ) ( ) ( ) iitiiiii dttdtt dt d txdttx Ψ=Ψ=→Ψ= then the Transversality Conditions Equations are: ( ) ( ) ( ) ( ) ( ) ( ) fitxttxtxtFtxtxtF iiiii T x iii ,00,,,, ==      −Ψ      +      ••• • ( ) ( )( ) ( ) ( ) fittptptxtH i T iiii ,00,, ==Ψ+−or (b) Suppose that ti and are not defined, and is not a function of ti , therefore d ti and are independent differentials and the Transversality Conditions will be: ix ix ixd ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,, 0,,,, =      =      −      • ••• • • iii x iiii T x iii txtxtF txtxtxtFtxtxtF ( ) ( )( ) ( ) 0 0,, = = i iii tp tptxtH or Return to Table of Content
  • 112.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –8) 12.3 Weierstrass-Erdmann Corner Conditions (Canonical Variables) At the corners c we found: ( ) 0 0000 =      −+               −−      −= + • − • + • − • •• c t T xt T x c t T x t T x txdFFdtxFFxFFJ cccc δ Using the Canonical Variables we obtain: ( ) ( )[ ] ( ) ( )[ ] ( ) 00000 =−++−= +−+− cc T c T ccc txdtptpdttHtHJδ (a) If they are apriori conditions at the corner like: ( ) ( ) ( ) ( ) ( ) cctccccc dttdtt dt d txdttx Ψ=Ψ=→Ψ= then the necessary conditions at the corner are: ( ) ( ) ( ) ( ) ( ) ( )0000 ++−− −Ψ=−Ψ cctc T cctc T tHttptHttp (b) If they are not apriori conditions at the corner; i.e. the function is not apriori defined then dti and independent variables and ( ) ( )cc ttx Ψ= cxd ( ) ( ) ( ) ( )0000 & +−+− == cccc tptptHtH Return to Table of Content
  • 113.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –9) 12.4 First Integrals of the Euler-Lagrange Equations A First Integral of a system of differential equations is a function which has a constant value along each integral curve of the system. We defined ( ) ( ) ( )yxtFypyxtFyppxtH T n i ii ,,,,,, 1 −=−= ∑= and ( ) ( ) ( ) ( ) td pd p pxtH td xd x pxtH t pxtH td pxtHd TT       ∂ ∂ +      ∂ ∂ + ∂ ∂ = ,,,,,,,, Using the Canonical Euler-Lagrange Equations x H td pd p H td xd ∂ ∂ −= ∂ ∂ = we obtain from ( ) ( ) ( )yxtFypyxtFyppxtH T n i ii ,,,,,, 1 −=−= ∑= t H x H p H p H x H t H td Hd TT ∂ ∂ = ∂ ∂       ∂ ∂ − ∂ ∂       ∂ ∂ + ∂ ∂ = If does not depend on t explicitly, H doesn’t depend on t explicitly and is is constant over the optimal path, i.e. is a First Integral of the Euler-Lagrange Equations. ( )yxtF ,, ( )pxH , 0= ∂ ∂ = t H td Hd
  • 114.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –10) First Integrals of the Euler-Lagrange Equations (continue – 1) Consider an arbitrary function ( )pxt ,,Φ=Φ and compute ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x pxtH p pxt p pxtH x pxt t pxt td pd p pxt td xd x pxt t pxt td pxtd TT TT ∂ ∂       ∂ Φ∂ − ∂ ∂       ∂ Φ∂ + ∂ Φ∂ =       ∂ Φ∂ +      ∂ Φ∂ + ∂ Φ∂ = Φ ,,,,,,,,,, ,,,,,,,, Define Poisson Bracket [ ] ( ) ( ) ( ) ( ) x pxtH p pxt p pxtH x pxt H TT ∂ ∂       ∂ Φ∂ − ∂ ∂       ∂ Φ∂ =Φ ,,,,,,,, :, Siméon Denis Poisson 1781-1840 From which [ ]H ttd d ,Φ+ ∂ Φ∂ = Φ is constant over the optimal path, i.e. is a First Integral of the Euler-Lagrange Equations iff 1.F and Φ do not depend on t explicitly. 2.[Φ,H] = 0. ( )px,Φ Return to Table of Content
  • 115.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –11) 12.5 Equivalence Between Euler-Lagrange and Hamilton Functionals The Euler-Lagrange Functional [ ] ( )∫= ft t dtxxtFxJ 0 ,,  is optimized by the solution of the Euler-Lagrange Equations: ( ) ( ) ( ) ( ) 0,,,, =      −      •• • txtxtF dt d txtxtF x x We set ( ) ( )      = • • txtxtFp x ,,: and the Hamiltonian ( ) ( ) ( ) ( ) xppxtHxxtFxxtFxppxtH TT  +−=⇒−= ,,,,,,:,, We define the Hamilton Functional [ ] ( )[ ]∫ +−= ft t T dtxppxtHpxJ 0 ,,:,  Since: ( ) ( )xxtFxppxtH T  ,,,, =+− ( )[ ] ( )∫∫ =+− ff t t t t T dtxxtFdtxppxtH 00 ,,,,  William Rowan Hamilton (1805-1855)
  • 116.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –12) Equivalence Between Euler-Lagrange and Hamilton Functionals (continue – 1) Hamilton Functional [ ] ( )[ ]∫ +−= ft t T dtxppxtHpxJ 0 ,,:,  William Rowan Hamilton (1805-1855) Let find the Euler-Lagrange Equations for the Hamilton Functional ( )[ ] ( )[ ] ( )[ ] ( )[ ] 0,,,, 0,,,, =       +− ∂ ∂ −+− ∂ ∂ =       +− ∂ ∂ −+− ∂ ∂ xppxtH ptd d xppxtH p xppxtH xtd d xppxtH x TT TT       or 0 0 =+ ∂ ∂ − =− ∂ ∂ − td xd p H td pd x H We recovered the Canonical Euler-Lagrange (Hamilton) Equations (William R. Hamilton 1835) Return to Table of Content
  • 117.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –13) 12.6 Equivalent Functionals Two functionals are said to be equivalent if they have the same extremal trajectories. Suppose we have an arbitrary continuous and differentiable function: ( )xtS , Define ( ) ( ) x x S t S xtS td d xxt T        ∂ ∂ + ∂ ∂ ==Ψ ,:,, Let compute ( ) x x S xt S xxt x  2 22 ,, ∂ ∂ + ∂∂ ∂ =Ψ ∂ ∂ ( ) x x S tx S xtd d x S xxt x     2 22 ,, ∂ ∂ + ∂∂ ∂ =      ∂ Ψ∂ ⇒ ∂ ∂ =Ψ ∂ ∂ Since tx S xt S ∂∂ ∂ = ∂∂ ∂ 22 0=      ∂ Ψ∂ − ∂ Ψ∂ xtd d x  Similar to Euler-Lagrange (E.-L.) Equations The functionals [ ] ( )∫= ft t dtxxtFxJ 0 ,,  0 .. =      ∂ ∂ − ∂ ∂ ⇒ − x F td d x FLE  [ ] ( ) ( )[ ]∫ Ψ−= ft t dtxxtxxtFxJ 0 ,,,, ~  0 .. =      ∂ ∂ − ∂ ∂ =      ∂ Ψ∂ + ∂ Ψ∂ −      ∂ ∂ − ∂ ∂ ⇒ − x F td d x F xtd d xx F td d x FLE  and have the same Euler-Lagrange Equations.
