SlideShare a Scribd company logo

2 classical field theories

•Download as PPT, PDF•
•
Solo Hermelin
Solo Hermelin

This presentation is intended for undergraduate students in physics and engineering. Please send comments to solo.hermelin@gmail.com. For more presentations on different subjects please visit my homepage at http://www.solohermelin.com. This presentation is in the Physics folder.

Science
1 of 273
04/08/15 1 Classical Field Theories SOLO HERMELIN Updated: 10.10.2013 16.11.2014 8.04.2015 http://www.solohermelin.com
04/08/15 2 SOLO Table of Content Classical Field Theories Generalized Coordinates 1. Newton’s Laws of Motion Analytic Dynamics Work and Energy Basic Definitions Constraints The Stationary Value of a Function and of a Definite Integral The Principle of Virtual Work 2. D’Alembert Principle 3. Hamilton’s Principle 4. Lagrange’s Equations of Motion 5. Hamilton’s Equations Introduction to Lagrangian and Hamiltonian Formulation Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  Second Method (Carathéodory) Equivalent Integrals Hamilton-Jacobi Theory Theorem of Noether for Single Integral
04/08/15 3 SOLO Table of Content (continue – 1) Classical Field Theories Fermat Principle in Optics Four-Dimensional Formulation of the Theory of Relativity Electromagnetic Field Introduction to Lagrangian and Hamiltonian Formulation for Continuous Systems and Fields Transition from a Discrete to Continuous Systems Extremal of the Functional ( ) ( )∫ ∫       ∂ ∂ ∂ ∂ = 2 1 ,,,,, t t V j k j kjk tdVd tx xtxtCI ψψ ψL Hamiltonian Formalism Theorem of Noether for Multiple Integrals Elasticity Vibrating String Vibrating Membrane Vibrating Beam Plate Theories Structural Model of the Solid Body The Inverse Square Law of Forces
04/08/15 4 SOLO Table of Content (continue – 2) Classical Field Theories Variational Principles of Hydrodynamics Part I – Lagrange Interpretation Part II: Euler’s Representation Part III: Simpler Eulerian Variational Principle Part IV: Problem of the Motion of Fluid Subjected to Gravity Forces and Surface Tension Forces Dynamics of Acoustic Field in Gases References Equation of Motion of a Variable Mass System – Lagrangian Approach
5 Analytic DynamicsSOLO Newton’s Laws of Motion “The Mathematical Principles of Natural Philosophy” 1687 First Law Every body continues in its state of rest or of uniform motion in straight line unless it is compelled to change that state by forces impressed upon it. Second Law The rate of change of momentum is proportional to the force impressed and in the same direction as that force. Third Law To every action there is always opposed an equal reaction. → =→= constvF  0 ( )vm td d p td d F  == 2112 FF  −= vmp  = td pd F  = 12F  1 2 21F  Return to Table of Content
6 Analytic DynamicsSOLO Work and Energy The work W of a force acting on a particle m that moves as a result of this along a curve s from to is defined by: F  1r  2r  ∫∫ ⋅      =⋅= •∆ 2 1 2 1 12 r r r r rdrm dt d rdFW      r  1r  2r  rd  rdr  + 1 2 F  m s rd  is the displacement on a real path. ••∆ ⋅= rrmT  2 1 The kinetic energy T is defined as: 1212 2 1 2 1 2 1 2 TTrrd m dtrr dt d mrdrm dt d W r r r r r r −=      ⋅=⋅      =⋅      = ∫∫∫ ⋅ ⋅ ••••⋅        For a constant mass m
7 Analytic DynamicsSOLO Work and Energy (continue) When the force depends on the position alone, i.e. , and the quantity is a perfect differential ( )rFF  = rdF  ⋅ ( ) ( )rdVrdrF  −=⋅ The force field is said to be conservative and the function is known as the Potential Energy. In this case: ( )rV  ( ) ( ) ( ) 212112 2 1 2 1 VVrVrVrdVrdFW r r r r −=−=−=⋅= ∫∫ ∆      The work does not depend on the path from to . It is clear that in a conservative field, the integral of over a closed path is zero. 12W 1r  2r  rdF  ⋅ ( ) ( ) 01221 21 1 2 2 1 =−+−=⋅+⋅=⋅ ∫∫∫ VVVVrdFrdFrdF path r r path r rC         Using Stoke’s Theorem it means that∫∫∫ ⋅×∇=⋅ SC sdFrdF  0=×∇= FFrot  Therefore is the gradient of some scalar functionF  ( ) rdrVdVrdF  ⋅−∇=−=⋅ ( )rVF  −∇=
8 Analytic DynamicsSOLO Work and Energy (continue) and • →∆→∆ ⋅−=⋅−= ∆ ∆ = rF dt rd F t V dt dV tt  00 limlim But also for a constant mass system •••••••••••• ⋅=⋅=      ⋅+⋅=      ⋅= rFrrmrrrr m rrm dt d dt dT  22 1 Therefore for a constant mass in a conservative field ( ) .0 constEnergyTotalVTVT dt d ==+⇒=+ Return to Table of Content
9 Analytic DynamicsSOLO 1.4 Basic Definitions Given a system of N particles defined by their coordinates: ( ) ( ) ( ) ( ) Nkktzjtyitxzyxrr kkkkkkkk ,,2,1,,   =++== where are the unit vectors defining any Inertial Coordinate Systemkji  ,, The real displacement of the particle mk : ( ) ( ) ( ) Nkktdzjtdyitdxrd kkkk ,,2,1   =++= is the infinitesimal change in the coordinates along real path caused by all the forces acting on the particle mk . The virtual displacements (Δxk , Δyk, Δzk, Δt) are infinitesimal changes in the coordinates; they are not real changes because they are not caused by real forces. The virtual displacements define a virtual path that coincides with the real one at the end points.
10 Analytic DynamicsSOLO Basic Definitions (continue) ( )trk  ( )1trk  ( )2trk  krd  1 2 F  km ),,,( dttdzzdyydxxPrdr kkkkkkkk ++++=+  ),,,( tzyxP kkk ),,,( ttzzyyxxP kkkkkk ∆+∆+∆+∆+ ),,,( tzzyyxxP kkkkkk ∆+∆+∆+ tvrd kk ∆=  i  j  k  True (Dynamical or Newton) Path Virtual Path Return to Table of Content
11 Analytic DynamicsSOLO 1.5 Constraints If the N particles are free the system has n = 3 N degrees of freedom.( ) Nkzyxr kkkk ,,2,1,,   = The constraints on the system can be of the following types: (1) Equality Constraints: The general form (the Pffafian form) ( ) ( ) ( )[ ] ( ) mldttradztradytradxtra l t N k k l zkk l ykk l xk ,,2,10,,,, 1   ==+++∑= or { } maaarankmldtarda l zk l yk l xk l t N k k l k ===+⋅∑= ,,,,2,10 1   We can classify the constraints as follows: (a) Time Dependency (a1) Catastatic mlal t ,,2,10 == (a2) Acatastatic mlal t ,,2,10 =≠ (1) Equality Constraints (2) Inequality Constraints
12 Analytic DynamicsSOLO Constraints (continue) Equality Constraints: The general form (the Pffafian form) (continue) { } maaarankmldtarda l zk l yk l xk l t N k k l k ===+⋅∑= ,,,,2,10 1   (b) Integrability (b1) Holonomic if the Pffafian forms are integrable; i.e.: mldt t f zd z f yd y f xd x f df N k l k k l k k l k k l l ,,2,1 1 = ∂ ∂ +      ∂ ∂ + ∂ ∂ + ∂ ∂ = ∑= or ( ) mltzyxzyxf NNNl ,,2,10,,,,,,, 111  == (b2) Non-holonomic if the Pffafian forms are not integrable (b2.1) Scleronomic: (b2.2) Rheonomic: ml l t f ,,2,1 0 = = ∂ ∂ or ( ) mlzyxzyxf NNNl ,,2,10,,,,,, 111  == ml l t f ,,2,1 0 = ≠ ∂ ∂
13 Analytic DynamicsSOLO Constraints (continue) (2) Inequality Constraints: (a) Stationary Boundaries (time independent): (b) Non-stationary Boundaries (time dependent): ( ) mlzyxzyxf NNNl ,,2,10,,,,,, 111  =≥ ( ) mltzyxzyxf NNNl ,,2,10,,,,,,, 111  =≥
14 Analytic DynamicsSOLO Constraints (continue) Displacements Consistent with the Constraints: The real displacement consistent with the General Equality Constraints (Pffafian form) is: The virtual displacement consistent with the General Equality Constraints (Pffafian form) is: dtkdzjdyidxrd kkkk ,  ++= [ ] mldtardadtadzadyadxa l t N k k l k l t N k k l zkk l ykk l xk ,,2,10 11   ==+⋅=+++ ∑∑ == tkzjyixr kkkk ∆∆+∆+∆=∆ ,  [ ] mltaratazayaxa l t N k k l k l t N k k l zkk l ykk l xk ,,2,10 11   ==∆+∆⋅=∆+∆+∆+∆ ∑∑ == Dividing the Pffafian equation by dt and taking the limit, we obtain: mlraa N k k l k l t ,,2,1 1   =⋅−= ∑= ⋅ Now replace in the virtual displacement equationl ta mltrra N k kk l k ,,2,10 1   ==      ∆−∆⋅∑= ⋅ Define the δ variation as: td d t∆−∆= ∆ δ
15 Analytic DynamicsSOLO Constraints (continue) Displacements Consistent with the Constraints (continue): Define the δ variation as: td d t∆−∆= ∆ δ ( )trk  kr  δ km ),,,( dttdzzdyydxxPrdr kkkkkkkk ++++=+  ),,,( tzyxP kkk ),,,( ttzzyyxxP kkkkkk ∆+∆+∆+∆+ ),,,( tzzyyxxP kkkkkk ∆+∆+∆+ dtrrd kk ⋅ =  i  j  k  True (Dynamical or Newton) Path Virtual Path kr  ∆ trr kk ∆=∆ ⋅ Then: kkk r td d trr  ∆−∆= ∆ δ From the Figure we can see that δ variation corresponds to a virtual displacement in which the time t is held fixed and the coordinates varied to the constraints imposed on the system. mlra N k k l k ,,2,10 1   ==⋅∑= δ For the Holonomic Constraints: ( ) mltzyxzyxf NNNl ,,2,10,,,,,,, 111  == mlrf N k klk ,,2,10 1   ==⋅∇∑= δ Return to Table of Content
16 Analytic DynamicsSOLO 1.6 Generalized Coordinates The motion of a mechanical system of N particles is completely defined by n = 3N coordinates . Quite frequently we may find it more advantageous to express the motion of the system in terms of a different set of coordinates, say . If we take in consideration the m constraints we can reduce the coordinates to n = 3N-m generalized coordinates. ( ) ( ) ( ) ( )Nktztytx kkk ,,2,1,, = ( )T nqqqq ,,, 21   = ( ) ( ) ( ) ( ) ( ) Nkktqzjtqyitqxtqqqrtqr kkknkk ,,2,1,,,,,,,, 21     =++== Nkkdzjdyidxdt t r dq q r rd kkk k j n j j k k ,,2,1 1    =++= ∂ ∂ + ∂ ∂ = ∑= Nk t r q q r td rd rv k j n j j kk kk ,,2,1 1      = ∂ ∂ + ∂ ∂ === ∑= ⋅ In the same way Nkkzjyixt t r q q r r kkk k j n j j k k ,,2,1 1    =∆+∆+∆=∆ ∂ ∂ +∆ ∂ ∂ =∆ ∑= and Nkt t r tq q r t t r q q r trrr k j n j j kk j n j j k kkk ,,2,1 11      ==∆ ∂ ∂ −∆ ∂ ∂ −∆ ∂ ∂ +∆ ∂ ∂ =∆−∆= ∑∑ == ⋅ δ
17 Analytic DynamicsSOLO Generalized Coordinates (continue) ( ) Nkq q r tqq q r r n j j j k jj n j j k k ,,2,1 11      = ∂ ∂ =∆−∆ ∂ ∂ = ∑∑ == δδ where tqqq jjj ∆−∆= ∆ δ The Generalized Equality Constraints in Generalized Coordinates will be: mldt t r aadq q r a dt t r aadq q r adtarda N k kl k l ti n i i k N k l k N k N k kl k l ti n i i kl k l t N k k l k ,,2,10 11 1 1 111          ==      ∂ ∂ ⋅++      ∂ ∂ ⋅= =      ∂ ∂ ⋅++ ∂ ∂ ⋅=+⋅ ∑∑ ∑ ∑ ∑∑∑ == = = === If we define ∑ ∑∑= = ∆ = ∆ ∂ ∂ ⋅+= ∂ ∂ ⋅= N k N k kl k l t l t n i i kl k l i t r aac q r ac 1 11 &     we obtain mldtcdqc l ti n i l i ,,2,10 1 ==+∑= and the virtual displacements compatible with the constraints are mlqc i n i l i ,,2,10 1 ==∑= δ
18 Analytic DynamicsSOLO Generalized Coordinates (continue) The number of degrees of freedom of the system is n = 3N-m. However, when the system is nonholonomic, it is possible to solve the m constraint equations for the corresponding coordinates so that we are forced to work with a number of coordinates exceeding the degrees of freedom of the system. This is permissible provided the surplus number of coordinates matches the number of constraint equations. Although in the case of a holonomic system it may be possible to solve for the excess coordinates, thus eliminating them, this is not always necessary or desirable. If surplus coordinates are used, the corresponding constraint equations must be retained. Return to Table of Content