  • 118.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –14) Equivalent Functionals (continue – 1) Two functionals are said to be equivalent if they have the same extremal trajectories. We can see that [ ] ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( )[ ] [ ] ( )[ ] ( )[ ]00 00 ,, ,,,, , ,,,,,, ~ 0 00 txtStxtSxJ txtStxtSdtxxtF dt td xtdS xxtFdtxxtxxtFxJ ff ff t t t t t t f ff +−= +−=       −=Ψ−= ∫ ∫∫   The functionals and are called equivalent functionals.( )xJ ( )xJ ~ Return to Table of Content
  • 119.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –15) 12.7 Canonical Transformations The functional [ ] ( )[ ]∫ +−= ft t T dtxppxtHpxJ 0 ,,:,  is optimized by the trajectories derived from the Canonical Euler-Lagrange (Hamilton) Equations x H td pd p H td xd ∂ ∂ −= ∂ ∂ = Let perform a change of variables, from to according topx, px ~,~ ( ) ( )pxpp pxxx ,~~ ,~~ = = Since                   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =      pd xd p p x p p x x x pd xd ~~ ~~ ~ ~ this is possible iff ( ) ( ) 0 , ~,~ :~~ ~~ det ≠ ∂ ∂ =             ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ px px p p x p p x x x
  • 120.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –16) Canonical Transformations (continue – 1) We look for transformations under which the Canonical Euler-Lagrange (Hamilton) Equations preserve their form. They are called Canonical Transformations. x H td pd p H td xd ~ ~~ ~ ~~ ∂ ∂ −= ∂ ∂ = and optimize the functional [ ] ( )[ ]∫ +−= ft t T dtxppxtHpxJ 0 ~~~,~, ~ :~,~  The two functional are equivalent if ( ) ( ) ( )xtS td d td xd ppxtH td xd ppxtH TT , ~ ~~,~, ~ ,, −+−=+− From which ( ) ( ) ( )( )pxxtSdxdptdpxtHxdptdpxtH TT ~,~,~~~,~, ~ ,, −+−=+− or ( )( ) ( ) ( )[ ] tdpxtHpxtHxdpxdppxxtSd TT ,, ~~,~,~~~,~, −+−=
  • 121.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –17) Canonical Transformations (continue – 2) ( )( ) ( ) ( )[ ] tdpxtHpxtHxdpxdppxxtSd TT ,, ~~,~,~~~,~, −+−= From xd x x p x xd p x pdpd p x xd x x xd ~ ~~~ ~~ ~ ~ ~ 11 ∂ ∂       ∂ ∂ −      ∂ ∂ =⇒ ∂ ∂ + ∂ ∂ = −− we can use instead of to obtainxx ~, px ~,~ ( )( ) ( ) ( ) ( )[ ] tdpxtHpxtHxdpxdp xd x S xd x S td t S xxtSdpxxtSd TT TT ,, ~~,~,~~ ~ ~ ~,,~,~, −+−=       ∂ ∂ +      ∂ ∂ + ∂ ∂ == Finally we obtain: ( ) ( ) t S pxtHpxtH x S p x S p ∂ ∂ =− ∂ ∂ −= ∂ ∂ = ,, ~~,~,~ ~ Return to Table of Content
  • 122.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –18) 12.8 Caratheodory's Lemma Consider the problem of minimizing the functional [ ] ( ) ( )∫       = ⋅fxt xt dttxtxtFxtJ , , 00 ,,, defined in a simple connected domain Ω in the plane. Given an initial point suppose that one and only one extremal of exists between the initial point and every point . ( )xt, ( ) Ω∈00 , xt [ ]xtJ , ( ) Ω∈xt, Assume that is continuous for all and all admissible . Define      ⋅ xxtF ,, ( ) Ω∈xt, x ( ) ( ) ( ) ( )∫       = ⋅ Ω∈ fxt xt xt dttxtxtFxtS , , , 00 ,,min:, called the geodesic distance (Hamilton called it optical distance) or the Hamilton's characteristic function. Since we have one and only one extremal connecting with along a curve Γ Ω. is a single-valued function for all .ϵ ( ) Ω∈00 , xt ( ) Ω∈xt, ( )xtS , ( ) Ω∈xt, Field of Extremals Starting from ( ) Ω∈00 , xt
  • 123.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –19) Caratheodory's Lemma (continue – 1) Field of Extremals Starting from ( ) Ω∈00 , xt Suppose that we calculate the functional along a neighbor curve Γε Ω that connects with .ϵ [ ]xtJ , ( )00 , xt ( )xt, By definition of ( )xtS , [ ] ( ) ( ) ( ) ( ) ( )∫ Γ ⋅ Γ Ω∈∀≥      −      =− fxt xt xtdtxtS td d txtxtFxtSxtJ , , 00 ,0,,,,, ε ε where we use the fact that ( ) ( ) ( ) ( )xtwithxtconnectingdtxtS td d xtS fxt xt ,,,, 00 , , 00 ε ε Γ∀      = ∫ Γ Along , if is continuous and differentiable relative to t and we haveΩ∈Γε ( )xtS , x ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )εεεεεεε ,,,,,,,,,,,, txtxtStxtStx t txtS x txtS t txtS td d T xt T += ∂ ∂       ∂ ∂ + ∂ ∂ = we obtain: [ ] ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )∫ Γ ⋅ Γ Ω∈∀≥      −−      =− fxt xt T xt xtdttxtxtStxtStxtxtFxtSxtJ , , 00 ,0,,,,,,,,,,, ε ε εεεεε  Since this is true for all and curve is the only optimal curve, the last equation is equivalent to: ( ) Ω∈xt, Ω∈Γ ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )εεεεεεε ε ,&,,0,,,,,,,,, txtxttxtxtStxtStxtxtF T xt  Γ≠Γ∈∀>−−      ⋅ ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) Γ∈=−−      ⋅ txtfortxtxtStxtStxtxtF T xt ,0,,,,  and
  • 124.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –20) Caratheodory's Lemma (continue – 2) Field of Extremals Starting from ( ) Ω∈00 , xt Since at any point the change in the curve direction is defined by its slope we can write ( ) Ω∈xt, ( )xtX , Carathéodory's Lemma If the Hamiltonian's characteristic function is defined on an admissible set of simple connected region of terminations Ω and if S is continuous and differentiable on it's arguments, then every element of an optimal trajectory that lies entirely in Ω is characterized by ( )xtS , ( )xxt ,, ( ) ( ) ( ) ( ) ( )[ ] 0,,,,min,,,, =−−=−−      ⋅ XxtSxtSXxtFxxtSxtSxxtF T xt X T xt   Constantin Carathéodory (1873-1950) R.E Bellman arrived to the Carathéodory's Lemma in a different way which will be described elsewhere. Return to Table of Content
  • 125.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –21) 12.9 Hamilton-Jacobi Equations From the Carathéodory's Lemma we can derive the following Theorem (a) If is continuous for all , is not constraint and if is continuous and differentiable on it's arguments, then for every element of an optimal trajectory that lies entirely in Ω, except for the corners of , the following two conditions have to hold       ⋅ xxtF ,, ( ) Ω∈xt, x ( )xtS , ( )xxt ,, x ( ) xxxtFxxtFxtS T xt        −      = ⋅⋅ ,,,,, ( )       = ⋅ xxtFxtS xx ,,,  (b) Here is the uniquely determined slope of the optimal trajectory at the point and is viewed as a function of t and . x ( ) Ω∈xt, x First Proof of (a) Using the Carathéodory's Lemma, let define: ( ) ( ) ( ) 0,,,,:,, ≥−−=      ⋅ XxtSxtSXxtFxxtE T xt  we see that if is not constraint, from the ordinary differential calculus, the necessary conditions for the minimum are, that on the optimal trajectory . x ( )xxt ,,
  • 126.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –22) Hamilton-Jacobi Equations (continue – 1) First Proof of (a) (continue – 1) ( ) ( ) ( )[ ] ( ) ( ) 0,,,,,,, =−=−− ∂ ∂ = ∂ ∂ xtSxxtFxxtSxtSxxtF xx E xx T xt    and this gives ( )       = ⋅ xxtFxtS xx ,,,  ( ) ( ) ( )[ ] ( ) 0,,,,,,22 2 ≥=−− ∂ ∂ = ∂ ∂ xxtFxxtSxtSxxtF xx E xx T xt    This is the Legendre's Necessary Condition. ( )       = ⋅ xxtFxtS xx ,,, If we substitute in Carathéodory's Lemma ( ) ( ) ( ) ( ) ( )[ ] 0,,,,min,,,, =−−=−−      ⋅ XxtSxtSXxtFxxtSxtSxxtF T xt X T xt   we obtain for the optimal trajectory ( )xxt ,, ( ) ( ) ( ) 0,,,,,,,,, =      −−      =−−      ⋅⋅⋅ xxxtFxtSxxtFxxtSxtSxxtF T xt T xt   If we substitute and in we obtain ( ) xxxtFxxtFxtS T xt        −      = ⋅⋅ ,,,,, ( )       = ⋅ xxtFxtS xx ,,,  ( ) ( ) ( ) 0,,,,:,, ≥−−=      ⋅ XxtSxtSXxtFxxtE T xt  ( ) ( ) ( ) ( ) 0,,,,,,,, ≥−−−=      ⋅ xXxxtFxxtFXxtFxxtE T x   Weierstrass' Condition E is called Weierstrass' Excess Function. End of Proof (a)
  • 127.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –23) Hamilton-Jacobi Equations (continue – 2) Second Proof of (a) (Geometrical) Suppose that is a point on the optimal trajectory .( )( ) Ω∈ii txt , Let plot ( ) ( )      = • txtxtF ,,η ( ) ( ) xxtSxtS T xtS ,, +=η • = xξas a function of and the hyper-plane From Carathéodory's Lemma those two functions intersect at and the hyper-plane must stay on one side of , therefore it is tangent to it. ( )( ) Ω∈ii txt , ( ) ( )      = • txtxtF ,,η
  • 128.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –24) Hamilton-Jacobi Equations (continue – 3) Second Proof of (a) (Geometrical) (continue – 1) On the other side the hyper-plane tangent at is given by( ) ( ) ( )      == •• iiiiii txtxtFtx ,,,ηξ ( ) ( ) ( ) ( ) ( )      +      −      = ••• iiiiiii T xT txtxtFtxtxtxtF ,,,, ξη  Since ηT ≡ ηS we must have ( ) xxxtFxxtFxtS T xt        −      = ⋅⋅ ,,,,, ( )       = ⋅ xxtFxtS xx ,,,  We can see from Figure that E is given by the difference between and the tangent to hyper-plane. We can see that the condition for minimality is that the tangent hyper-plane remains bellow the surface . ( ) ( )       == • XtxtxtF ,,η ( ) ( )      = • txtxtF ,,η End of Geometrical Proof of (a) Geometric Representation of , and Weierstrass Condition( ) ( )      = • txtxtF ,,η ( ) ( ) xxtSxtS xtS ,, +=η
  • 129.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –25) Hamilton-Jacobi Equations (continue – 4) Proof of (b) Let use the Canonical Forms. Start with definition ( )xxtF x F p x    ,,: = ∂ ∂ = If the problem is regular, i.e. is nonsingular, we proved, that according to the Implicit Function Theorem we obtain a unique function in the interval t ϵ (t0, tf ). ( )xxtF xx  ,, ( )pxtxx ,,:  = End of Proof of (b) (b) Here is the uniquely determined slope of the optimal trajectory at the point and is viewed as a function of t and . x ( ) Ω∈xt, x Theorem (continue) Note ( )       = ⋅ xxtFxtS xx ,,,  ( )xxtF x F p x    ,,: = ∂ ∂ =Using and ( ) ( ) ( )xtppxtSxxtF x F p xx ,,,,: =⇒== ∂ ∂ =    End Note
  • 130.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –26) Hamilton-Jacobi Equations (continue – 5) We defined, also, the function H (Hamiltonian) of the variables t, px, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )xtHpxtHxxxtFxxtFxxxtpxxtFH xtpppxtxx T x T , ~ ,,,,,,,,,,: ,,,: == ==+−=+−=    If we compare this expression with , we obtain:( ) xxxtFxxtFxtS T xt        −      = ⋅⋅ ,,,,, ( ) ( )pxtHxtSt ,,, −= If we compare with , we obtain:( )xxtF x F p x    ,,: = ∂ ∂ = ( )       = ⋅ xxtFxtS xx ,,,  ( ) pxtSx =, Therefore the Carathéodory's Lemma equation ( ) ( ) ( ) ( ) ( )[ ] 0,,,,min,,,, =−−=−−      ⋅ XxtSxtSXxtFxxtSxtSxxtF T xt X xt   can be rewritten as ( ) ( ) 0,,, =+ xt SxtHxtS Hamilton-Jacobi Equation William Rowan Hamilton (1805-1855) Carl Gustav Jacob Jacobi 1804-1851 The Hamilton-Jacobi Equation is a Partial Differential Equation in which is in general nonlinear.( )xtS , Return to Table of Content
  • 131.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –27) Jacobi’s Theorem Carl Gustav Jacob Jacobi 1804-1851 Let be a general solution of the Hamilton-Jacobi equation:( )α,, xtS ( ) ( ) 0,,,,,, =      ∂ ∂ + ∂ ∂ αα xt x S xtHxt t S depending on the parameters ( )n T ααα ,,1 = Assume also that ( ) 0,,detdet 2 ≠      ∂∂ ∂ = α α α xt x S S x Let n arbitrary constants.( )n T βββ ,,1 = The two-parameter family of solutions of the Hamilton Equations ( ) ( )βαβα ,,,,, tpptxx == x H td pd p H td xd ∂ ∂ −= ∂ ∂ = are obtained from ( ) βα α = ∂ ∂ ,, xt S together with ( )α,, xt x S p ∂ ∂ =
  • 132.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –28) Proof of Jacobi’s Theorem Carl Gustav Jacob Jacobi 1804-1851 Since det Sαx ≠ 0 using the Implicit Function Theorem (see Appendix 1) we can use to uniquely find as a function of( ) βα α = ∂ ∂ ,, xt S x ( )βα ,,t ( ) ( )βαβα α α ,,,, Im 0det txxxt S Theorem Function plicit S x =⇒= ∂ ∂ ≠ Substitute this back in and differentiate with respect to t( ) βαα =,, xtS ( )( ) 0,,,, 22 = ∂∂ ∂ + ∂∂ ∂ =      ∂ ∂ td xd x S t S txt S td d αα αβα α Now, take the partial differential of the Hamilton-Jacobi equation with respect to α ( ) ( ) 0,,,,,, 22 = ∂ ∂ ∂∂ ∂ + ∂∂ ∂ =      ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ xS H x S t S xt x S xtHxt t S αα α α α α If we use in this equation the fact that Sαt = Stα and , we obtainxSp = ( ) ( ) 0,,,,,, 22 = ∂ ∂ ∂∂ ∂ + ∂∂ ∂ =      ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ p H x S t S xt x S xtHxt t S αα α α α α therefore 0 2 =      ∂ ∂ − ∂∂ ∂ p H td xd x S α + - Since det Sαx ≠ 0 the previous equation is satisfied only if 0= ∂ ∂ − p H td xd
  • 133.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –29) Proof of Jacobi’s Theorem (continue – 1) Carl Gustav Jacob Jacobi 1804-1851 ( )α,, xt x S p ∂ ∂ =Let differentiate with respect to t ( ) p H xx S xt S td xd xx S xt S xt x S td d td pd ∂ ∂ ∂∂ ∂ + ∂∂ ∂ = ∂∂ ∂ + ∂∂ ∂ = ∂ ∂ = 2222 ,, α Now, take the partial differential of the Hamilton-Jacobi equation with respect to x ( ) ( ) 0,,,,,, 22 = ∂ ∂ ∂∂ ∂ + ∂ ∂ + ∂∂ ∂ =      ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ xS H xx S x H tx S xt x S xtH x xt t S x αα If we use in this equation the fact that Sαt = Stα and , we obtainxSp = x H p H xx S xt S ∂ ∂ −= ∂ ∂ ∂∂ ∂ + ∂∂ ∂ 22 x H td pd ∂ ∂ −= We obtain q.e.d. Return to Table of Content
  • 134.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –30) Example: Geometrical Optics and Fermat’s Principle ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += For the Geometrical Optics we obtained: Define ( ) [ ] ( ) [ ] 2/122 2/122 1 ,, : 1 ,, : zy zzyxn z F p zy yzyxn y F p z y       ++ = ∂ ∂ = ++ = ∂ ∂ = Adding the square of those two equations gives ( )( ) ( )2222222 1 zynzypp zy  +=+++ from which ( ) ( )222 2 22 1 zy ppn n zy +− =++  Substitute this equation in that of F ( ) ( )222 2 ,,,, zy zy ppn n ppzyxF +− = ( ) ( )222 222 zy z zy y ppn p z ppn p y +− = +− =   solve for zy ,
  • 135.