19 Analytic DynamicsSOLO 1.7 The Stationary Value of a Function and of a Definite Integral In problems of dynamics is often sufficient to find the stationary value of functions instead of the extremum (minimum or maximum). Definition: A function is said to have a stationary value at a certain point if the rate of change in every direction of the point is zero. Examples: (1) ( ) ni u f du u f dfuuuf i n i i i n ,,2,100,,, 1 21  == ∂ ∂ →= ∂ ∂ =→ ∑= By solving those n equations we obtain for which f is stationary ( )nuuu ,,, 21 
20 Analytic DynamicsSOLO The Stationary Value of a Function and of a Definite Integral (continue) Examples (continue): (2) ( )nuuuf ,,, 21  with the constraints { } marankmldua l k N k k l k ===∑= ,,2,10 1  Lagrange’s multipliers solution gives: 0 1 1 =      + ∂ ∂ = ∑ ∑= = i n i m l l il i dua u f df λ By choosing the m Lagrange’s multipliers λl to annihilate the coefficients of the m dependent differentials dui we have equationsmn mldua nia u f n l i l i m l l il i +        == ==+ ∂ ∂ ∑ ∑ = = ,,2,10 ,,2,10 1 1  λ
21 Analytic DynamicsSOLO The Stationary Value of a Function and of a Definite Integral (continue) Examples (continue): (3) The functional ( ) ( ) ∫       = 2 1 ,, x x dx xd xyd xyxFI We want to find such that I is stationary, when the end points and are given. ( )xy ( )1xy ( )2xy ( )xy ( ) ( ) ( ) ( )xxyxyxy ηεδ +=+ ( )11, yx ( )22 , yx x y The variation of is( )xy ( ) ( ) ( ) ( ) ( ) ( ) ( ) 021 ==+=+= xxxxyxyxyxy ηηηεδ and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫∫       ++=      = 2 1 2 1 ,,,, x x x x dx xd xd xd xyd xxyxFdx xd xyd xyxFI η εηεε ( ) ( ) +++== == 2 0 2 2 0 2 1 0 ε ε ε ε εε εε d d Id d d Id II
22 Analytic DynamicsSOLO The Stationary Value of a Function and of a Definite Integral (continue) Examples (continue): Continue: The functional ( ) ( ) ∫       = 2 1 ,, x x dx xd xyd xyxFI The necessary condition for a stationary value is ( ) ( ) ( ) ( ) ( )[ ] 0 0 12 0 2 1 2 1 =−       ∂ ∂ +                     ∂ ∂ − ∂ ∂ =                     ∂ ∂ + ∂ ∂ = ∫ ∫ =    xx xd yd F dxx xd yd F xd d y F dx xd xd xd yd F x y F d Id x x nintegratio partsby x x ηηη η η ε ε Since this must be true for every continuous function η(x) we have 210 xxx xd yd F xd d y F ≤≤=                     ∂ ∂ − ∂ ∂ Euler-Lagrange Differential Equation By solving this differential equation, y(x),for which I is stationary is found. JOSEPH-LOUIS LAGRANGE 1736-1813 LEONHARD EULER 1707-1783
23 Leonhard Euler (1707-1783) generalized the brothers Bernoulli methods in “Me tho dus inve nie ndi line as curvas m axim i m inim ive pro prie tate g aude nte s sive so lutio pro ble m atis iso pe rim e trici latissim o se nsu acce pti” (“Me tho d fo r finding plane curve s that sho w so m e pro pe rty o f m axim a and m inim a”) published in 1744. Euler solved the Ge o de sic Pro ble m , i.e. the curves of minimum length constrained to lie on a given surface. Joseph-Louis Lagrange (1736-1813) gave the first analytic methods of Calculus of Variations in "Essay on a new method of determining the maxima and minima of indefinite integral formulas" published in 1760. Euler-Lagrange Equation: SOLO CALCULUS OF VARIATIONS Return to Table of Content
24 Analytic DynamicsSOLO 1.8 The Principle of Virtual Work This is a statement of the Static Equation of a mechanical system. If the system of N particles is in dynamic equilibrium the resultant force on each particle is zero; i.e.: 0=iR  0 1 =⋅= ∑= N i ii rRW  δδ Because of this, for a virtual displacement the Virtual Work of the system is ir  δ If the system is subjected to the constraints: { } maaarankmldtarda l zk l yk l xk l t N k k l k ===+⋅∑= ,,,,2,10 1   Then we denote the external forces on particle i by and the constraint’s forces by . The resultant force on i is: iF  iF'  0' =+= iii FFR 
25 Analytic DynamicsSOLO The Principle of Virtual Work (continue) We want to find the Virtual Work of the constraint forces. There are two kind of constraints: (1) The particle i is constrained to move on a definite surface. We assume that the motion is without friction and therefore the constraint forces must be normal to the surface. The virtual variation compatible with the constraint must be on the surface, therefore . iF  ir  δ 0' =⋅ ii rF  δ ir  δ iF'  (2) The particle i is acting on the particle j and the distance between them is l(t). . iF'  i j jF'  ir  jr  ( )tl
26 Analytic DynamicsSOLO The Principle of Virtual Work (continue) ( ) ( ) ( )tlrrrr jiji 2 =−⋅−  ( ) ( ) tllrrrr jiji ∆=∆−∆⋅−  ( ) llrrrr jiji  =      −⋅− ⋅⋅ ( ) ( ) ( ) ji rr jiji jjiiji rrrrrr trrtrrrr ji    δδδδ =→=−⋅−→ →=            ∆−∆−      ∆−∆⋅− ≠ ⋅⋅ 0 0 ji FF ''  −= If we compute the virtual variation and differential and we multiply the second equation by and add to the first we obtaint∆− Because is a real (not a generalized) force we can use Newton’s Third Law: i.e.:iF'  and the virtual work of the constraint forces of this system is: ( ) 0''''' =⋅−+⋅=⋅+⋅= rFrFrFrFW iijjii  δδδδδ We can generalized this by saying that: 0' 1 =⋅∑= N i ii rF  δ The work done by the constraint forces in virtual displacements compatible with the constraints (without dissipation) is zero.
27 Analytic DynamicsSOLO The Principle of Virtual Work (continue) From equation we obtain:0' =+= iii FFR  ∑∑∑∑ ==== ⋅=⋅+⋅=⋅= N i ii N i ii N i ii N i ii rFrFrFrR 1 0 111 '0    δδδδ or 0 1 =⋅= ∑= N i ii rFW  δδ The Principle of Virtual Work The work done by the applied forces in infinitesimal virtual displacements compatible with the constraints (without dissipation) is zero
28 Analytic DynamicsSOLO The Principle of Virtual Work (continue) { } mjNimaaarank mjra j zi j yi j xi N k k j k ,,2,1&,,1,, ,,2,10 1    === ==⋅∑= δ We found that the General Equality Constraint the virtual displacement compatible with the constraint must be: ir  δ Let adjoin the m constraint equations by the m Lagrange’s multipliers and add to the virtual work equation: jλ 0 1 11 11 =⋅        +=      ⋅+⋅= ∑ ∑∑ ∑∑ = == == N i i m j j iji m j N i i j ij N i ii raFrarFW  δλδλδδ There are 3N virtual displacements from which m are dependent of the constraint relations and 3N-m are independent. We will choose the m Lagrange’s multipliers to annihilate the coefficients of the m dependent variables: jλ    −+= = =+ ∑= iationsmNtindependenNmi mtheofbecausemi aF j m j j iji var33,,1 ,,2,1 0 1   λ λ
29 Analytic DynamicsSOLO The Principle of Virtual Work (continue) From we obtain:0' =+= iii FFR  ∑= = m j j iji aF 1 '  λ where are chosen such thatjλ mjforaF m j j iji ,,2,10 1   ==+ ∑= λ Since , we obtain:k n k k i i q q r r δδ ∑= ∂ ∂ = 1   0 1 1 11 1 111 1 =                       ∂ ∂ +      ∂ ∂ = = ∂ ∂ ⋅        +=⋅        += ∑ ∑ ∑∑ ∑ ∑∑∑ ∑ = = == = === = n k k m j N i k ij ij N i k i i N i n k k k i m j j iji N i i m j j iji q q r a q r F q q r aFraFW δλ δλδλδ      We define: nk q r FQ N i k i ik ,,2,1 1   = ∂ ∂ =∑= ∆ nkc q r aQ m j j kj m j N i k ij ijk ,,2,1' 11 1    ==      ∂ ∂ = ∑∑ ∑ == = ∆ λλ nk q r ac N i k ij i j k ,,2,1 1    = ∂ ∂ =∑= ∆ Generalized Forces Generalized Constraint Forces Return to Table of Content
30 Analytic DynamicsSOLO 2. D’Alembert Principle Jean Le Rond d’ Alembert 1717-1783 Newton’s Second Law for a particle of mass and a linear momentum vector can be written as im iii vmp  = D’Alembert Principle: 0' =−+ • iii pFF  D’Alembert Principle enables us to trait dynamical problems as if they were statical. Let extend the Principle of Virtual Work to dynamic systems: 0' 1 =⋅      −+∑= •N i iiii rpFF  δ Assuming that the constraints are without friction the virtual work of the constraint force is zero . Then we have Generalized D’Alembert Principle: 0 1 =⋅      −∑= •N i iii rpF  δ 0' 1 =⋅∑= N i ii rF  δ The Generalized D’Alembert Principle The total Virtual Work performed by the effective forces through infinitesimal Virtual Displacement, compatibile with the system constraints are zero. 0=− • ii pF  is the effective force. “Traité de Dynamique” 1743 Return to Table of Content where and are the applied and constraint forces, respectively.iF  iF' 
31 Analytic DynamicsSOLO 3. Hamilton’s Principle William Rowan Hamilton 1805-1865 Let write the D’Alembert Principle: in integral form0 1 =⋅      −∑= •N i iii rpF  δ 0 2 1 1 =⋅      −∫∑= • t t N i iii dtrpF  δ But ( )∑∑∑ === • ⋅+      ⋅−=⋅− N i iii N i iii N i iii r td d vmrvm td d rvm 111  δδδ Let find ( )ir td d  δ iiiii r td d ttvrr        ∆−∆=∆−∆=δ are the virtual displacements compatible with the constraints mjra N i i j i ,,2,10 1   ==⋅∑= δ ( )tri  ir  δ tvi ∆  ( )tPi( )tP i' ( )ttP i ∆+' ir  ∆ Virtual Path True Path (P) Newtonian or Dynamic Path The Constraint Space at t mjra N i i j i ,,10 1   ==⋅∑= δ ( ) ( ) ( ) ( ) ( ) ( )     =∆=∆ =∆=∆ == 0 0 0 21 21 21 tttt trtr trtr ii ii   δδ 1t 2t
32 Analytic DynamicsSOLO Hamilton’s Principle (continue) Since td rd vv i Pi i   == ( ) ( ) ( ) t td d vr td d vt td d r td d v t td d r td d td rd tdtd rdrd ttd rrd vvv iiiii i i iiii ttPii i ∆−∆+≈      ∆−      ∆+≈ ∆+ ∆+ = = ∆+ ∆+ = ∆+ ∆+ ==∆+ ∆+      1 1 ' ( ) ( ) tar td d tatvr td d t td d vr td d v iiiiiiii ∆+=∆+∆−∆=∆−∆=∆  δ Therefore ( ) ( ) ecommutativare td d and r td d vv td d ttavr td d iiiiii δ δδδ → →==      ∆−∆=∆−∆= 
33 Analytic DynamicsSOLO Hamilton’s Principle (continue) Now we can develop the expression: tavmvvmrvm td d ram N i iii N i iii N i iii N i iii ∆⋅−      ∆⋅+      ⋅−=⋅− ∑∑∑∑ ==== 1111  δδ But the Kinetic Energy T of the system is: ∑= ⋅= N i iii vvmT 12 1  ∑= ∆⋅=∆ N i iii vvmT 1  ∑∑∑ === ⋅ ⋅=⋅=⋅= N i iii N i iii N i iii vFmavmvvmT 111  Therefore Trvm td d tTTrvm td d ram N i iii N i iii N i iii δδ δδ +      ⋅−= =∆−∆+      ⋅−=⋅− ∑ ∑∑ = • == 1 11  
34 Analytic DynamicsSOLO Hamilton’s Principle (continue) From the integral form of D’Alembert Principle we have: ( ) ∫ ∑∫ ∑∑ ∫ ∑∫ ∑ ∫∑       ⋅+=      ⋅++⋅−= =      ⋅++      ⋅−= =⋅+−= === == = 2 1 2 1 2 1 2 1 2 1 2 1 111 0 11 1 0 t t N i ii t t N i ii N i t tiii t t N i ii t t N i iii t t N i iiii dtrFTdtrFTrvm dtrFTdtrvm td d dtrFam      δδδδδ δδδ δ We obtained ( ) 0 2 1 2 1 1 =+=      ⋅+ ∫∫ ∑= dtWTdtrFT t t t t N i ii δδδ  Extended Hamilton’s Principle