    SOLO Calculus of Variations CanonicalForm of Euler-Lagrange Equations (continuous –31) Example: Geometrical Optics and Fermat’s Principle (continue – 1) Define the Hamiltonian ( ) ( ) ( ) ( ) ( ) ( ) ( )222 222 2 222 2 222 2 ,, ,,,,:,,,, zy zy z zy y zy zyzyzy ppzyxn ppn p ppn p ppn n zpypppzyxFppzyxH +−−= +− + +− + +− −=++−=  The canonical equations are ( ) ( )222 222 zy z z zy y y ppn p p H xd zd z ppn p p H xd yd y +− = ∂ ∂ == +− = ∂ ∂ ==   ( ) ( )222 222 zy z zy y ppn z n n z H xd pd ppn y n n y H xd pd +− ∂ ∂ −= ∂ ∂ −= +− ∂ ∂ −= ∂ ∂ −= We recover the previous equations. We can also see that if n is constant, than H is not an explicit function of x, y, z and is also constant since from previous equations both py and pz are constant. Return to Table of Content
  • 136.
    136 [1] C. Carathéodory,“Calculus of Variations and Partial Differential Equations of the First Order”, Part I and Part II, Holden-Day Inc, 1965, English translation from German 1935 References SOLO Calculus of Variations [2]O. Bolza, “Lectures on the Calculus of Variations”, Dover Publications, New York, 1961, Republication of a work published by Univ. of Chicago 1904 [3] G.A. Bliss, “Lectures on the Calculus of Variations”, Univ. of Chicago Press, Chicago, 1946 [4] W.S. Kimball, “Calculus of Variations, by Parallel Displacement”, Butterworths Scientific Publications, 1952 [5] L.E. Elsgolc, , “Calculus of Variations”, Pergamon Press, Addison-Wesley, 1962 [6] I.M. Gelfand, S.V. Fomin, “Calculus of Variations”, Prentice-Hall, 1963 [7] G. Leitmann, “Calculus of Variations and Optimal Control, An Introduction”, Plenum Press, 1981 [8] H. Sagan, “Introduction to Calculus of Variations”, Dover Publication, New York, 1969 [9] D. Lovelock, H. Rund, “Tensors Differential Forms, and Variational Principles”, Dover Publication, New York, 1975, 1989 [10] J.L. Troutman, “Variational Calculus with Elementary Convexity”, Springer-Verlag, 1983 [11] R. Weinstock, “Calculus of Variations with Applications to Physics and Engineering”, Dover Publication, New York, 1952, 1974
  • 137.
  • 138.
    138 SOLO References (continue –1) Return to Table of Content Calculus of Variations
  • 139.
    February 20, 2015139 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA
  • 140.
    SOLO Appendix: Useful MathematicalTheorems Calculus of Variations The following mathematical theorems are useful in the Calculus of Variations: •Implicit Function Theorem •Heine-Borel Theorem •Ordinary Differential Equations Theorems •Euler-Lagrange Ordinary Differential Equations Theorems •Partial Differential Equations of the First Order Theorems
  • 141.
    SOLO Appendix 1: ImplicitFunctions Theorem Calculus of Variations Let continuous functions on a domain D of the parameters:( ) ( ) 0,,:, 1 == T nffuxf  ( ) n T n Rxxx ∈= ,,: 1  ( ) m T m Ruuu ∈= ,,: 1  having continuous first partial derivatives                 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ =                 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ = m nn m u n nn n x u f u f u f u f u f f x f x f x f x f x f f ,, ,, : ,, ,, : 1 1 1 1 1 1 1 1       Consider an interior point of the domain of definition of for which( )00 ,uxP ( )uxf , ( ) ( ) ( ) 0, 00 =uxf and the following Jacobian is nonzero: ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) 0 ,, ,, 00 00 00 ,1 1 1 1 , , ≠ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ = uxn nn n ux uxx x f x f x f x f x f f   
  • 142.
    SOLO Appendix 1: ImplicitFunctions Theorem (continue – 1) Calculus of Variations Then there exists a certain neighborhood of this point a unique system of continuous functions that satisfies the conditions: ( ) ( ) ( ) { }δδ ≤−= 00 : uuuuN ( )ux ϕ= ( ) ( ) ( )00 ux ϕ=a ( )( ) ( ) ( )0 0:, uNuuuf δϕ ∈∀=b u∂ ∂ϕc exists in the same neighborhood, are continuous and are found by solving: ( )( ) ( ) ( ) 0 ,, : , = ∂ ∂ + ∂ ∂ ∂ ∂ = u uxf ux uxf ud uufd ϕϕ If are of class C(p) in than is also of class C(p) .u( )uxf , ( )uϕ Proof Existenc e ( ) ( ) 0,:, 1 2 ≥= ∑= n i i uxfuxFDefine the scalar and the neighborhoods: ( ) ( ) ( ) { } ( ) ( ) ( ) { }    ≤−= ≤−= δ ρ δ ρ 00 00 : : uuuuN xxxxN ( )0 u ( )0 x ( ) δ+0 u( ) δ−0 u ( ) ρ+0 x ( ) ρ−0 x ( ) σ−0 x ( ) σ+0 x u ( ) 0, =uxf D
  • 143.
    SOLO Appendix 1: ImplicitFunctions Theorem (continue – 2) Calculus of Variations Proof of Existence (continue – 1) ( )0 u ( )0 x ( ) δ+0 u( ) δ−0 u ( ) ρ+0 x ( ) ρ−0 x ( ) σ−0 x ( ) σ+0 x u ( ) 0, =uxf D Let be the set of boundary points of defined as( ) ( )0 xNρ∂ ( ) ( )0 xNρ ( ) ( ) ( ) { }ρρ =−=∂ 00 : xxxxN ( ) ( ) ( ) 0, 00 =uxfAccording to we have ( ) ( ) ( ) ( ) ( ) ( ) 0,:, 1 00200 == ∑= n i i uxfuxF Let choose such thatu ( ) ( )0 uNu δ∈ Since is continuous and compact (bounded and closed) in and for and chosen . ( )uxf , ( ) ( )0 xNρ ( ) ( )0 uNδ ( ) ( )0 xNx ρ∂∈ ( ) ( )0 uNu δ∈ ( ) ( ) 0,, 1 2 >= ∑= n i i uxfuxF and attains its minimum value m on ( ) ( )0 xNρ∂ ( )( ) ( )( ) ( )uxFm uNu xNx ,inf 0 0 δ ρ ∈ ∂∈ = Since we can find a σ < ρ such that and( ) ( ) ( ) 0, 00 =uxF ( ) ( ) ( ) ( )00 xNxN ρσ ⊂ ( )( )( )( ) ( ) ( ) ( ) ( )0 2 , 2 ,inf 0 0 xNxfor m uxF m uxF uNu xNx σ δ σ ∂∈>→= ∈ ∂∈
  • 144.
    SOLO Appendix 1: ImplicitFunctions Theorem (continue – 3) Calculus of Variations Proof of Existence (continue – 2) ( )0 u ( )0 x ( ) δ+0 u( ) δ−0 u ( ) ρ+0 x ( ) ρ−0 x ( ) σ−0 x ( ) σ+0 x u ( ) 0, =uxf D Also since we can diminish σ < ρ such that( ) 0, 00 =uxF ( ) ( ) ( )0 2 , xNinsidexfor m uxF σ< Since on the boundary of and , at some point inside , the scalar attains its minimum inside and ( ) 2 , m uxF > ( ) ( )0 xNσ ( ) 2 , m uxF < ( ) ( )0 xNσ ( )uxF , ( ) ( )0 xNσ ( ) ( ) ( ) ( ) ( ) 0 , , , ,, 1 11 1 =         ∂ ∂ ⋅= ∂ ∂ ⋅= ∑ ∑∑∑ = == = n j j n i j i i n i n j j j i i dx x uxf uxfdx x uxf uxfuxFd This must hold for each , therefore ( ) ( )0 xNxd j σ∈ ( ) ( ) nj x uxf uxf n i j i i ,,10 , , 1 =∀= ∂ ∂ ⋅∑= ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0, , , , ,, ,, 1 1 1 1 1 =      ∂ ∂ =                           ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ uxf x uxf uxf uxf x uxf x uxf x uxf x uxf n n nn n     or ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) 0 ,, ,, 00 00 00 ,1 1 1 1 , , ≠ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ = uxn nn n ux uxx x f x f x f x f x f f    Since it follows that the previous equality is possible only if: ( ) 0, =uxf We proved that for every , exist at least one such that . ( ) ( )0 uNu δ∈ ( ) ( )0 xNx σ∈ ( ) 0, =uxf
  • 145.
    SOLO Appendix 1: ImplicitFunctions Theorem (continue – 4) Calculus of Variations Proof of Uniqueness ( )0 u ( )0 x ( ) δ+0 u( ) δ−0 u ( ) ρ+0 x ( ) ρ−0 x ( ) σ−0 x ( ) σ+0 x u ( ) 0, =uxf D Suppose that for a given we have two values such that . ( ) ( )0 uNu δ∈ ( ) ( ) ( ) ( )021 , xNxx σ∈ ( ) ( ) ( ) ( ) 0,, 21 == uxfuxf We can write this equation as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )uxxxfuxxxfuxfuxf niniii ,,,,,,,,,, 11 2 1 1 22 2 2 1 12  −=− ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )uxxxxfuxxxxf uxxxxfuxxxxf uxxxxfuxxxxf uxxxxfuxxxxf nini nini nini nini ,,,,,,,,,, ,,,,,,,,,, ,,,,,,,,,, ,,,,,,,,,, 11 3 1 2 1 1 21 3 1 2 1 1 11 3 1 2 1 1 22 3 1 2 1 1 22 3 1 2 1 1 22 3 2 2 1 1 22 3 2 2 1 1 22 3 2 2 2 1      −+ −+ −+ −= ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )12 1 0 1221211 1 211 1 221 1 ,,,,, ,,,,,,,,,, jjijjjnjjj j i njinji xxBxxduxxxxx x f uxxxfuxxxf −=−−+ ∂ ∂ = − ∫ θθ  We can write where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ −+ ∂ ∂ = +− 1 0 22 1 1211 1 1 1 21 ,,,,,,,:,, θθ duxxxxxxx x f uxxB njjjjj j i ij  We can see that ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ux x f duxxxxx x f uxxB j i njjj j i ij ,,,,,,,,,, 1 1 0 22 1 11 1 1 1 11 ∂ ∂ =∫ ∂ ∂ = +− θ
  • 146.
    SOLO Appendix 1: ImplicitFunctions Theorem (continue – 5) Calculus of Variations Proof of Uniqueness (continue – 1) ( )0 u ( )0 x ( ) δ+0 u( ) δ−0 u ( ) ρ+0 x ( ) ρ−0 x ( ) σ−0 x ( ) σ+0 x u ( ) 0, =uxf D Therefore we can write ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,, ,,,,,,,,,, 1 1221 11 2 1 1 22 2 2 1 12 =−= −=− ∑= n j jjij niniii xxuxxB uxxxfuxxxfuxfuxf  Those equations hold for i= 1,2,…,j, therefore we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) 0,, 1221 12 1 2 2 2 1 1 2 1 21 22221 11211 =−=               − − −             xxuxxB xx xx xx BBB BBB BBB ij nn nnnn n n      Since we have ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )( ) ( ) ( )( ) 0 ,, ,, ,, 00 00 00 ,1 1 1 1 , 000 , ≠ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ == uxn nn n ux ijuxx x f x f x f x f x f uxxBf    and is continuous in , if we choose σ small enough we can assure that and the equation is satisfied only if . ( )uxfx , ( )ux, ( ) ( ) ( )[ ] 0,det 21 ≠xxBij ( ) ( ) ( )[ ] ( ) ( ) ( ) 0, 1221 =− xxxxBij ( ) ( )12 xx = This proves that for every , exist at least one (σ small enough) such that we can write . ( ) ( )0 uNu δ∈ ( ) ( )0 xNx σ∈ ( ) 0, =uxf ( )ux ϕ=
  • 147.
    SOLO Appendix 1: ImplicitFunctions Theorem (continue – 6) Calculus of Variations Proof of Continuity ( )0 u ( )0 x ( ) δ+0 u( ) δ−0 u ( ) ρ+0 x ( ) ρ−0 x ( ) σ−0 x ( ) σ+0 x u ( ) 0, =uxf D By taking the derivative of with respect to , we obtain( )( )uuf ,ϕ u ( )( ) ( ) ( ) 0 ,,, = ∂ ∂ + ∂ ∂ ∂ ∂ = u uxf ux uxf ud uufd ϕϕ from which we can see that exists in the same neighborhood, and are continuous because and exist and are continuous. u∂ ∂ϕ ( ) x uxf ∂ ∂ , ( ) u uxf ∂ ∂ , If are of class C(p) in than is also of class C(p) .u( )uxf , ( )uϕ q.e.d.
  • 148.
    SOLO Appendix 1: ImplicitFunctions Theorem (continue – 7) Calculus of Variations Extension of the Implicit Functions Theorem to all Domain of Definition of ( ) ( ) 0,,:, 1 == T nffuxf  We found the unique function that satisfies the conditions:( )ux ϕ= ( ) ( ) ( )00 ux ϕ= ( )( ) ( ) ( )0 0, uNuuuf δϕ ∈∀= ( )uxf ,We want to extend this result to the domain D where the functions are defined provided that ( ) ( ){ }0,&,0 ,, ,, 1 1 1 1 =∈∀≠ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ = uxfDux x f x f x f x f x f f n nn n x    a) b) D is compact (closed and bounded) For this purpose we use the Heine-Borel Theorem ( see Appendix 2 for proof): A compact domain S can be covered by a given finite number of open covering sub-domains.
  • 149.
    SOLO Appendix 1: ImplicitFunctions Theorem (continue – 8) Calculus of Variations Extension of the Implicit Functions Theorem to all Domain of Definition of (continue – 1) ( ) ( ) 0,,:, 1 == T nffuxf  We are using the following procedure: Choose( ) ( ) ( )12 uNu δ∈ Find the unique ( ) ( ) ( ) ( ) ( ) ( )01 1 1 1 xNux σ ϕ ∈= 2) Define a neighborhood ( ) ( )1 uNδ 1) Choose Find the unique ( ) ( ) ( ) ( ) ( ) ( )12 2 2 2 xNux σ ϕ ∈= Choose( ) ( ) ( )1− ∈ NN uNu δ N) Define a neighborhood ( ) ( )N uNδ Find the unique ( ) ( ) ( ) ( ) ( ) ( )1 2 − ∈= NNN xNux N σ ϕ ( )0 u ( )0 x ( )1 u ( ) 0, =uxf D ( )1 x According to Heine-Borel Theorem the compact domain D for which is covered by a finite number N of sets. ( ) 0, =uxf Return to Table of Content
  • 150.
    150 Heinrich Eduard Heine ( 1821- 1881) Félix Édouard Justin Émile Borel (1871 –1956) Appendix 2: Heine–Borel Theorem SOLO Calculus of Variations A compact domain S can be covered by a given finite number of open covering sub-domains. Proof of Heine–Borel Theorem Both S and the sub-domains Ti are given beforehand, since it is no hard to pick out a single open interval which completely covers a bounded set S. Let S be contained in the interval -N ≤ x ≤ N (S is bounded). Now divide this closed interval into two equal intervals (1) -N ≤ x ≤ 0 (2) 0 ≤ x ≤ N Any element x of S will belong to either (1) or (2). If the theorem is false, it will not be possible to cover the points of S in both (1) and (2), by a finite number of sub-domains of T, so the points of S in either (1) or (2) require an infinite covering. Assume that the elements of S in (1) still require an infinite covering, We subdivide this interval into two equal parts and repeat the above argument. In this way we construct a sequence of sets such that each Si is closed and bounded and such that the diameters of the  ⊃⊃⊃⊃⊃ i SSSS 321 0lim →∞→ ii S
  • 151.
    151 Heinrich Eduard Heine ( 1821- 1881) Félix Édouard Justin Émile Borel (1871 –1956) Appendix 2: Heine–Borel Theorem (continue – 1) SOLO Calculus of Variations A compact domain S can be covered by a given finite number of open covering sub-domains. Proof of Heine–Borel Theorem (continue – 1) Because the sub-domains are nested there exists a unique point P which is contained in each Si. Since P is in Si, one of the open intervals of T, say Tp, will cover P. This Tp has a finite nonzero diameter so that eventually one of the Si will be contained in Tp, since . But by assumption all the elements of this Si require an infinite number of the sub-domains in T to cover them. This is a direct contradiction to the fact that a single Tp covers them. Hence our original assumption is wrong, and the theorem is proved. 0lim → ∞→ i i S Return to Table of Content
  • 152.