35 Analytic DynamicsSOLO Hamilton’s Principle (continue) If we develop and we can writetTTT ∆−∆= δ tvrr iii ∆−∆=  δ 0 2 1 2 1 111 =      ∆      ⋅+−      ∆⋅+∆=      ⋅+ ∫ ∑∑∫ ∑ === dttvFTrFTdtrFT t t N i ii N i ii t t N i ii   δδ and because ∑= ⋅= N i ii vFT 1   02 2 1 1 =      ∆−      ∆⋅+∆∫ ∑= dttTrFT t t N i ii  The pair and is arbitrary but compatible with the constraints:ir  ∆ t∆ mjtara j t N i i j i ,,2,10 1   ==∆+∆⋅∑=
36 Analytic DynamicsSOLO Hamilton’s Principle (continue) For a Conservative System VF ii −∇=  VrVrFW N i ii N i ii δδδδ −=⋅∇−=⋅= ∑∑ == 11  We have ( ) ( ) 0 2 1 2 1 2 1 ==−=+ ∫∫∫ t t t t t t dtLdtVTdtWT δδδ where WTVTL +=−= ∆       =−∇=−== ∆ ∫ NiVFVTLdtL ii t t ,,2,1;0 2 1   δ Hamilton’s Principle for Conservative Systems Hamilton’s Principle for Conservative Systems: The actual path of a conservative system in the configuration space renders the value of the integral stationary with respect to all arbitrary variations (compatible with the constraints) of the path between the two instants and provided that the path variations vanish at those two points. ∫= 2 1 t t dtLI 1t 2t Return to Table of Content
37 Analytic DynamicsSOLO 4. Lagrange’s Equations of Motion Joseph Louis Lagrange 1736-1813 “Mecanique Analitique” 1788 The Extended Hamilton’s Principle states: 0 2 1 1 =      ⋅+∫ ∑= dtrFT t t N i ii  δδ where are the virtual displacements compatible with the constraints: ir  δ mjqcq q r ara n k k k i n k k N i k ij i N i i j i ,,2,10 11 11   ===      ∂ ∂ =⋅ ∑∑ ∑∑ == == δδδ T the kinetic energy of the system is given by: ∑ ∑∑∑ = ⋅ === ⋅⋅       =      ∂ ∂ + ∂ ∂ ⋅      ∂ ∂ + ∂ ∂ =⋅= N j n i j i i j n i j i i j j N j jjj tqqT t r q q r t r q q r mrrmT 1 111 ,, 2 1 2 1        where is the vector of generalized coordinates.( )nqqqq ,,, 21   =       −∆ ∂ ∂ +      ∆ ∂ ∂ +∆ ∂ ∂ +      =      −      ∆+∆+∆+=∆ ⋅ = ⋅⋅⋅⋅ ∑ tqqTt t T q q T q q T tqqTtqqTttqqqqTT n i i i i i ,,,,,,,, 1     t T q q T q q T T n i i i i i ∂ ∂ +      ∂ ∂ + ∂ ∂ = ∑=1    ( ) ( ) ∑∑ ==       ∂ ∂ + ∂ ∂ =      ∆−∆ ∂ ∂ +∆−∆ ∂ ∂ =∆−∆= n i i i i i n i ii i ii i q q T q q T tqq q T tqq q T tTTT 11      δδδ
38 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue) ( ) ( ) ∑∑ ==       ∂ ∂ + ∂ ∂ =      ∆−∆ ∂ ∂ +∆−∆ ∂ ∂ =∆−∆= n i i i i i n i ii i ii i q q T q q T tqq q T tqq q T tTTT 11      δδδ But because δ and are commutative and:td d ( )ii q dt d q δδ = ∑=       ∂ ∂ + ∂ ∂ = n i i i i i q dt d q T q q T T 1 δδδ  This is an expected result because the variation δ keeps the time t constant. We found that , therefore∑= ∂ ∂ = n i i i j j q q r r 1 δδ   ∫∑∫∑ ∑∫ ∑ ∑∫ ∑ == == == =        ∂ ∂ ⋅=        ∂ ∂ ⋅=        ⋅ 2 1 2 1 2 1 2 1 11 11 11 t t n i ii t t n i i N j i j j t t N j n i i i j j t t N j jj dtqQdtq q r Fdtq q r FdtrF δδδδ     where ForcesdGeneralizeni q r FQ N j i j ji ,,2,1 1    = ∂ ∂ ⋅=∑= ∆ Now ( ) ∫∑∑ ∫∑∫ ∑ == ==       − ∂ ∂ −      ∂ ∂ − ∂ ∂ =       + ∂ ∂ + ∂ ∂ =        ⋅+= 2 1 2 1 2 1 2 1 11 0 .int 11 0 t t n i iii i i i n i t ti i partsby t t n i iii i i i t t N j jj dtqQq q T q q T td d q q T dtqQq q T q td d q T dtrFT δδδδ δδδδδ    
39 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue) 0 2 1 1 =      − ∂ ∂ −      ∂ ∂ ∫∑= t t i n i i ii dtqQ q T q T td d δ  where the virtual displacements must be consistent with the constraints . Let adjoin the previous equations by the constraints multiplied by the Lagrange’s multipliers iqδ mkqc n i i k i ,,2,10 1 ==∑= δ ( )mkk ,,2,1 =λ 0 1 11 1 =      =      ∑ ∑∑ ∑ = == = n i i m k k ik m k n i i k ik qcqc δλδλ to obtain 0 2 1 1 1 =      −− ∂ ∂ −      ∂ ∂ ∫∑ ∑= = t t i n i m k k iki ii dtqcQ q T q T td d δλ  While the virtual displacements are still not independent, we can chose the Lagrangian’s multipliers so as to render the bracketed coefficients of equal to zero. The remaining being independent can be chosen arbitrarily, which leads to the conclusion that the coefficients of are zero. It follows iqδ iqδ ( )mkk ,,2,1 =λ ( )nmiqi ,,2,1 +=δ ( )miqi ,,2,1 =δ nicQ q T q T dt d m k k iki ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λ
40 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue) nicQ q T q T dt d m k k iki ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λ We have here n equations with n+m unknowns . To find all the unknowns we must add the m equations defined by the constraints, to obtain ( ) ( ) mn tqtq λλ ,,,,, 11  nicQ q T q T dt d m k k iki ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λLagrange’s Equations: mkcqc k t n i i k i ,,2,10 1  ==+∑= Let define Generalized Constraint Forces: nicQ m k k iki ,,2,1' 1 == ∑= λ
41 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue) If the system is acted upon by some forces which are derivable from a potential function and some forces which are not, we can write:( ) ( )nn qqqVrrrV ,,,,,, 2121     −=− n jF  n jjj FVF  +−∇= ( ) ∑∑ ∑∑ ∑∑ == == == =              ∂ ∂ ⋅+ ∂ ∂ ⋅∇−=      ∂ ∂ ⋅+∇−=⋅ n i ii n i i N j i jn j i j j N j n i i i jn jj N j jj qQq q r F q r Vq q r FVrF 11 11 11 δδδδ     But where∑= ∂ ∂ ⋅∇= ∂ ∂ N j i j j i q r V q V 1  k z V j y V i x V V jjj j  ∂ ∂ + ∂ ∂ + ∂ ∂ =∇ Therefore: niQ q V q r F q r VQ in i N j i jn j N j i j ji ,,2,1 11     =+ ∂ ∂ −= ∂ ∂ ⋅+ ∂ ∂ ⋅∇−= ∑∑ == Generalized External Forces: Generalized External Nonconservative Forces: ni q r FQ N j i jn jin ,,2,1 1    = ∂ ∂ ⋅=∑= ∆
42 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue) The Lagrange’s Equations nicQ q T q T dt d m k k iki ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λ Define: ( ) ( ) ( )qVtqqTtqqL  −= ∆ ,,,, Because we assume that , we have( ) i q qV i ∀= ∂ ∂ 0   Lagrange’s Equations: nicQ q L q L dt d m k k ikin ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λ mkcqc k t n i i k i ,,2,10 1  ==+∑= We proved       =−∇=−== ∆ ∫ NiVFVTLdtL ii t t ,,2,1;0 2 1   δ Hamilton’s Principle for Conservative Systems Lagrange’s Equations for a Conservative System without Constraints: ( )0,,,,2,10 =−=−∇=== ∂ ∂ −      ∂ ∂ k iii ii cVTLVFni q L q L dt d    If they are no constraints, from the Lagrange’s Equations, or from Euler- Lagrange Equation for a stationary solution of , we obtain:∫= 2 1 t t dtLI
Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue) The Lagrangian gives unique Euler-Lagrange Equations, but the inverse is not true, there more than one Lagrangian that gives the same Euler-Lagrange Equations. Example 1: ( ) ( ) ( ) td tqqd tqqLtqqL ,, ,,:,,1   Φ += ( ) ( ) ( ) ( )   ( )∫∫∫∫ = ∂ Φ∂ + ∂ Φ∂ +=Φ+= 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ,,,,,,,,:,, 00 1 t t t t t t t t t t t t t t tdtqqLq q q q tdtqqLtqqtdtqqLtdtqqL     δδδδδδδ Example 2: This is not the most general case ( ) ( ) [ ]2 2 2 1 2 2 2 1221211 2 1 :,,&:,, qqqqtqqLqqqqtqqL −−+=−=          =−= ∂ ∂ −      ∂ ∂ =−= ∂ ∂ −      ∂ ∂ 0 0 11 2 1 2 1 22 1 1 1 1 qq q L q L dt d qq q L q L dt d            =−= ∂ ∂ −      ∂ ∂ =−= ∂ ∂ −      ∂ ∂ 0 0 22 2 2 2 2 11 1 2 1 2 qq q L q L dt d qq q L q L dt d     Return to Table of Content
44 Analytic DynamicsSOLO 5. Hamilton’s Equations The Lagrange’s Equations nicQ q T q T dt d m k k iki ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λ can be rewritten as: nicQ q T tq T q qq T q qq T q T dt d m k k iki i n i i j ji j jii ,,2,1 11 222       =++ ∂ ∂ =         ∂∂ ∂ + ∂∂ ∂ + ∂∂ ∂ =      ∂ ∂ ∑∑ == λ therefore consist of a set of n simultaneous second-order differential equations. They must be solved tacking in consideration the m constraint equations. mkcqc k t n i i k i ,,2,10 1  ==+∑= A procedure for the replacement of the n second-order partial-differential equations by 2n first-order ordinary-differential equations consists of formulating the problem in terms of 2n Hamilton’s Equations. We define first: General Momentum: ni q T p i i ,,2,1   = ∂ ∂ = ∆ We want to find the transformation from the set of variables to the set by the Legendre’s Dual Transformation. ( )tqq ,,  ( )tpq ,, 
45 Analytic DynamicsSOLO Hamilton’s Equations (continue) Legendre’s Dual Transformation. Adrien-Marie Legendre 1752-1833 Let consider a function of n variables , m variables and time t.ix iy ( )tyyxxF mn ,,,,,, 11  and introduce a new set of variables defined by the transformation:iu ni x F u i i ,,2,1 = ∂ ∂ = ∆ We can see that:                             ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ +                             ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ ∂ =             m mnn m n nn n n dy dy dy yx F yx F yx F yx F dx dx dx x F xx F xx F x F du du du          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 by the new variables . We can see that the new n variables are independent if the Hessian Matrix is nonsingular. ( )nidxi ,,2,1 = ni njji xx F ,,1 ,,1 2   = =         ∂∂ ∂
46 Analytic DynamicsSOLO Hamilton’s Equations (continue) Legendre’s Dual Transformation (continue-1) Let define a new function G of the variables , and t.iu iy ( )tyyuuGFxuG mn n i ii ,,,,,, 11 1 =−=∑= ∆ Then: ( ) dt t F dy y F dx x F udux dt t F dy y F dx x F dxuduxdG m j j j n i i i iii n i m j j j i i n i iiii ∂ ∂ − ∂ ∂ −                   ∂ ∂ −+= = ∂ ∂ − ∂ ∂ − ∂ ∂ −+= ∑∑ ∑ ∑∑ == = == 11 0 1 11  But because: ( )tyyuuGG mn ,,,,,, 11 = dt t G dy y G du u G dG n i m j j j i i ∂ ∂ + ∂ ∂ + ∂ ∂ = ∑ ∑= =1 1 Because all the variations are independent we have: t F t G mj y F y G ni u G x jji i ∂ ∂ −= ∂ ∂ = ∂ ∂ −= ∂ ∂ = ∂ ∂ = ;,,1;,,1 
47 Analytic DynamicsSOLO Hamilton’s Equations (continue) Legendre’s Dual Transformation (continue-2) Now we can define the Dual Legendre’s Transformation from ( )tyyxxF mn ,,,,,, 11  ( ) FxutyyuuG n i iimn −= ∑=1 11 ,,,,,, to by using ni x F u i i ,,2,1 = ∂ ∂ = ni u G x i i ,,2,1 = ∂ ∂ = End of Legendre’s Dual Transformation
48 Analytic DynamicsSOLO Hamilton’s Equations (continue) Following the same pattern to find the transformation from the set of variables to the set , we introduce the Hamiltonian:( )tqq ,,  ( )tpq ,,  ( )tqqTqpH n i ii ,, 1   −=∑= ∆ where ni q T p i i ,,2,1   = ∂ ∂ = Then ( )tpqHH ,,  = dt t H dp p H dq q H dt t T dq q T dpq dt t T dq q T qd q T qdpdpqdH n i i i i i n i i i ii n i i i i i iiii ∂ ∂ +      ∂ ∂ + ∂ ∂ = ∂ ∂ −      ∂ ∂ −= = ∂ ∂ −      ∂ ∂ − ∂ ∂ −+= ∑∑ ∑ == = 11 1     and 0 1 =      ∂ ∂ + ∂ ∂ +            − ∂ ∂ +      ∂ ∂ + ∂ ∂ ∑= dt t T t H dpq p H dq q T q Hn i ii i i ii 
49 Analytic DynamicsSOLO Hamilton’s Equations (continue) If the Hessian Matrix is nonsingular, all the are independent, but not the that must be consistent with the constraints: ni njji qq T ,,1 ,,1 2    = =         ∂∂ ∂ ( )nidpi ,,2,1 = ( )nidqi ,,2,1 = mjdtcdqc j t n i i j i ,,2,10 1 ==+∑= Let adjoin the previous equations by the constraint equations multiplied by the m Lagrange’s multipliers :j'λ 0''' 11 11 1 =+        =      + ∑∑ ∑∑ ∑ == == = m j j ij n i i m j j ij m j j t n i i j ij dtcdqcdtcdqc λλλ We have 0'' 11 1 =        + ∂ ∂ + ∂ ∂ +               − ∂ ∂ +        + ∂ ∂ + ∂ ∂ ∑∑ ∑ == = dtc t T t H dpq p H dqc q T q H m j j tj n i ii i i m j j ij ii λλ 
50 Analytic DynamicsSOLO Hamilton’s Equations (continue) By proper choosing the m Lagrange’s multipliers λ’j ,the remainder differentials and dt are independent and therefore we have:ii dpdq , ni c t H t T c q H q T p H q m j j tj m j j ij ii i i ,,2,1 ' ' 1 1   =           − ∂ ∂ −= ∂ ∂ − ∂ ∂ −= ∂ ∂ ∂ ∂ = ∑ ∑ = = λ λ Legendre’s Dual Transformation By differentiating the General Momentum Equation and using Lagrange’s Equations we obtain: ( )∑∑ == −++ ∂ ∂ −=++ ∂ ∂ =      ∂ ∂ = m j j ijji i m j j iji ii i cQ q H cQ q T q T dt d p 11 ''''' λλλ   ni cQ q H p p H q m j j iji i i i i ,,2,1 1    =        ++ ∂ ∂ −= ∂ ∂ = ∑= λ mkcqc k t n i i k i ,,2,10 1  ==+∑= Extended Hamilton’s Equations Constrained Differential Equations
51 Analytic DynamicsSOLO Hamilton’s Equations (continue) For Holonomic Constraints (constraints of the form ) we can (theoretically) reduce the number of generalized coordinates to n-m and we can assume that and n represents the number of degrees of freedom of the system (this reduction is not possible for Nonholonomic Constraints). Then: ( ) mjtqqf nj .,10,,,1  == 0== j t j i cc ni Q q H p p H q i i i i i ,,2,1    =        + ∂ ∂ −= ∂ ∂ = Extended Hamilton’s Equations for Holonomic Constraints ni q V Q i i ,,2,1 = ∂ ∂ −= ( ) ( ) ( )qVtqqTtqqL  −= ∆ ,,,, ni q L q T p ii i ,,2,1   = ∂ ∂ = ∂ ∂ = Extended Hamilton’s Equations for Holonomic Constraints and a Conservative System Conservative System
52 Analytic DynamicsSOLO Hamilton’s Equations (continue) Define: Hamiltonian for Conservative Systems ( ) ( ) ( ) ( )qVtqqTtqqLqptqqH n i ii   −=−=∑= ∆ ,,,,,, 1 Hamilton’s Canonical Equations for Conservative Systems with Holonomic Constraints ni q H p p H q i i i i ,,2,1    =        ∂ ∂ −= ∂ ∂ = We have: Return to Table of Content
04/08/15 53 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Second Method (Carathéodory) Constantin Carathéodory (1873-1950) Carathéodory developed another approach to the Euler-Lagrange Equations ( ) ( )          Covariant k l l jj k jj k q q q qqtL q qqtL p ∂ ∂ ∂ ∂ = ∂ ∂ = ,,,, :Define Canonical Momentum We assume that is a one to one correspondence between the n components of pj and the n components of and from the definition of pk (by the Inverse Function Theorem) we have j q ( )k kjj pqtx ,,φ= From we see that the one to one correspondence is possible only if ( ) ( ) ( ) td tq qqtL qd qq qqtL qd qq qqtL pd k jj j jk jj j jk jj k ∂∂ ∂ + ∂∂ ∂ + ∂∂ ∂ =        ,,,,,, : 222 ( ) 0 ,, det 2 ≠      ∂∂ ∂ jk jj qq qqtL   Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
04/08/15 54 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Second Method (Carathéodory) ( ) ( ) k l l jj k jj k q q q qqtL q qqtL p       ∂ ∂ ∂ ∂ = ∂ ∂ = ,,,, : Canonical Momentum ( )k jj ptq ,φ= Define ( ) ( )[ ] ( )k j ji kk k k ptpptqtLpqtH ,,,,:,, φφ +−= Hamiltonian Let compute j q q L p j j k k j k k j qq p p p q q L p H kk kk      φ φ = ∂ ∂ = =+ ∂ ∂ + ∂ ∂ ∂ ∂ −= ∂ ∂ j q L p jjjjjj q L q p qq L q L q H jj ∂ ∂ −= ∂ ∂ + ∂ ∂ ∂ ∂ − ∂ ∂ −= ∂ ∂ ∂ ∂ =   φφ t L t p t q q L t L t H kk kk q q L p j j j j ∂ ∂ −= ∂ ∂ + ∂ ∂ ∂ ∂ − ∂ ∂ −= ∂ ∂ = ∂ ∂ = φ φ     Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  William Rowan Hamilton 1805-1865
04/08/15 55 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Second Method (Carathéodory) ( ) k jj k q qqtL p   ∂ ∂ = ,, : Canonical Momentum ( ) ( )[ ] ( )k j ji kk k k ptpptqtLpxtH ,,,,:,, φφ +−= Hamiltonian j j q p H = ∂ ∂ jj q L q H ∂ ∂ −= ∂ ∂ t L t H ∂ ∂ −= ∂ ∂ Let compute ( ) j j jjj q H td pd q L q L td d LE ∂ ∂ += ∂ ∂ −      ∂ ∂ =  Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  Return to Table of Content
04/08/15 56 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Equivalent Integrals Given a Scalar S = S (t,qk ) є C2 . Along a curve C: qj = qj (t) we can form the Total Derivative: j j q q S t S td Sd  ∂ ∂ + ∂ ∂ = With the aid of this Scalar ,S, we may construct an alternative Lagrangian by writing ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) td tqtSd tqtqtLcq q S t S tqtqtLctqtqtL j jjk k jjjj , ,,,,:,,* −= ∂ ∂ − ∂ ∂ −=  We obtain a new Integral ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )12 ** 2 1 2 1 ,,,, SSCIctd td Sd tqtqtLctdtqtqtLCI t t jj t t jj −−=      −== ∫∫  ( ) ( ) 21 * SSCIcCI −=− where c > 0 is a constant. Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
04/08/15 57 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Equivalent Integrals Alternative Lagrangian ( ) ( )( ) ( ) ( )( ) ( )( ) td tqtSd tqtqtLctqtqtL j jjjj , ,,:,,* −=  ( ) ( ) ( )( ) ( ) ( )12 ** 2 1 ,, SSCIctdtqtqtLCI t t jj −−== ∫  We obtain That is independent of the choice of the curve C joining the points P1 and P2. It follows that C will be an extremal of the integral I* , if and only if, it is an extremal of the integral I. Accordingly I and I* are called Equivalent Integrals. ( ) ( ) 21 * SSCIcCI −=− Therefore the Lagrangian that gives the Equations of Motions is not necessarily L = T – V but it can be any function L* = c (T – V) – dS/dt. Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
04/08/15 58 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Equivalent Integrals Alternative Lagrangian that gives the same equation of Motion doesn’t have necessarily to differ by a total time derivative. For example Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  ( ) ( ) ( ) ( )[ ]2 2 2 1 2 2 2 1 2121 2 1 qqqqL qqqqL b a −−+= −=   0&0 0&0 22 22 11 11 11 22 22 11 =+=      ∂ ∂ −      ∂ ∂ =+=      ∂ ∂ −      ∂ ∂ =+=      ∂ ∂ −      ∂ ∂ =+=      ∂ ∂ −      ∂ ∂ qq q L q L td d qq q L q L td d qq q L q L td d qq q L q L td d bbbb aaaa         ( ) ( )[ ]2 21 2 21 2 1 qqqqLL ab −−−=−  Return to Table of Content
A Geodesic Field is defined as a Field of Contravariant Vectors , given at each point of a finite Region G in the n+1 Space (t, q1 , q2 ,…,qn ) by a set of n class C2 functions ψj (t,qk ) that are such that, for a suitably chosen function S = S (t,qk ), the following conditions are satisfied while One other condition (Jacobi) that the curves that define a Geodesic Field, is that they don’t intersect for t1 < t < t2. ( )kjj qtq ,ψ= ( ) ( ) ( )kjjkkkk qtqwhenever td Sd qqtLqqtL ,0,,:,,* ψ==−=  ( ) otherwiseqqtL kk 0,,* > ( )kjj qtq ,ψ= 04/08/15 59 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Geodesic Field See “Calculus of Variation” for a detailed exposition. Note: Since we see that it is equivalent to “Action”.( ) ( )∫= t t kk tdqqtLtS 1 ,,  Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
04/08/15 60 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Geodesic Field (continue – 1) For any other curve K in G that joins the points P1 and P2 we have ( ) ( ) FieldGeodesictqqtdqqtL jj t t kk ⊂=Γ=∫ :0,, 2 1 0 *   ( ) ( ) FieldGeodesictqqKtdqqtL jj t t kk ⊄=>∫ > :0,, 2 1 0 *   Therefore the Integral I* (Γ) is a local minimum, and since this is an Equivalent Integral to I (Γ), this is also a local minimum in G. ( )kjj qtq ,ψ= is a System of n First-Order Differential Equations, that may be integrated to cover the region G. Let Γ: qj = qj (t) be a member of this family and Let P1 and P2 the points on Γ, corresponding to t1 and t2, It follows that Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
04/08/15 61 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Basic Properties of Geodesic Field Since ( ) ( ) ( ) FieldGeodesictqqq q S t S qqtLqqtL jjl l kkkk ⊂=Γ= ∂ ∂ − ∂ ∂ −= :0,,:,,*  ( ) ( ) ( )kjj jj p q L jj kk j kk qtqwhenever q S p qd S q qqtL q qqtL jj , ,,,, 0 * ψ= ∂ ∂ −= ∂ − ∂ ∂ = ∂ ∂ = = ∂ ∂       Therefore ( ) FieldGeodesictqq q S p jj jj ⊂=Γ ∂ ∂ = : ( ) ( )kj k kjj qtpqtq ,,, ψφ == This defines the Covariant Vector Field pj = pj (t, qk ) as a function of position in G. Assuming the one-to-one correspondence between pj and (j,k=1,2,…,n), we have k q We get ( ) l l q S p q l l kk p t S q q S t S qqtL ll ll φ φ + ∂ ∂ = ∂ ∂ + ∂ ∂ = ∂ ∂ = =     ,, ( ) ( ) ( )       ∂ ∂ =+−= = ∂ ∂ = k k pqtq q S p l l kkkk q S qtHpqqtLqqtH k kkk kk ,,,,,, ,,φ φ   0,, =      ∂ ∂ + ∂ ∂ k k q S qtH t S and Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
04/08/15 62 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory William Rowan Hamilton 1805-1865 Carl Gustav Jacob Jacobi (1804-1851) 0,, =      ∂ ∂ + ∂ ∂ k k q S qtH t S Hamilton-Jacobi Equation Hamilton-Jacobi is a First-Order Nonlinear Partial Differential Equation for the Scalar Function S = S (t,qk ), that defines the Geodesic Field through from which a Unique Vector Field is obtained jj q S p ∂ ∂ = ( ) ( )kj k kjj qtpqtq ,,, ψφ == Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
04/08/15 63 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory William Rowan Hamilton 1805-1865 Carl Gustav Jacob Jacobi (1804-1851) 0,, =      ∂ ∂ + ∂ ∂ k k q S qtH t S Start with jj q S p ∂ ∂ = ( ) l ljjj k j q qq S xt S q qtS td d td pd  ∂∂ ∂ + ∂∂ ∂ =      ∂ ∂ = 22 , Partially Differentiate the Hamilton Jacobi Equation with respect to qj ( ) ( ) 0 ,,,, 222 = ∂∂ ∂ + ∂ ∂ + ∂∂ ∂ = ∂ ∂ ∂ ∂ + ∂ ∂ + ∂∂ ∂ ∂ ∂ = = ∂ ∂ l ljjj q S p x p Hj l l k k j k k j q qq S q H tq S q p p pqtH q pqtH tq S ll l l   we obtain j j q H td pd ∂ ∂ −= Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
04/08/15 64 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory William Rowan Hamilton 1805-1865 Carl Gustav Jacob Jacobi (1804-1851) we obtain        = ∂ ∂ −= = ∂ ∂ = nj q H td pd nj p H td qd j j j j ,,2,1 ,,2,1   ( ) 0= ∂ ∂ += ∂ ∂ −      ∂ ∂ = = ∂ ∂ ∂ ∂ = ∂ ∂ j j td pd q L q H q Ljjj q H td pd q L q L td d LE j j jj   A System of 2n First-Order Ordinary Differential Equations which the curve Γ must satisfy. This is equivalent to Euler-Lagrange n Second-Order Partial Differential Equations, since ( ) k jj k q qqtL p   ∂ ∂ = ,, : Canonical Momentum ( ) ( )[ ] ( )k kj ji ikk k k pqtppqtqtLpqtH ,,,,,,:,, φφ +−= Hamiltonian Hamilton’s Equations Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