    152 Appendix 3: OrdinaryDifferential Equations (ODE) SOLO Calculus of Variations is a normal system of ordinary differential equations. Well-Posed Problems A differential equation problem is well-posed if: •A solution exists. •The solution is unique. •The solution depends continuously on the initial values. The well-posedness requires proving theorems of existence (there is a solution), uniqueness (there is only one solution), and continuity (the solution depends continuously on the initial value). ( ) givenConditionsInitialn nitxxxf td xd nii i ,,1,,,,,1  ==
  • 153.
    153 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 1) SOLO Calculus of Variations ( ) givenConditionsInitialn nitxxxf td xd nii i ,,1,,,,,1  == is a normal system of ordinary differential equations. Definitions: • Solution of an ODE means an explicit solution xi = φi (t) defined in a region R. • A neighborhood of a point is defined as a sphere contained this point as is center, satisfying (r some positive constant) ( )00 ,tx ( ) ( ) 22 0 2 0 rttxx <−+− • A point is an interior point of R if it contains a neighborhood that is wholly contained in R. It is an exterior point of R if it contains a neighborhood that doesn’t contain any point of R. ( )00 ,tx • A point is a boundary point of R if every neighborhood has both interior and exterior points in R. ( )00 ,tx ( ) ( )txxxtx ni ,,,,,:, 1 =• Neighborhood Interior Point Boundary Point
  • 154.
    154 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 2 SOLO Calculus of Variations ( ) givenConditionsInitialn txf td xd ,= is a normal system of ordinary differential equations. Definitions continue – 1): Interior Point Boundary Point • Limit of a Sequence is equivalent to for each ε >0 exist an integer N such that . ( ) ( )00 ,, txtx i ii ∞→ ⇒ ( ) ( ) Nittxx ii >∀<−+− 22 0 2 0 ε • Limit of a Function in a region R. We say that if for every sequence in R such that converges to the same limit A. ( )txf , ( ) ( ) ( ) Atxf txtx = → ,lim 00 ,, ( ) ( )00 ,, txtx i ii ∞→ ⇒ ( )txf , • A Function is Continuous at a point if( )txf , ( )00 ,tx ( ) ( ) ( ) ( )00 ,, ,,lim 00 txftxf txtx = → • Uniform Convergence A sequence Uniform Converge to a limit if for each positive ε there is an integer N, independent on , such that ( )ii tx , ( )tx, x ( ) ( ) Nitxtxi >∀<− ε
  • 155.
    155 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 3) SOLO Calculus of Variations Derivation of a Solution of the Ordinary Differential Equations. ( ) givenConditionsInitialntxf td xd ,= Theorem I (Existence) (Cauchy- Peano)If the functions are continuous in a closed and bounded region R of n+1 dimensional space , then through each interior point of the region there exists at least one continuously derivable curve which is defined in an interval |t-t0| < a ( )txf , ( )tx, ( )00 ,tx ( )txx = Giuseppe Peano 1858 - 1932 Augustin-Louis Cauchy 1789 –1857 Pro ofSince R is closed and bounded, the functions are uniformly bounded in R and there exists a positive number M such that ( )txf , ( ) ( ) RtxniMtxfi ∈∀=< ,,,1,  ( ) ( ) ( ) ( ) ( ) ( )1111 121112 010001 , , , −−−− −+= −+= −+= NNNNNN tttxfxx tttxfxx tttxfxx  A solution can be build by dividing the interval |t-t0| < a in small intervals t0 < t1 <… <tj<… < tN < t0+a & |tj+1-tj| <δ. Start from and perform( )00 ,tx Interior Point
  • 156.
    156 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 4) SOLO Calculus of Variations Derivation of a Solution of the Ordinary Differential Equations (continue – 1). Proof (continue -1)This construction defines a polygon, that for small enough, approximate a solution ( )txx = ( ) ( )∫+= t dxfxtx 0 0 , ττ We can see that ( ) ( ) ( ) ( ) ( ) ( ) ( )001211 01211012110 ttMttMttMttM xxxxxxxxxxxxxx jjjjj jjjjjjjjj −=−++−+−≤ −++−+−≤−++−+−=− −−− −−−−−−   Therefore the solution is bounded in the region ( )00 ttMxx −≤− We can obtain also this result from ( ) ( ) ( ) ( )0 000 0 ,, ttMdMdxfdxfxtx ttt −=≤≤=− ∫∫∫ τττττ Interior Point Slope +M Slope -M The Theorem that proves only “Existence”, not “Uniqueness”, was first discovered by Giuseppe Peano in 1890. This solution is due to Augustin Cauchy and the polygon is called “Cauchy Polygon”.
  • 157.
    157 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 5) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Rudolf Lipschitz (1832 – 1903) ( ) givenConditionsInitialn txf td xd ,= To obtain Uniqueness of a Solution of the Ordinary Differential Equations we need in addition to the condition the condition ( ) ( ) RtxniMtxfi ∈∀=< ,,,1,  ( ) ( ) ( ) ( ) Rtxtxconstknixxktxftxf iiii ∈∀==−<− ,~&,.,,,1~,~,  Lipschit z Conditio nUnder suitable hypothesis the Mean Value Theorem, gives ( ) ( ) ( ) ( ) xxxx x tf txftxf i ii ~~, ,~, ≥≥− ∂ ∂ =− η η This means that we can replace the Lipschitz Condition with the more restrictive condition ( ) ( ) Rtxconstknik x txf ii i ∈∀==≤ ∂ ∂ ,.,,,1 , 
  • 158.
    158 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 6) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. ( )txf td xd ,= Theorem (Existence and Uniqueness) (Picard–Lindelöf Theorem) In a Bounded region R , let be continuous and satisfying( )txf , ( ) ( ) RtxniMtxf ii ∈∀=< ,,,1,  as well as ( ) ( ) ( ) ( ) Rtxtxconstknixxktxftxf iiii ∈∀==−<− ,~&,.,,,1~,~,  or . ( ) ( ) Rtxconstknik x txf ii i ∈∀==≤ ∂ ∂ ,.,,,1 ,  The ODE has one, and only one, solution containing the internal point . The solution lies in the shadow region (defined by ) and can be extended to the right and the left of until it meets the boundary of R. ( )txx = ( )00 ,tx Interior Point Slope +M Slope -M ( ) ii Mtxf <, 0x Lipschit z Conditio n Those conditions are “sufficient” but not “necessary” for existence and uniqueness of solutions. There are cases when those conditions are not satisfied and a unique solution exists.
  • 159.
    159 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 7) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (Picard–Lindelöf Theorem) We introduce the Picard Successive Approximations, called also the Picard Iterations, after the French Mathematician Charles Picard: Slope +M Slope -M Shaded Region Charles Émile Picard 1856 - 1941 ( ) ( ) ( )txf td xd x txf td xd x txf td xd x n n n ,:' ,:' ,:' 1 1 2 2 0 1 1 −== == ==  Ernst Leonard Lindelöf 1870 - 1946 Let show first that all those curves are defined for a ≤ t ≤ b and lie in the Shaded Region defined by . ( )txfx ii ,' 1−= 00 ttxx −≤− Assume that the graph is defined for a ≤ t ≤ b and lies in the Shaded Region, then for a ≤ t ≤ b ; hence , and ( )txx n= ( ) mtxf n ≤, ( ) mtxfx nn ≤=+ ,' 1 ( ) ( ) 0110 00 '' ttmdttxdttxxx t t n t t n −≤≤=− ∫∫ ++ Therefore lies in the Shadow Region.( )txx n=
  • 160.