04/08/15 65 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory William Rowan Hamilton 1805-1865 Carl Gustav Jacob Jacobi (1804-1851) Let return to the condition ( ) ( ) ( ) ( ) 0 ,, ,,,,* ≥ ∂ ∂ − ∂ ∂ −= k k jj jjjj q q qtS t qtS qqtLqqtL  where the equality holds for ( ) ( )kj k kjj qtpqtq ,,, ψφ == We want to eliminate S from this inequality. For this use ( ) ( )[ ] ( )l lk kl ljj p q S j j j pqtppqtqtL q S qtH t qtS jj ,,,,,,:,, , φφ +−=      ∂ ∂ = ∂ ∂ − = ∂ ∂ ( ) ( )[ ] ( ) ( ) 0 , ,,,,,,,, ≥ ∂ ∂ −+− k k j l lk kl ljjjj q q qtS pqtppqtqtLqqtL  φφ ( )jjj j qqtE ,,, φ : Weierstrass Excess Function Weierstrass Sufficiency Condition for a Local Minimum ( ) ( ) ( ) ( ) 0,,,,:,,, ≥− ∂ ∂ +−= kk k jjjjjjj j q q L qtLqqtLqqtE  φφφ and ( ) ( ) k j k jj k q qtS q qqtL p ∂ ∂ = ∂ ∂ = ,,, :   Karl Theodor Wilhelm Weierstrass (1815-1897) Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  Return to Table of Content
04/08/15 66 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtxtxtLCI  Example: Recovering Newton Equation VF −∇= - Force on Mass m, due to External Potential V rrmT  ⋅= 2 1 - Kinetic Energy of Mass m ( )rV  - Potential Energy of the External Force rm r L p    = ∂ ∂ = - Canonical Momentum - HamiltonianEVTVrrmrrmVrrmxpLH =+=+⋅=⋅++⋅−=+−= : 2 1 2 1 :   VrrmVTL −⋅=−=  2 1 : - Lagrangian
04/08/15 67 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtxtxtLCI  Example: Recovering Newton Equation (continue – 1) S r S rm r L p ∇= ∂ ∂ == ∂ ∂ =    - Canonical Momentum - HamiltonianVTVrrmH +=+⋅=  2 1 Hamilton-Jacobi Equation where S is given by VSS m VrrmH +∇⋅∇=+⋅= 2 1 2 1  0 2 1 =+∇⋅∇+ ∂ ∂ VSS mt S Hamilton-Jacobi Equation ( ) 0,, =∇+ ∂ ∂ SrtH t S 
04/08/15 68 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtxtxtLCI  Example: Recovering Newton Equation (continue – 2) 0 2 1 =+∇⋅∇+ ∂ ∂ VSS mt S Hamilton-Jacobi Equation Let take the Gradient of this equation ( ) ( ) Vrr r m p t VSS mt S rmpS ∇+⋅ ∂ ∂ + ∂ ∂ =∇+∇⋅∇∇+ ∂ ∂ ∇= ==∇      0 22 1 0 FVp t p td d  =−∇= ∂ ∂ = Newton Equation ( ) p t rm r rp t p td d      ∂ ∂ = ∂ ∂ ⋅+ ∂ ∂ = 0
04/08/15 69 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtxtxtLCI  Example: Classical Harmonic Oscillator In Equilibrium Displaced from Equilibrium xmxkF =−= - Force of Spring on Mass m 2 2 1 xmT = - Kinetic Energy of Mass m 2 2 1 xkV = - Potential Energy of the Spring xm x L p   = ∂ ∂ = - Canonical Momentum - HamiltonianEVTxkxmxxmxkxmxpLH =+=+=⋅++−=+−= : 2 1 2 1 2 1 2 1 : 2222  22 2 1 2 1 : xkxmVTL −=−=  - Lagrangian 02 2 >= ∂ ∂ m x L 
04/08/15 70 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtxtxtLCI  Example: Classical Harmonic Oscillator (continue – 1) In Equilibrium Displaced from Equilibrium- Hamiltonian( ) 2 2 2 1 2 1 :, xk m p pxH += x m k td pd mtd xd m p p H td xd xk x H td pd −==⇒        = ∂ ∂ = −= ∂ ∂ −= 1 2 2 Initial Conditions: ( ) ( )     == == 00 0 t td xd Atx ( ) ( ) ( ) m k tAmtxmtp tAtx =    −== = : sin cos ω ωω ω  : Solution On the minimizing curve Γ ( ) . 2 1 cos 2 1 sin 1 2 1 2 1 2 1 :, 2222222 2 constAktAkt m k Am m xk m p pxH ==+      =+= ωω Hamilton’s Equations:
04/08/15 71 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtxtxtLCI  Example: Classical Harmonic Oscillator (continue – 2) In Equilibrium Displaced from Equilibrium ( ) ( ) td Sd tAktAktA m k m m xk m p VTpxL =−=−        =−=−= ωωω 2cos 2 1 cos 2 1 sin 1 2 1 2 1 2 1 , 22 2 2 2 ( ) t t tA k ttA k tA k S ω ω ω ω ωω ω ω ω cos sin cos 2 1 cossin 2 1 2sin 4 1 222 −=−=−= ( ) t t x k xtS ω ω ω cos sin 2 1 , 2 −= ( ) HAk t xk t xtS tAx −=−=−= ∂ ∂ = 2 cos 2 2 2 1 cos 1 2 1, ω ω ( ) ptA k t t x k x xtS tAmp m k tAx ωω ω ω ω ω ωω ω ω sincos sin cos sin, −= = = =−=−= ∂ ∂ ω ω m m k mkm mk kk ==== / Return to Table of Content
04/08/15 72 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  Invariance Properties of the Fundamental Integral Theorem of Noether for Single Integral Amalie Emmy Noether (1882 –1935) Consider the Functional ( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  Symmetry of Lagrangian is a Geometric Property in which quantities remain unchanged under coordinate transformation. Invariance is an Algebraic or Analytic Property in which “Integrals of Motion” are constant along System Trajectories.. Consider also a continuous family of coordinate transformation qj (t) → qj (t,ε) parameterized by a single quantity ε. We assume that the Lagrangian is invariant to this parameter which implies a Symmetry of the System. Thus ( ) ( )( ) ( ) ( )( )0,,0,,,,,, === εεεε tqtqtLtqtqtL jjjj  ( ) ( )( ) nj q q Lq q Lq q Lq q L tqtqtL d d j j j j n j j j j j jj ,,100,,,, 1       == ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ⇒=         ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∑ = αααα εε α
73 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  Invariance Properties of the Fundamental Integral Theorem of Noether for Single Integral (continue – 1) Amalie Emmy Noether (1882 –1935) nj q q Lq q L j j j j ,,10    == ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ αα Since qi(ε) is a solution of Euler-Lagrange’s Equation we have nj q L q L td d jj ,,10   == ∂ ∂ −         ∂ ∂ Substitute this in previous equation we obtain ( ) ( ) ( ) nj tq q L td dtq q Ltq q L td d j j j j j j ,,10 ,,,     ==         ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ + ∂ ∂         ∂ ∂ α α α α α α The quantity is a Constant of Motion. ( ) ( ) α α α α ∂ ∂ = ∂ ∂ ∂ ∂ ,, tq p tq q L j j j j  Noether Theorem is not of great importance in Particle Dynamics. It becomes extremely important in Field Theory and in Quantum Mechanics. nj q L p j j ,,1 = ∂ ∂ = Return to Table of Content
74 SOLO Foundation of Geometrical Optics Fermat’s Principle (1657) in Optics 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). The idea that the light travels in the shortest path was first put forward by Hero of Alexandria in his work “Catoptrics”, cc 100B.C.-150 A.C. Hero showed by a geometrical method that the actual path taken by a ray of light reflected from plane mirror is shorter than any other reflected path that might be drawn between the source and point of observation.
75 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations 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  ++=      +      +=
76 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 1) 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   
77 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 2) 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 ∂ ∂ =     
78 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 3) 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
79 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 4) 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 ∂ ∂ =     
80 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 5) 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 ∇=       The equation is called Eikonal Equation Eikonal (from Greek έίκων = eikon → image) . See “Geometrical Optics” Presentation
81 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 6) Hamilton’s Canonical Equations Define ( ) [ ] ( ) ( ) [ ] ( ) sd zd zyxn zy zzyxn z F p sd yd zyxn zy yzyxn y F p z y ,, 1 ,, : ,, 1 ,, : 2/122 2/122 = ++ = ∂ ∂ = = ++ = ∂ ∂ =       ( )( ) ( )2222222 1 zynzypp zy  +=+++ Adding the square of twose two equations gives ( ) ( ) 2 222 2 22 1       = +− =++ xd sd ppn n zy zy  from which Substituting in ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += gives ( ) ( )222 2 ,,,, zy zy ppn n ppzyxF +− = ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      +=
82 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 7) Hamilton’s Canonical Equations (continue – 1) From ( ) [ ] ( ) ( ) [ ] ( ) sd zd zyxn zy zzyxn z F p sd yd zyxn zy yzyxn y F p z y ,, 1 ,, : ,, 1 ,, : 2/122 2/122 = ++ = ∂ ∂ = = ++ = ∂ ∂ =       solve for ( ) ( )222 2 ,,,, zy zy ppn n ppzyxF +− =and ( ) ( )222 222 zy z zy y ppn p z ppn p y +− = +− =   Define the Hamiltonian ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sd xd zyxnppzyxn ppn p ppn p ppn n zpypppzyxFppzyxH zy zy z zy y zy zyzyzy ,,,, ,,,,:,,,, 222 222 2 222 2 222 2 −=+−−= +− + +− + +− −= ++−= 
83 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 8) Hamilton’s Canonical Equations (continue – 2) From We obtain the Hamilton’s Canonical Equations ( ) ( ) ( ) ( ) sd xd zyxnppzyxnppzyxH zyzy ,,,,,,,, 222 −=+−−= ( ) ( )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 +− ∂ ∂ −= ∂ ∂ −= +− ∂ ∂ −= ∂ ∂ −=
84 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 9) Hamilton’s Canonical Equations (continue – 3) From ( ) ( ) ( ) ( ) sd xd zyxnppzyxnppzyxH zyzy ,,,,,,,, 222 −=+−−= ( )222 zy ppn n sd xd +− = By similarity with ( ) sd yd zyxnpy ,,= define ( ) ( ) ( ) ( )222 ,,,,,,,,: zyzyx ppzyxnppzyxH sd xd zyxnp +−=−== Let differentiate px with respect to x ( ) x H xd Hd ppn x n n xd pd zy x ∂ ∂ −=−= +− ∂ ∂ = 222 Let compute ( ) ( ) x n n ppn ppn x n n sd xd xd pd sd pd zy zy xx ∂ ∂ = +− +− ∂ ∂ == 222 222
85 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 10) Hamilton’s Canonical Equations (continue – 4) and ( ) ( ) x n n ppn ppn x n n sd xd xd pd sd pd zy zy xx ∂ ∂ = +− +− ∂ ∂ == 222 222 ( ) ( ) y n n ppn ppn y n n sd xd xd pd sd pd zy zy yy ∂ ∂ = +− +− ∂ ∂ == 222 222 ( ) ( ) z n n ppn ppn z n n sd xd xd pd sd pd zy zy zz ∂ ∂ = +− +− ∂ ∂ == 222 222 n sd pd ∇=  xp nsd xd 1 = ( ) ( ) y zy zy y p nn ppn ppn p sd xd xd yd sd yd 1 222 222 = +− +− == ( ) ( ) z zy zy z p nn ppn ppn p sd xd xd zd sd zd 1 222 222 = +− +− == p nsd rd ray   1 = We recover the result from Geometrical Optics Return to Table of Content
04/08/15 86 SOLO Classical Field Theories The Inverse Square Law of Forces The Two Body Central Force Problem Assume a system of two mass points m1 and m2 located at , respectively, subject to an interaction potential V, where V is any function of the range vector between particles. 21 randr  Define: ( ) ( )212211 12 / : mmrmrmR rrr c ++= −=   - range vector between particles - system center of mass 21 1 2 21 2 1 , mm rm Rr mm rm Rr cc + += + −=   We obtain: The kinetic energy of the system:         + +⋅        + ++        + −⋅        + −=⋅+⋅= 21 1 21 1 2 21 2 21 2 1222111 2 1 2 1 2 1 2 1 mm rm R mm rm Rm mm rm R mm rm RmrrmrrmT cccc       ( ) rr mm mm RRmmT M cc    ⋅ + +⋅+= 21 21 21 2 1 2 1or ( ) ( ) ,, 2 1 2 1 21 rrVrrMRRmmVTL cc −⋅+⋅+=−= The Lagrangian of the system is
04/08/15 87 SOLO Classical Field Theories The Inverse Square Law of Forces ( ) ( ) ( ) ( )2121 222 /: 2 1 2 1 mmmmM rVrrMrVrrMVTL += −+=−⋅=−= θ When V is a function of r only like in classical gravitation and electromagnetic fields, we can write, in polar coordinates and ignoring the term describing the motion of center of mass The Lagrange’s Equations are ,2,10 =∀= ∂ ∂ −       ∂ ∂ ∂ ∂ i q L t q L td d ii 2. r - independent variable r V rM r L rM r L ∂ ∂ −= ∂ ∂ = ∂ ∂ 2 , θ  1. θ - independent variable 0,2 = ∂ ∂ = ∂ ∂ θ θ θ L rM L   ( ) ( ) 02 == ∂ ∂ − ∂ ∂ θ θθ   rM td dLL td d ( ) 02 = ∂ ∂ +−= ∂ ∂ − ∂ ∂ r V rMrM r L r L td d θ 
04/08/15 88 SOLO Classical Field Theories The Inverse Square Law of Forces The Specific Angular Momentum and Angular Momentum are defined as θθ  22 :&: rMhMlrh === ( ) ( ) 02 == ∂ ∂ − ∂ ∂ θ θθ   rM td dLL td d The Equation means that Angular Momentum is constant (Conservation of Angular Momentum ) This Equation can be rewritten as θdrMdtl 2 = that implies       =⇒= θθθ d d rM l d d rM l dt d d d rM l dt d 222 2 2 a b The Area swept out by one of the body when moving around the other (see Figure) is given by ( )θdrrAd 2 1 = It follows that . 22 const h d Ad rM l td Ad === θ This is Kepler Second Law.
04/08/15 89 SOLO Classical Field Theories The Inverse Square Law of Forces 02 = ∂ ∂ +− r V rMrM θReturn to the equation θ2 : rMl =and We have ( ) ( ) 0 22 2 2 2 2 3 2 =      ++=      + ∂ ∂ += ∂ ∂ +− rV rM l rd d rMrV rM l r rM r V rM l rM  or ( ) ( ) ( ) ( ) 2 2 2 2 2 : 2 r l rVrV rV rd d rV rM l rd d rM eff eff µ += −=      +−= Multiplying the differential equation by we obtainr ( ) ( )rV td d rV rd d td rd rM td d rrM effeff −=−=      = 2 2 1  or   0 2 1 2 ==               + energy total energy potential eff energy kinetic E td d VrM td d   The total Energy is conservedeffVrME += 2 2 1 : 
04/08/15 90 SOLO Classical Field Theories The Inverse Square Law of Forces The total Energy is conservedeffVrME += 2 2 1 :  Solving for we obtainr ( )       −−±=−±= 2 2 2 22 rM l VE M VE M r eff  or       −− = 2 2 2 2 rM l VE M rd td Using we obtain θd d rM l dt d 2 = 2 222 122 1 r rd rl VM l EM d −− =θ Consider the case where V (r) = -k/r = - k u ud uu l kM l EM d 2 22 22 1 −− − =θ Integrating this expression gives 1 2 1 1 sin 28 2 2 sin 2 2 2 1 2 22 2 1 +       − =       + +− =− −− kM lE rkM l l kM l EM l kM u iθθ
04/08/15 91 SOLO Classical Field Theories The Inverse Square Law of Forces 1 2 1 1 sin 28 2 2 sin 2 2 2 1 2 22 2 1 +       − =       + +− =− −− kM lE rkM l l kM l EM l kM u iθθ Inverting this expression we obtain ( )         −+−= i kM lE l kM r θθsin 2 11 1 2 2 2 This is usually written as ( ){ } 2 2 2 2 1: sin1 1 kM lE e e l kM r i += −−= θθ This is a equation of Conic Section: 1.If e > 1 and E > 0, the trajectory is a Hyperbola. 2. If e = 1 and E = 0, the trajectory is a Parabola. 3. If e < 1 and E < 0, the trajectory is a Ellipse. 4. If e = 0 and E = -mk2 /(2l2 ), the trajectory is a Circle. Return to Table of Content
04/08/15 92 SOLO Four-Dimensional Formulation of the Theory of Relativity Special Relativity Theory We introduce a 4-dimensional space-time or four-vector x with components: ( ) ( ) ctxxxxxxxx == 003210 ,,,,: µ The differential length element is defined as: ( ) ( ) ( ) ( ) ( ) ( ) 220232221202 : xdxdxdxdxdxds  −=−−−= or ( ) νµ µν µ ν νµ µν dxdxgdxdxgds summationsEimstein convention '3 0 3 0 2 == ∑∑= = The metric corresponding to this differential length is given by g: ( )µνgg =               − − − = 1000 0100 0010 0001 therefore: ( )νµµν ≠=−==== 0,1,1 33221100 ggggg We can see that Igg =               =               − − −               − − − = 1000 0100 0010 0001 1000 0100 0010 0001 1000 0100 0010 0001
04/08/15 93 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 1) Therefore since we have γ µ νγ µν δ=gg µν µν gg = If a 4-vector has the contravariant components A0 ,A1 ,A2 ,A3 we have ( ) ( ) ( )32103210 ,,,,,, AAAAwhereAAAAAAA === α Using the g metric we get the same 4-vector described by the covariant components: ( )AA A A A A A A A A AgA  −=               − − − =                             − − − == , 1000 0100 0010 0001 0 3 2 1 0 3 2 1 0 β αβα The Scalar Product of two 4-vectors is: ( )( ) ( )( ) αα αβαβ αβα αα α ABgABgABABAABBABAABBAB ==⋅−=−==−=  000000 ,,,, Special Relativity Theory
04/08/15 94 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 1) Therefore since we have γ µ νγ µν δ=gg µν µν gg = If a 4-vector has the contravariant components A0 ,A1 ,A2 ,A3 we have ( ) ( ) ( )32103210 ,,,,,, AAAAwhereAAAAAAA === α Using the g metric we get the same 4-vector described by the covariant components: ( )AA A A A A A A A A AgA  −=               − − − =                             − − − == , 1000 0100 0010 0001 0 3 2 1 0 3 2 1 0 β αβα The Scalar Product of two 4-vectors is: ( )( ) ( )( ) αα αβαβ αβα αα α ABgABgABABAABBABAABBAB ==⋅−=−==−=  000000 ,,,, Special Relativity Theory
95 SOLO Special Relativity Theory Four-Dimensional Formulation of the Theory of Relativity (continue – 2) Let introduce the following definitions:       ∇ ∂ ∂ =      ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ =∂       −∇ ∂ ∂ =      ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ = ∂ ∂ =∂ ,,,,: ,,,,: 03210 03210 xxxxxx xxxxxx αα α α The 4-divergence of a 4-vector A is the invariant: ( ) ( ) A x A AA x AAA x A  ⋅∇+ ∂ ∂ =      ∇ ∂ ∂ =∂=−      −∇ ∂ ∂ =∂ 0 0 0 0 0 0 ,,,, α αα α The four-dimensional Laplacian operator (d’Alembertian) is defined as:         ∇− ∂ ∂ =      ∇ ∂ ∂       −∇ ∂ ∂ =∂∂=      −∇ ∂ ∂       ∇ ∂ ∂ =∂∂= 2 20 2 0000 ,,,,: xxxxx α αα α
04/08/15 96 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 3) We have: ( ) ( ) ( ) ( ) ( )         −=−=−−−= 2 2 22222232221202 1 1 dt xd c dtcxddtcdxdxdxdxds   We define - the velocity vector of the particle in the inertial frame. td xd u   = 2 1 1 :&: u uu c u β γβ − == ( ) ( ) 22222222 2 2 222 /11 τγβ dcdtcdtc c u dtcds uu ==−=      −= where ( ) ( )t dt dtd u u γ βτ =−= 2 1 is the differential of the proper time τ. Special Relativity Theory
04/08/15 97 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 4) 4-Coordinates Vector The 4-coordinates vector is: ( ) ( ) ( ) ( )xxxxxxxxxxxxxx  −=−−−=== ,,,,&,,,, 0321003210 α α ( )( ) ( ) ( ) 23210000 22222 ,, sdxdxdxdxdxdxdxdxdxdxdxdxdxd =−−−=⋅−=−=  α α Special Relativity Theory
04/08/15 98 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 5) 4-Velocity Vector ( ) ( ) td xd uUUuc d dt dt xd d dt dt dx d xd d dx U uu    ===      =      = ,,,,: 0 00 γγ ττττ α ( ) ( ) 20 0 u 1 1 :,,,:       − =−=−=      −= c UUuc d xd d dx U uuu γγγ ττ α   The 4-Velocity vector is defined as: ( ) ( ) ( ) 2 2 2 2 2 2 222 1 1 1 ,, c c u c c u ucucucUU uuuuu =      −       − =−=−= ∆ γγγγγα α  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )         −=−        = =−=        −+−        = =−+−    === ττττ ττττττ ττττ α α d Ud d dU UUUU d Ud d dU d Ud U d dU U d Ud d dU UUUU d Ud d dU UU d d UUUUUU d d UU d d d dc          ,,2,,2 22,,,, ,,,,0 0 00 0 0 0 0 00 0 0000 2 ( ) ( ) ( ) ( ) 0==== α αα αα αα α ττττ UU d d U d d UU d d UUU d d Special Relativity Theory
04/08/15 99 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 5) 4-Moment Vector The 4-Momentum vector of the mass m is defined as: ( )UUmUmp  ,000 == αα 2 02 2 0 000 0 1 1 :       − == ==       − === c u m mofEnergymcE c E mc c u cm cmUmp uγ pumu c u m umUm u  ==       − == 2 0 00 1 γ Therefore: ( ) ( ) ( ) ( )UUmUmpppUUmUmppp  −==−==== ,,&,, 000 0 000 0 αα αα 22 0 2 0 2 2 2 ,, cmUUmp c E p c E p c E pp ==      −=      −      = α α α α  from which we get: ( ) ( )22 0 22222 cmcpcmE +== Special Relativity Theory
100 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 6) 4-Force Vector The 4-Force vector on the mass m is defined as:       =      == ττττ µµ d pd d Ed c p c E d d p d d F   , 1 ,: From the relations: ( ) u u u d dtdt dtd γ τγ βτ =→=−= 2 1 ( ) ( ) c uF d td td rd c F d Ed c EdcmEdrdFdT u   ⋅ =⋅=→=−=⋅= γ ττ 12 0 F td pd d td d pd u  γ ττ == Special Relativity Theory 2 u 1 1 :       − = c uγ ( ) ( )       −⋅=        − ⋅ ==       ⋅=        ⋅ == F c u FF c uF d pd F F c u FF c uF d pd F uu u uu u   ,,: ,,: γγ γ τ γγ γ τ µ µ µ µ
101 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue –7) 4-Force Vector (continue – 1) Assuming that the rest mass doesn’t change :00 = τd dm ( ) τττ µ µµµ d dU mUm d d p d d F 00 === Using the relation: ( ) ( ) ( ) ( ) 0==== α αα αα αα α ττττ UU d d U d d UU d d UUU d d we get: 00 == τ µ µ µ µ d dU UmFU Special Relativity Theory
102 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue –8) 4-Force Vector (continue – 2) Assuming that the rest mass changes ,:00 ≠ τd dm ( ) µµµ µ µµ ττττ Π+=+== FU d md d Ud mUm d d p d d 0 00 Using the relation: ( ) ( ) ( ) ( ) 0==== α αα αα αα α ττττ UU d d U d d UU d d UUU d d where: 00 == τ µ µ µ µ d dU UmFU Special Relativity Theory µµ µ µ ττ U d md d Ud mF 0 0 :,: =Π= ( ) ( )         − ⋅ ==         ⋅ == = = F c uF d pd F F c uF d pd F u u constm u u constm   γ γ τ γ γ τ µ µ µ µ ,: ,: . . 0 0
103 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue –9) 4-Force Vector (continue – 3) Assuming that the rest mass changes ,:00 ≠ τd dm Special Relativity Theory ( ) ( ) dt xd uUUuc d dt dt xd d dt dt dx d xd d dx U uu    ===      =      = ,,,,: 0 00 γγ ττττ α ( ) ( ) ( )2 0 0 /1/1:,,,: cuUUuc d xd d dx U uuu −=−=−=      −= γγγ ττ α   The 4-Velocity vector is defined as: Let define:       Π− Φ =Π      Π Φ =Π uuuu cc γγγγ α α  ,:,,: ( ) ( ) 02 :,, Φ=Π⋅−Φ=      Π− Φ =Π  u c ucU uuuuu γγγγγα α µµµ µ µ ττ Π+=+ FU d md d Ud mU 0 0  02 0 0 0 0 Φ Π+=+ µ µ µ µ µ µ µ µ ττ UFUUU d md d Ud Um c τ α α d md cU 020 =Π=Φ The 4-Velocity Momentum vector that the particle loses per unit proper time τ.
104 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue –10) 4-Force Vector (continue – 4) Assuming that the rest mass changes ,:00 ≠ τd dm Special Relativity Theory       µ µ µµµ µ ττ D R U d md F d Ud m 0 0 −Π+= µ µ τ D d Ud m =0 µµ τ µµµ µµµ τ U c U d md R RFD c d md 2 0 0 020 : : Φ −Π=−Π= += Φ= Fμ – External Force Rμ – Reaction Force Dμ – Driving Force 0 0 00 2 0 20 =+= = Φ −Π= Φ   µ µ µ µ µ µ µ µ µ µ µ µ RUFUDU UU c URU c       Π+Φ+⋅=Π+=      =  F c uF c Fp c E d d p d d u , 11 , 0 γ ττ µµµ ( ) u u u d dtdt dtd γ τγ βτ =→=−= 2 1 Also       Π+Φ+⋅=      =  F c uF c p c E td d p td d , 11 , 0µ