    160 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 8) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (continue – 1) Slope +M Slope -M Shaded Region Let estimate now the difference between two successive approximations. Define: ( ) ( ) ( ) ( ) ( ) ( ) 0&: 0 01 0 001 =−=−= −−  txtxtwtxtxtw nnnnnn ( ) ( )txfxandtxfx nnnn ,',' 11 −+ ==We have Subtract and take the norm ( ) ( ) 111 ,,'' −−+ −≤−=− nn Lipschitz nnnn xxktxftxfxx or ( ) ( ) ( )twktwtw td d n Lipschitz nn ≤= ++ 11 ' ( ) ( ) ( ) ( ) 00000011 00 , ttmtdmtdtxftxtxtw t t t t −=≤=−= ∫∫Let compute: ( ) ( ) ( )twktw td d twk nnn ≤≤− +1 ( ) ( ) ( ) 00121 1 ttmktwktw td d twk n −=≤≤−⇒ = Integrating from t0 to t (since the integrand doesn’t change sign the inequalities are preserved after integration): ( ) ( ) ( ) 22 2 0 0000221 2 0 0 1 0 tt mkttmktwtwtwk tt mk t t n − =−≤−≤−= − −⇒ ∫ =
  • 161.
    161 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 9) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (continue – 2) Slope +M Slope -M Shaded Region ( ) ( ) ( )twktw td d twk nnn ≤≤− +1Start from: ( ) ( ) 22 2 0 0 0 022 2 0 0 1 tt kmtwtw tt km n − ≤−≤ − −⇒ =  ( ) ( ) !3!3 3 02 0 0 033 3 02 0 2 tt kmtwtw tt km n − ≤−≤ − −⇒ =  ( ) ( ) !! 01 0 0 0 01 0 1 n tt kmtwtw n tt km n n nn n n n − ≤−≤ − −⇒ −− −  ( ) ( ) ( ) 22 2 02 0232 2 02 0 2 tt kmtwktw td d twk tt km n − =≤≤−= − −⇒ = Integration Induction
  • 162.
    162 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 10) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (continue – 2) Slope +M Slope -M Shaded Region We obtained: ( ) ( ) !! 01 0 0 0 01 0 1 n tt kmtwtw n tt km n n nn n n n − ≤−≤ − −⇒ −− −  Therefore: ( ) ( ) ! 00 n ttk k m tw n n − ≤ Let compute: ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ] 11012110 wwwtxtxtxtxtxtxtxtx nnnnnnn +++=−++−+−=− −−−−  ( ) ( ) ( ) ( )1 ! 00 1 00 110 −≤ − ≤+++≤− − = − ∑ ttk n i i nnn e k m i ttk k m wwwtxtx  ( ) ( ) ( )1lim 00 0 −≤− − ∞→ ttk n n e k m txtxLet take the limit n→∞ of the previous expression: Therefore the limit : ( ) ( )txtxn n = ∞→ lim exists as an Uniform Limit on the interval a ≤ t ≤ b. Using again Lipschitz Condition: ( ) ( ) ( ) ( ) btatxtxktxftxf nn ≤≤−≤− ,, we obtain: ( ) ( ) btatxftxf n n ≤≤= ∞→ ,,lim q.e.d. Existence
  • 163.
    163 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 11) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (continue – 3) Slope +M Slope -M Shaded RegionUniqueness: Assume the existence of another solution, such that: ( ) ( ) ( )00 ~,,~ ~ txtxtxf td xd == ( ) ( ) ( )001 ,, txtxtxf td xd nn n == −Use the sequence: ( ) ( ) ( ) 000 00 ,~~ ttmtdmtdtxftxtx t t t t −=≤=− ∫∫Compute: Subtracting those equations and taking the norm, we obtain: ( ) ( ) 11 ~,,~ ~ −− −≤−=− n Lipschitz n n xxktxftxf td xd td xd 00 1 1 ~ ~ ttkmxxk td xd td xdn −≤−≤−⇒ =
  • 164.
    164 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 12) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (continue – 4) Slope +M Slope -M Shaded Region Uniqueness (continue – 1): 0 1 0 1 ~ ttkm td xd td xd ttkm n −≤−≤−−⇒ = ( ) ( ) 2 ~ 2 2 0 1 2 0 1 tt kmtxtx tt km n − ≤−≤ − −⇒ = 11 ~ ~ ~ −− −≤−≤−− n n n xxk td xd td xd xxk Integration We obtained: 2 ~ ~ ~ 2 2 02 1 2 1 2 02 2 tt kmxxk td xd td xd xxk tt km n − ≤−≤−≤−−≤ − −⇒ = Integration !3 ~~~ !3 3 02 121 3 02 2 tt kmxxkxxxxk tt km n − ≤−≤−≤−−≤ − −⇒ = ( ) ( ) ( ) ( ) ( ) ( )!1 ~ !1 1 0 1 0 + − ≤−≤ + − −⇒ ++ n ttk k m txtx n ttk k m n n n n Induction
  • 165.
    165 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 13) SOLO Calculus of Variations Uniqueness of a Solution of the Ordinary Differential Equations. Proof of Theorem (Existence and Uniqueness) (continue – 5) Slope +M Slope -M Shaded Region Uniqueness (continue – 1): We obtained: ( ) ( ) ( ) ( )!1 ~ 1 0 + − ≤− + n ttk k m txtx n n We see that: ( ) ( ) 0~ ∞→ ⇒− n n txtx Therefore the limit : ( ) ( )txtxn n ~lim = ∞→ is Unique and exists as an Uniform Limit on the interval a ≤ t ≤ b. q.e.d.
  • 166.
    166 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 14) SOLO Calculus of Variations Continuous Dependence of Solution of the Ordinary Differential Equations. Thomas Hakon Grönwall (1877 – 1932) We want to show that the Solution of the Ordinary Differential Equations depends continuously on the Initial Values. For this let state the: Grönwall Inequality Let u and v be continuous function satisfying u (x) > 0 and v (x) ≥ 0 on [a,b]. Let c ≥ 0 be a constant. If ( ) ( ) ( ) bxatdtvtucxv x a ≤≤+≤ ∫ , then ( ) ( ){ } bxatdtucxv x a ≤≤≤ ∫ ,exp Prof of Grönwall Inequality : First assume c > 0 and define ( ) ( ) ( ) bxatdtvtucxV x a ≤≤+= ∫ ,: Then V (x) ≥ v (x), and since u and v are nonnegative, V (x) ≥ V (a) = c on [a,b]. Moreover, V’(x) = u (x) v (x0 ≤ u (x) V (x). Dividing by V (x), we get
  • 167.
    167 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 15) SOLO Calculus of Variations Continuous Dependence of Solution of the Ordinary Differential Equations. Thomas Hakon Grönwall (1877 – 1932) Prof of Grönwall Inequality (continue – 1): If we take c → 0+, we get v (x) ≤ 0, which is the same result obtained from Grönwall Inequality with c = 0. ( ) ( ) ( ) bxa xV xV xu ≤≤≥ , ' Integrate both sides of this equation, from a to x ( ) ( ) ( ) ( ) ( )( )cxVsVsd sV sV sdsu x a x a x a /lnln ' ==≥ ∫∫ Since logarithmic and exponential function are increasing function with their argument, we can resolve and preserve the inequality ( ) ( ){ } bxasdsucxV x a ≤≤≤ ∫ ,exp Since V (x) ≥ v (x) we proved that .( ) ( ){ } 0&,exp >≤≤≤ ∫ cbxatdtucxv x a q.e.d. Grönwall Inequality
  • 168.
    168 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 16) SOLO Calculus of Variations Continuous Dependence of Solution of the Ordinary Differential Equations. Let use Grönwall Inequality to prove the following Continuous Dependence of ODE on Initial Value ( ) ( ) hk exxtxtx 0000 ~,,~ −≤−φφ ( )tx ,~ 0φ where k is any positive constant such that for all . Moreover as approaches , the solution approaches uniformly in . kxf ≤∂∂ / ( ) 0, Rtx ⊂ 0x0 ~x ( )tx ,0φ htt ≤− 0 Let continuous functions on an open rectangle containing the point . Assume that for all sufficiently close to , the solution of the ODE exists on the interval and the graph lies within a closed region . Then, for ( ) xfandtxf ∂∂ /, ( ){ }dxcbtatxR <<<<= ,:, ( )00 ,tx 0 ~x 0x ( )tx ,~ 0φ htt ≤− 0RR ⊂0 htt ≤− 0
  • 169.