105 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue –11) 4-Force Vector (continue – 5) Assuming that the rest mass changes ,:00 ≠ τd dm Special Relativity Theory ( ) ( ) ( )PcQdtcdt dt cdQ u uu dtd u  δδ γ γγτδ γτ µµ ,/,/,/ 00 / =ΠΦ=ΠΦ=Π= = In special cases the rate of change of Πμ may be due to emission or absorption of Heat by the particle where     Π= Φ= tdP tdQ  δ δ 0
04/08/15 106 SOLO Four-Dimensional Formulation of the Theory of Relativity Special Relativity Theory Principle of Last Action for 4-Vector for a Free Particle (zero external forces) We want to define the Action Integral for 4-Vectors, similar to Fermat Principle in Optics, to recover the 4-Momentum and 4-Force Vectors. Define ∫∫ =−= 2 1 : t t b a tdLsdαA α is a constant, to be defined ( )22 /1 cudtcsd −=Using we obtain ∫∫ =−−= 2 1 2 1 22 /1: t t t t tdLtdcucαA Therefore the Lagrangian is c u ccucL 2 /1 2 22 α αα +−≈−−= Since the constant α c in the Lagrangian doesn’t affect the equations of motion, we concentrate on the second term that must be equal to Energy. We choose α to obtain the non-relativistic kinetic energy m0u2 /2 (m0 – rest mass), i.e.: cm0=α 222 0 /1 cucmL −−= ∫∫ −−=−−= 2 1 2 1 0 222 0 /1: s s t t sdcmtdcucmA ( ) ( ) uzyxqzyxq   === ,,&,,
04/08/15 107 SOLO Four-Dimensional Formulation of the Theory of Relativity Special Relativity Theory Canonical Momentum 222 0 /1 cucmL −−= ( )( ) um cu um czyxcm z y x q L p         = − =++−−           ∂∂ ∂∂ ∂∂ = ∂ ∂ = 22 022222 0 /1 /1 / / / ( )( ) 0/1 / / / 22222 0 =++−−           ∂∂ ∂∂ ∂∂ = ∂ ∂ czyxcm z y x q L  Particle Equation of Motion Q td pd cu um td d q L q L td d   ==        − = ∂ ∂ −      ∂ ∂ 22 0 /1 - are the External Forces acting on the particle (zero for free particle)Q  Principle of Last Action for 4-Vector for a Free Particle 22 0 /1 : cu m m − = - relativistic mass of Particle at velocity u Free Particle Lagrangian
04/08/15 108 SOLO Four-Dimensional Formulation of the Theory of Relativity Special Relativity Theory 222 0 /1 cucmL −−= Principle of Last Action for 4-Vector for a Free Particle The Free Particle Hamiltonian is defined as: LpH −⋅= u:  ( )( ) um cu um czyxcm z y x q L p         = − =++−−           ∂∂ ∂∂ ∂∂ = ∂ ∂ = 22 022222 0 /1 /1 / / / where: therefore: 2 22 2 0222 022 0 /1 /1u /1 u: cm cu cm cucm cu um LpH = − =−+⋅ − =−⋅=    We van see that the Hamiltonian H derived from the Lagrangian L is equal to the Total Energy E of the Free Particle Ecm cu cm LpH == − =−⋅= 2 22 2 0 /1 u:  Free Particle Lagrangian
04/08/15 109 SOLO Four-Dimensional Formulation of the Theory of Relativity Special Relativity Theory α αUUcucmcucmL 22 0 222 0 /1/1 −−=−−= Principle of Last Action for 4-Vector for a Free Particle (Covariant Treatment) Let use: UαUα = c2 to write ∫∫∫ −=−=−−= 2 1 2 1 2 1 00 222 0 /1: τ τ α α τ τ ττ dUUcmdccmtdcucm t t A We defined ( ) u u u d dtdt dtd γ τγ βτ =→=−= 2 1 2 u 1 1 :       − = c uγ therefore the Action Integral is To use this for a variational calculation we must add the constraint 2 cUU =α α 0= τ α α d Ud Uor equivalently This can be added using Lagrange’s multipliers technique, but here we use a different approach. ττ ττ τ βα αβα α α αα α β αβα dcxdxdgxdxdd d xd d xd dUU xdgxd ==== =
04/08/15 110 SOLO Four-Dimensional Formulation of the Theory of Relativity Special Relativity Theory Principle of Last Action for 4-Vector for a Free Particle (Covariant Treatment) ∫∫∫ −=−=−−= 2 1 2 1 2 1 00 222 0 /1: τ τ α α τ τ ττ dUUcmdccmtdcucm t t A Action Integral is where s = s (τ) is any monotonically increasing function of τ. ττ βααβ βα αβα α dcsd sd xd sd xd gxdxdgdUU === ∫−= 2 1 0: s s sd sd xd sd xd gcm βααβ A sd xd sd xd gcmL βααβ 0: −= ( ) ( )γδ αβ gg =               − − − = 1000 0100 0010 0001 sd xd sd xd sd xd cm sd xd sd xd g sd xd g sd xd g cm sd xd L x L jj β β α βααβ αββα αα 2 2 2 ,0 00 −= + −=       ∂ ∂ = ∂ ∂ Euler-Lagrange Equation             −=               −= ∂ ∂ −                   ∂ ∂ = sd d c sd xd d d sd d cm sd xd sd xd sd xd sd d cm x L sd xd L sd d sd d ττ τ α β β α αα  000 02 2 0 = τ α d xd m 4-Vector Free Particle Equation of MotionReturn to Table of Content
04/08/15 111 SOLO Classic Electrodynamics Start with Microscopic Maxwell’s Equations in Gaussian Coordinates: eE ρπ4=⋅∇  Gauss’ Law (Electric) j ct E c B    π41 = ∂ ∂ −×∇ Ampère’ Law (with Maxwell’s Extension) 0 1    = ∂ ∂ +×∇ t B c EFaraday’ Law of Induction Gauss’ Law (Magnetic) 0=⋅∇ B  Lorentz Force Equation: Electric Field Intensity [statV . cm-1 ] E  Magnetic Induction [statV . sec . cm-2 = gauss] B  ρe Charge density [statC . cm-3 ] j  Current Density [statA . cm-3 ] eF  Electromagnetic force [dynes] u  Charge velocity [cm . sec] 1 V = 1/3x10-2 statV 1 C = 3x109 statC 1 A= 3x109 statA       ×+= B c u EqFe  Classical Electromagnetic Theory q electric charge [statC]
04/08/15 112 SOLO        = ∂ ∂ −×∇⋅∇ =⋅∇ ∂ ∂ j ct E c B E tc e     π ρπ 41 4 1 Classical Electromagnetic Theory Continuity Equation0=⋅∇+ ∂ ∂ j t e ρ Classic Electrodynamics
04/08/15 113 SOLO 0 111      =        ∂ ∂ +×∇=×∇ ∂ ∂ +×∇= ∂ ∂ +×∇ t A c EA tc E t B c E Therefore we can define a Potential φ, such that We can define a vector such that0=⋅∇ B  0≡×∇⋅∇⇐×∇= AAB  A  0 1 ≡∇×∇⇐−∇= ∂ ∂ + ϕϕ t A c E   and φ are not uniquely defined since will give the same results for . Such a transformation that doesn’t change the results is called a “Gauge Transformation” on the Potentials. ( )BE  , A  t f c andfgradAA ∂ ∂ −=+= 1 11 ϕϕ  ϕ∇− ∂ ∂ −= ×∇= t A c E AB    1 We obtained Classical Electromagnetic Theory Classic Electrodynamics Electrodynamic Potentials and φA 
04/08/15 114 SOLO Classical Electromagnetic Theory j ct E c B    π41 = ∂ ∂ −×∇From we obtain ϕ∇− ∂ ∂ −= ×∇= t A c E AB    1 ( ) j c A t A ctc A tct A c A      πϕϕ 41111 2 2 2 22 2 2 =        ∇− ∂ ∂ +      ∂ ∂ +⋅∇∇= ∂ ∂ ∇− ∂ ∂ +×∇×∇ From we obtaineE ρπ4=⋅∇  eA tc ρπϕ 4 1 2 =∇−⋅∇ ∂ ∂ −  Since and φ are not uniquely defined, let add the following relation: A  0 1 = ∂ ∂ +⋅∇ tc A ϕ We obtain Lorenz Condition e tc j c A t A c ρπϕ ϕ π 4 1 41 2 2 2 2 2 2 2 2 =∇− ∂ ∂ =∇− ∂ ∂   Waveform Equations for and φ.A  Ludwig Valentin Lorenz 1829-1891 Classic Electrodynamics
04/08/15 115 SOLO 0 1 = ∂ ∂ +⋅∇ tc A ϕ Lorenz Condition e tc j c A t A c ρπϕ ϕ π 4 1 41 2 2 2 2 2 2 2 2 =∇− ∂ ∂ =∇− ∂ ∂   Waveform Equations for and φ.A  If we define the new Potentials through the relations t f c andfAA ∂ ∂ −=∇+= 1 11 ϕϕ  To satisfy the Lorenz Condition 0 111 2 2 2 2 0 1 1 = ∂ ∂ −∇+ ∂ ∂ +⋅∇= ∂ ∂ +⋅∇ t f c f tc A tc A   ϕϕ 0 1 2 2 2 2 = ∂ ∂ −∇ t f c fTherefore f must satisfy Classical Electromagnetic Theory Classic Electrodynamics
116 SOLO Energy and Momentum ( ) EJB c u Eup e Buu Ju e ee    ⋅=      ×+⋅= =×⋅ = 0 ρ ρ or ( ) ( ) ( ) ( ) ( )[ ] ( )BE c t B BE t E E t E BEEB c E t E c B c EJp t B c E BEBEEB J ct E c B e e                 ×⋅∇−        ∂ ∂ ⋅+⋅ ∂ ∂ −= ⋅ ∂ ∂ −×⋅∇−×∇⋅= ⋅        ∂ ∂ −×∇=⋅= ∂ ∂ −=∇× ×⋅∇=∇×⋅−∇×⋅ + ∂ ∂ =∇× ππ ππ π π 44 1 4 1 4 1 4 1 41 ( ) ( ) ( ) ( )          =⋅∇ =⋅∇ + ∂ ∂ =×∇ ∂ ∂ −=×∇ 0 4 41 1 BGM EGE J ct E c BA t B c EF e e        ρπ π Classical Electromagnetic Theory ( )BE cBBEE t EJp e    ×⋅∇−        ⋅ + ⋅ ∂ ∂ −=⋅= ππ 4224 1 The power density of the Lorentz Force the charge ρe and velocity isu 
117 SOLO Energy and Momentum (continue -1) We identify the following quantities EJe  ⋅       ⋅ ∂ ∂ =⋅= BB t pBBw mm  ππ 8 1 , 8 1       ⋅ ∂ ∂ =⋅= EE t pEEw ee  ππ 8 1 , 8 1 ( )BE c pR  ×⋅∇= π4 - Magnetic energy and power densities, respectively (Energy transferred from the field to the particles) - Electric energy and power densities, respectively - (Energy transferred from the field to the particles) - Radiation power density (Power lost through the boundaries ( )EE tt E E    ⋅ ∂ ∂ = ∂ ∂ ⋅ 2 1 ( )BB tt B B    ⋅ ∂ ∂ = ∂ ∂ ⋅ 2 1 -Power density of the current density eJ  Classical Electromagnetic Theory ( )BE cBBEE t EJp e    ×⋅∇−        ⋅ + ⋅ ∂ ∂ −=⋅= ππ 4224 1 Note: The minus sign means transfer of power from (loss) the electromagnetic field. ( )BE c S  ×= π4 : - Poynting power flux vector
118 SOLO Energy and Momentum (continue – 2) Let integrate this equation over a constant volume V         = ∂ ∂ ∫∫ VV td d t Classical Electromagnetic Theory ( )BE cBBEE t EJp e    ×⋅∇−        ⋅ + ⋅ ∂ ∂ −=⋅= πππ 488               PowerRadiationEM V PowerMagnetic V PowerEletrical V Powet V e vd c BE cvd BB t vd EE t vdEJ ∫∫∫∫         × ⋅∇− ⋅ ∂ ∂ − ⋅ ∂ ∂ −=⋅ πππ 488 2
04/08/15 119 SOLO Electromagnetic Stress Lorentz Force Density ( ) ( ) ( ) ( )          =⋅∇ =⋅∇ + ∂ ∂ =×∇ ∂ ∂ −=×∇ 0 4 41 1 BGM EGE J ct E c BA t B c EF e e        ρπ π Electric Field Intensity [statV . cm-1 ]E  Magnetic Induction [statV . sec . cm-2 = gauss]B  ρe Charge density [statC . cm-3 ] eJ  Current Density [statA . cm-3 ] Maxwell Electromagnetic Stress Start with Maxwell Equations in Gaussian Coordinates uJ BJ c Ef ee eee   ρ ρ = ×+= 1 ef  Electromagnetic force density [dynes . cm-3 ] u  Charge velocity [cm . sec] ( ) ( ) ( ) ( ) ( ) ( ) B t E c BBBB t B E c EEEE B t E c BBEEfe              × ∂ ∂ −         ⋅∇+××∇+                     ∂ ∂ ×−×∇×−+⋅∇= × ∂ ∂ −××∇+⋅∇= 1 4 1 4 11 4 1 1 4 1 4 1 4 1 0 0 πππ πππ ( ) ( )[ ] ( ) ( )         ∂ ∂ ×+× ∂ ∂ −         ⋅∇−×∇×−⋅∇−×∇×−= t B EB t E c BBBBEEEEfe      1 4 1 4 1 4 1 0 πππ 1 V = 1/3x10-2 statV 1 C = 3x109 statC 1 A= 3x109 statA uJ BJ c Ef ee eee   ρ ρ = ×+= 1
04/08/15 120 SOLO Electromagnetic Stress Maxwell Electromagnetic Stress ( ) ( )[ ] ( ) ( )[ ]         ∂ ∂ ×+× ∂ ∂ −×∇×−⋅∇+×∇×−⋅∇= t B EB t E c BBBBEEEEfe     1 4 1 4 1 4 1 πππ Use to get( ) ( ) ( )BABABA B  ∇⋅−⋅∇=×∇× ( ) ( ) ( )EEEEEE E  ∇⋅−⋅∇=×∇× ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )EEIEEEEEEEEEEEE EE  ⋅∇⋅−⋅∇=⋅∇−∇⋅+⋅∇=×∇×−⋅∇ and ( ) ( )[ ] ( ) ( )[ ] ( ) ( )    ⋅−⋅∇=⋅∇−∇⋅+⋅∇=×∇×−⋅∇ EEIEEEEEEEEEEEE  2 1 2 1 therefore where the unit dyadic.           = 100 010 001 I  ( ) ( )[ ] ( ) ( )[ ] ( ) ( )      ⋅−⋅∇=⋅∇−∇⋅+⋅∇=×∇×−⋅∇ BBIBBBBBBBBBBBB  2 1 2 1 On the same
04/08/15 121 SOLO Electromagnetic Stress Maxwell Electromagnetic Stress ( ) ( )         × ∂ ∂ −      ⋅−⋅∇+      ⋅−⋅∇= c BE t BBIBBEEIEEfe πππ 42 1 4 1 2 1 4 1   Define ( )    ⋅−= EEIEETe  2 1 4 1 : π ( )    ⋅−= BBIBBTm  2 1 4 1 : π ( )      ⋅+⋅−+=+= BBEEIBBEETTT me  2 1 4 1 : π [ ]2− ⋅cmdynesTm  . - Magnetic Stress Tensor [ ]2− ⋅cmdynesTe  . - Electric Stress Tensor [ ]2− ⋅+≡ cmdynesTTT me  . - Electromagnetic Stress Tensor c BE G π4 :   × = [ ]213 secsec −−− ⋅⋅=⋅⋅ cmergcmdynesG  - Momentum Density Vector t S Tfe ∂ ∂ −⋅∇=   Return to Relativistic EM Stress
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories
2 classical field theories