    169 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 17) SOLO Calculus of Variations Continuous Dependence of Solution of the Ordinary Differential Equations. Proof of Continuous Dependence of ODE on Initial Value: The Solution of ODE satisfies the Integral Equation:( )tx ,0φ ( ) ( )( ) httsdssxfxtx t t ≤−+= ∫ 0000 ,,,, 0 φφ Similarly the Solution of ODE satisfies the Integral Equation:( )tx ,~ 0φ ( ) ( )( ) httsdssxfxtx t t ≤−+= ∫ 0000 ,,,~~,~ 0 φφ Subtracting the second equation from the first gives: ( ) ( ) ( )( ) ( )( )[ ]∫ −+−=− t t sdssxfssxfxxtxtx 0 ,,~,,~,~, 00000 φφφφ Assume that t > t0 and tacking the norm of both sides ( ) ( ) ( )( ) ( )( )∫ −+−≤− t t sdssxfssxfxxtxtx 0 ,,~,,~,~, 00000 φφφφ kxf ≤∂∂ /Since or satisfies the Lipschitz Condition( )txf , ( )( ) ( )( ) ( ) ( )sxsxkssxfssxf ,~,,,~,, 0000 φφφφ −≤− ( ) ( ) ( ) ( )∫ −+−≤− t t sdsxsxkxxtxtx 0 ,~,~,~, 00000 φφφφWe obtain:
  • 170.
    170 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 18) SOLO Calculus of Variations Continuous Dependence of Solution of the Ordinary Differential Equations. Proof of Continuous Dependence of ODE on Initial Value (continue – 1): ( ) ( ) ( ) ( )∫ −+−≤− t t sdsxsxkxxtxtx 0 ,~,~,~, 00000 φφφφWe obtained: Use Grönwall Inequality: ( ) ( ) ( ) ( ) ( ) 0&0,0 , ≥≤≤≥> ≤≤+≤ ∫ cbxaonxvxu bxatdtvtucxv x a ( ) ( ){ } bxatdtucxv x a ≤≤≤ ∫ ,exp ( ) ( ) ( ) ( ) 0~:&0,~,:,0: 000 ≥−=≥−=>= xxctxtxtvktu φφwith: ( ) ( ) { } ( ) hk htt ttkt t exxexxsdkxxtxtx ~~exp~,~, 00000 0 0 0 −≤−=−≤− ≤− − ∫φφ For t < t0, we can use t0 – t instead of t. ( )tx ,~ 0φWe see that as approaches , the solution approaches uniformly in . 0 ~x 0x ( )tx ,0φ htt ≤− 0 q.e.d.
  • 171.
    171 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 19) SOLO Calculus of Variations Let use Grönwall Inequality to prove the following Continuous Dependence on of Solutions of ODE ( ) ( ) ( ) RtxtxftxF ⊂∀≤− ,,,, ε Continuous Dependence on of Solution of the Ordinary Differential Equations.( )txf , ( )txf , Let continuous functions on an open rectangle containing the point . Let be continuous in R and assume that ( ) xfandtxf ∂∂ /, ( ){ }dxcbtatxR <<<<= ,:, ( )00 ,tx ( )txF , Let be the solution to the initial value problem:( )tx,φ ( ) ( ) 00,, xtxtxf td xd == Let be the solution to the initial value problem:( )tx,ψ ( ) ( ) 00,, xtxtxF td xd == RR ⊂0 Assume both solutions exist on [t0 – h, t0 +h] and their graphs lie in a closed region . Then for |t-t0| ≤ h, ( ) ( )txandtx ,, ψφ ( ) ( ) hk ehtxtx εψφ ≤− ,, where k is any positive constant such that for all . Moreover as approaches uniformly on R, that is, as ε→0+, the solution approaches uniformly in . kxf ≤∂∂ / ( ) 0, Rtx ⊂ fF htt ≤− 0 ( )tx,ψ ( )tx,φ
  • 172.
    172 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 20) SOLO Calculus of Variations Continuous Dependence on of Solution of the Ordinary Differential Equations.( )txf , Proof of Continuous Dependence on of Solutions of ODE( )txf , ( ) ( )( ) httsdssxfxtx t t ≤−+= ∫ 00 ,,,, 0 φφ The integral representation of , is( ) ( )txandtx ,, ψφ ( ) ( )( ) httsdssxFxtx t t ≤−+= ∫ 00 ,,,, 0 ψψ Subtracting those equations gives ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )[ ] ( )( ) ( )( )[ ] httsdssxFssxfsdssxfssxf sdssxFsdssxftxtx t t t t t t t t ≤−−+−= −=− ∫∫ ∫∫ 0 00 00 ,,,,,,,, ,,,,,, ψψψφ ψφψφ Applying the norm and using the Triangle Inequality, we obtain ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) httsdssxFssxfsdssxfssxftxtx t t t t ≤−−+−≤− ∫∫ 0 00 ,,,,,,,,,, ψψψφψφ ( )( ) ( )( ) ( ) ( )sxsxkssxfssxf kxf Lipschitz or ,,,,,, / ψφψφ −≤− ≤∂∂ use and ( )( ) ( )( ) εψψ ≤− ssxFssxf ,,,, ( ) ( ) ( ) ( ) ( ) ( ) htthsdsxsxksdsdsxsxktxtx t t t t t t ≤−+−≤+−≤− ∫∫∫ 0 000 ,,,,,, εψφεψφψφ
  • 173.
    173 Appendix 3: OrdinaryDifferential Equations (ODE) (continue – 21) SOLO Calculus of Variations Return to Table of Content We obtained: Use Grönwall Inequality: ( ) ( ) ( ) ( ) ( ) 0&0,0 , ≥≤≤≥> ≤≤+≤ ∫ cbxaonxvxu bxatdtvtucxv x a ( ) ( ){ } bxatdtucxv x a ≤≤≤ ∫ ,exp ( ) ( ) ( ) ( ) hctxtxtvktu εψφ =≥−=>= :&,0,,:,0:with: ( ) ( ) { } ( ) hk htt ttkt t ehehsdkhtxtx εεεψφ ≤− − ≤=≤− ∫ 0 0 0 exp,, Continuous Dependence on of Solution of the Ordinary Differential Equations.( )txf , Proof of Continuous Dependence on of Solutions of ODE (continue – 1)( )txf , ( ) ( ) ( ) ( ) httsdsxsxkhtxtx t t ≤−−+≤− ∫ 0 0 ,,,, ψφεψφ In particular as approaches uniformly on R, that is, as ε→0+, the solution approaches uniformly in . fF htt ≤− 0 ( )tx,ψ ( )tx,φ q.e.d.

Editor's Notes

  • #12 http://curvebank.calstatela.edu/brach
  • #151 http://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem http://en.wikipedia.org/wiki/Eduard_Heine http://en.wikipedia.org/wiki/%C3%89mile_Borel
  • #152 http://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem http://en.wikipedia.org/wiki/Eduard_Heine http://en.wikipedia.org/wiki/%C3%89mile_Borel
  • #153 http://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem http://en.wikipedia.org/wiki/Eduard_Heine http://en.wikipedia.org/wiki/%C3%89mile_Borel
  • #154 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #155 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #156 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #157 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #158 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #159 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #160 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 – 158 Picard, É., “Mémoires sur la théorie des équatons aux dérivées partielles et le méthode des approximations successives”, J. math. Pures appl., (4), 6 (1890), pp. 145-210 Lindelöf, E., “Démonstration élémentaire de l’existence des intégrales d’un système d’équations différentielles ordinaires”, Acta Soc. Sci. Fenn., 21, No. 7 (1897)
  • #161 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #162 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #163 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #164 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #165 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #166 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #167 Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems”, 4th Ed., Pearson Addison Wesley, 2004, pp. 827 - 833 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #168 Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems”, 4th Ed., Pearson Addison Wesley, 2004, pp. 827 - 833 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #169 Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems”, 4th Ed., Pearson Addison Wesley, 2004, pp. 827 - 833 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #170 Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems”, 4th Ed., Pearson Addison Wesley, 2004, pp. 827 - 833 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #171 Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems”, 4th Ed., Pearson Addison Wesley, 2004, pp. 827 - 833 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #172 Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems”, 4th Ed., Pearson Addison Wesley, 2004, pp. 827 - 833 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #173 Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems”, 4th Ed., Pearson Addison Wesley, 2004, pp. 827 - 833 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158
  • #174 Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems”, 4th Ed., Pearson Addison Wesley, 2004, pp. 827 - 833 I.S. Sokolnikoff, ”Mathematics of Physics and Modern Engineering”, McGraw-Hill, 2nd Edition, International Student Edition, 1966, pp. 138 - 158