Recommended

Generation of optical harmonics
Generation of optical harmonicsGeneration of optical harmonics
Generation of optical harmonicsRaphael Bouskila
 
Nonlinear optics
Nonlinear opticsNonlinear optics
Nonlinear opticsGandhimathi Muthuselvam
 
Many-body Green functions theory for electronic and optical properties of or...
Many-body Green functions theory for electronic and optical properties of or...Many-body Green functions theory for electronic and optical properties of or...
Many-body Green functions theory for electronic and optical properties of or...Claudio Attaccalite
 
Power point slides_chapter_03
Power point slides_chapter_03Power point slides_chapter_03
Power point slides_chapter_03Surbhi Jain
 
APS_March_Meeting2016
APS_March_Meeting2016APS_March_Meeting2016
APS_March_Meeting2016James Lee
 
Problems and solutions statistical physics 1
Problems and solutions statistical physics 1Problems and solutions statistical physics 1
Problems and solutions statistical physics 1Alberto de Mesquita
 
Wien2k getting started
Wien2k getting startedWien2k getting started
Wien2k getting startedABDERRAHMANE REGGAD
 
UCSD NANO106 - 01 - Introduction to Crystallography
UCSD NANO106 - 01 - Introduction to CrystallographyUCSD NANO106 - 01 - Introduction to Crystallography
UCSD NANO106 - 01 - Introduction to CrystallographyUniversity of California, San Diego
 

Recommended

Generation of optical harmonics
Generation of optical harmonicsGeneration of optical harmonics
Generation of optical harmonicsRaphael Bouskila
 
Nonlinear optics
Nonlinear opticsNonlinear optics
Nonlinear opticsGandhimathi Muthuselvam
 
Many-body Green functions theory for electronic and optical properties of or...
Many-body Green functions theory for electronic and optical properties of or...Many-body Green functions theory for electronic and optical properties of or...
Many-body Green functions theory for electronic and optical properties of or...Claudio Attaccalite
 
Power point slides_chapter_03
Power point slides_chapter_03Power point slides_chapter_03
Power point slides_chapter_03Surbhi Jain
 
APS_March_Meeting2016
APS_March_Meeting2016APS_March_Meeting2016
APS_March_Meeting2016James Lee
 
Problems and solutions statistical physics 1
Problems and solutions statistical physics 1Problems and solutions statistical physics 1
Problems and solutions statistical physics 1Alberto de Mesquita
 
Wien2k getting started
Wien2k getting startedWien2k getting started
Wien2k getting startedABDERRAHMANE REGGAD
 
UCSD NANO106 - 01 - Introduction to Crystallography
UCSD NANO106 - 01 - Introduction to CrystallographyUCSD NANO106 - 01 - Introduction to Crystallography
UCSD NANO106 - 01 - Introduction to CrystallographyUniversity of California, San Diego
 
Presentation laser
Presentation laserPresentation laser
Presentation laserDr. M. K. Deore
 
Non linear optics and SHG
Non linear optics and SHGNon linear optics and SHG
Non linear optics and SHGsahil rajput
 
Poisson's equation 2nd 4
Poisson's equation 2nd 4Poisson's equation 2nd 4
Poisson's equation 2nd 4HIMANSHU DIWAKAR
 
Highly efficient organic devices.
Highly efficient organic devices.Highly efficient organic devices.
Highly efficient organic devices.Sociedade Brasileira de Pesquisa em Materiais
 
Fullprof Refinement
Fullprof RefinementFullprof Refinement
Fullprof RefinementSaurav Ch. Sarma
 
Materials Modelling: From theory to solar cells (Lecture 1)
Materials Modelling: From theory to solar cells (Lecture 1)Materials Modelling: From theory to solar cells (Lecture 1)
Materials Modelling: From theory to solar cells (Lecture 1)cdtpv
 
Phonons & Phonopy: Pro Tips (2015)
Phonons & Phonopy: Pro Tips (2015)Phonons & Phonopy: Pro Tips (2015)
Phonons & Phonopy: Pro Tips (2015)Jonathan Skelton
 
Lagrangian mechanics
Lagrangian mechanicsLagrangian mechanics
Lagrangian mechanicsAmeenSoomro1
 
TRANSMISSION ELECTRON MICROSCOPY
TRANSMISSION ELECTRON MICROSCOPYTRANSMISSION ELECTRON MICROSCOPY
TRANSMISSION ELECTRON MICROSCOPYSutapa Saha
 
Pulsed Laser Ablation
Pulsed Laser AblationPulsed Laser Ablation
Pulsed Laser AblationArun Aravind
 
6933.laser p pts
6933.laser p pts6933.laser p pts
6933.laser p ptsAnkush Saini
 
QUANTITIVE EXPRESSION FOR DEBYE SCREENING AND QUASINEUTRAL PLASMA
QUANTITIVE EXPRESSION FOR DEBYE SCREENING AND QUASINEUTRAL PLASMAQUANTITIVE EXPRESSION FOR DEBYE SCREENING AND QUASINEUTRAL PLASMA
QUANTITIVE EXPRESSION FOR DEBYE SCREENING AND QUASINEUTRAL PLASMAAnitaMalviya
 
1次元平均値フィルタ（フィルタサイズ3）
1次元平均値フィルタ（フィルタサイズ3）1次元平均値フィルタ（フィルタサイズ3）
1次元平均値フィルタ（フィルタサイズ3）Toru Tamaki
 
BoltzTrap webinar116_David_J_Singh.pdf
BoltzTrap webinar116_David_J_Singh.pdfBoltzTrap webinar116_David_J_Singh.pdf
BoltzTrap webinar116_David_J_Singh.pdfDrSanjaySingh13
 
Origin of quantum mechanics
Origin of quantum mechanicsOrigin of quantum mechanics
Origin of quantum mechanicsMUHAMMED ABDURAHMAN
 
VASP: Some Accumulated Wisdom
VASP: Some Accumulated WisdomVASP: Some Accumulated Wisdom
VASP: Some Accumulated WisdomJonathan Skelton
 
Methods available in WIEN2k for the treatment of exchange and correlation ef...
Methods available in WIEN2k for the treatment of exchange and correlation ef...Methods available in WIEN2k for the treatment of exchange and correlation ef...
Methods available in WIEN2k for the treatment of exchange and correlation ef...ABDERRAHMANE REGGAD
 
Ellipsometry- non destructive measuring method
Ellipsometry- non destructive measuring methodEllipsometry- non destructive measuring method
Ellipsometry- non destructive measuring methodViji Vijitha
 
Alpha decay - physical background and practical applications
Alpha decay - physical background and practical applicationsAlpha decay - physical background and practical applications
Alpha decay - physical background and practical applicationsAndrii Sofiienko
 
Laser lecture 05
Laser lecture 05Laser lecture 05
Laser lecture 05Ibb University, Yemen + Jazan University, KSA
 
Kuhn tucker y lagrange
Kuhn tucker y lagrangeKuhn tucker y lagrange
Kuhn tucker y lagrangeandressparada
 
De Alembert’s Principle and Generalized Force, a technical discourse on Class...
De Alembert’s Principle and Generalized Force, a technical discourse on Class...De Alembert’s Principle and Generalized Force, a technical discourse on Class...
De Alembert’s Principle and Generalized Force, a technical discourse on Class...Manmohan Dash
 

More Related Content

What's hot

Non linear optics and SHG
Non linear optics and SHGNon linear optics and SHG
Non linear optics and SHGsahil rajput
 
Materials Modelling: From theory to solar cells (Lecture 1)
Materials Modelling: From theory to solar cells (Lecture 1)Materials Modelling: From theory to solar cells (Lecture 1)
Materials Modelling: From theory to solar cells (Lecture 1)cdtpv
 
Phonons & Phonopy: Pro Tips (2015)
Phonons & Phonopy: Pro Tips (2015)Phonons & Phonopy: Pro Tips (2015)
Phonons & Phonopy: Pro Tips (2015)Jonathan Skelton
 
Lagrangian mechanics
Lagrangian mechanicsLagrangian mechanics
Lagrangian mechanicsAmeenSoomro1
 
TRANSMISSION ELECTRON MICROSCOPY
TRANSMISSION ELECTRON MICROSCOPYTRANSMISSION ELECTRON MICROSCOPY
TRANSMISSION ELECTRON MICROSCOPYSutapa Saha
 
Pulsed Laser Ablation
Pulsed Laser AblationPulsed Laser Ablation
Pulsed Laser AblationArun Aravind
 
QUANTITIVE EXPRESSION FOR DEBYE SCREENING AND QUASINEUTRAL PLASMA
QUANTITIVE EXPRESSION FOR DEBYE SCREENING AND QUASINEUTRAL PLASMAQUANTITIVE EXPRESSION FOR DEBYE SCREENING AND QUASINEUTRAL PLASMA
QUANTITIVE EXPRESSION FOR DEBYE SCREENING AND QUASINEUTRAL PLASMAAnitaMalviya
 
1次元平均値フィルタ（フィルタサイズ3）
1次元平均値フィルタ（フィルタサイズ3）1次元平均値フィルタ（フィルタサイズ3）
1次元平均値フィルタ（フィルタサイズ3）Toru Tamaki
 
BoltzTrap webinar116_David_J_Singh.pdf
BoltzTrap webinar116_David_J_Singh.pdfBoltzTrap webinar116_David_J_Singh.pdf
BoltzTrap webinar116_David_J_Singh.pdfDrSanjaySingh13
 
VASP: Some Accumulated Wisdom
VASP: Some Accumulated WisdomVASP: Some Accumulated Wisdom
VASP: Some Accumulated WisdomJonathan Skelton
 
Methods available in WIEN2k for the treatment of exchange and correlation ef...
Methods available in WIEN2k for the treatment of exchange and correlation ef...Methods available in WIEN2k for the treatment of exchange and correlation ef...
Methods available in WIEN2k for the treatment of exchange and correlation ef...ABDERRAHMANE REGGAD
 
Ellipsometry- non destructive measuring method
Ellipsometry- non destructive measuring methodEllipsometry- non destructive measuring method
Ellipsometry- non destructive measuring methodViji Vijitha
 
Alpha decay - physical background and practical applications
Alpha decay - physical background and practical applicationsAlpha decay - physical background and practical applications
Alpha decay - physical background and practical applicationsAndrii Sofiienko
 

What's hot (20)

Presentation laser
Presentation laserPresentation laser
Presentation laser
 
Non linear optics and SHG
Non linear optics and SHGNon linear optics and SHG
Non linear optics and SHG
 
Poisson's equation 2nd 4
Poisson's equation 2nd 4Poisson's equation 2nd 4
Poisson's equation 2nd 4
 
Highly efficient organic devices.
Highly efficient organic devices.Highly efficient organic devices.
Highly efficient organic devices.
 
Fullprof Refinement
Fullprof RefinementFullprof Refinement
Fullprof Refinement
 
Materials Modelling: From theory to solar cells (Lecture 1)
Materials Modelling: From theory to solar cells (Lecture 1)Materials Modelling: From theory to solar cells (Lecture 1)
Materials Modelling: From theory to solar cells (Lecture 1)
 
Phonons & Phonopy: Pro Tips (2015)
Phonons & Phonopy: Pro Tips (2015)Phonons & Phonopy: Pro Tips (2015)
Phonons & Phonopy: Pro Tips (2015)
 
Lagrangian mechanics
Lagrangian mechanicsLagrangian mechanics
Lagrangian mechanics
 
TRANSMISSION ELECTRON MICROSCOPY
TRANSMISSION ELECTRON MICROSCOPYTRANSMISSION ELECTRON MICROSCOPY
TRANSMISSION ELECTRON MICROSCOPY
 
Pulsed Laser Ablation
Pulsed Laser AblationPulsed Laser Ablation
Pulsed Laser Ablation
 
6933.laser p pts
6933.laser p pts6933.laser p pts
6933.laser p pts
 
QUANTITIVE EXPRESSION FOR DEBYE SCREENING AND QUASINEUTRAL PLASMA
QUANTITIVE EXPRESSION FOR DEBYE SCREENING AND QUASINEUTRAL PLASMAQUANTITIVE EXPRESSION FOR DEBYE SCREENING AND QUASINEUTRAL PLASMA
QUANTITIVE EXPRESSION FOR DEBYE SCREENING AND QUASINEUTRAL PLASMA
 
1次元平均値フィルタ（フィルタサイズ3）
1次元平均値フィルタ（フィルタサイズ3）1次元平均値フィルタ（フィルタサイズ3）
1次元平均値フィルタ（フィルタサイズ3）
 
BoltzTrap webinar116_David_J_Singh.pdf
BoltzTrap webinar116_David_J_Singh.pdfBoltzTrap webinar116_David_J_Singh.pdf
BoltzTrap webinar116_David_J_Singh.pdf
 
Origin of quantum mechanics
Origin of quantum mechanicsOrigin of quantum mechanics
Origin of quantum mechanics
 
VASP: Some Accumulated Wisdom
VASP: Some Accumulated WisdomVASP: Some Accumulated Wisdom
VASP: Some Accumulated Wisdom
 
Methods available in WIEN2k for the treatment of exchange and correlation ef...
Methods available in WIEN2k for the treatment of exchange and correlation ef...Methods available in WIEN2k for the treatment of exchange and correlation ef...
Methods available in WIEN2k for the treatment of exchange and correlation ef...
 
Ellipsometry- non destructive measuring method
Ellipsometry- non destructive measuring methodEllipsometry- non destructive measuring method
Ellipsometry- non destructive measuring method
 
Alpha decay - physical background and practical applications
Alpha decay - physical background and practical applicationsAlpha decay - physical background and practical applications
Alpha decay - physical background and practical applications
 
Laser lecture 05
Laser lecture 05Laser lecture 05
Laser lecture 05
 

Viewers also liked

Kuhn tucker y lagrange
Kuhn tucker y lagrangeKuhn tucker y lagrange
Kuhn tucker y lagrangeandressparada
 
De Alembert’s Principle and Generalized Force, a technical discourse on Class...
De Alembert’s Principle and Generalized Force, a technical discourse on Class...De Alembert’s Principle and Generalized Force, a technical discourse on Class...
De Alembert’s Principle and Generalized Force, a technical discourse on Class...Manmohan Dash
 
Reduced order observers
Reduced order observersReduced order observers
Reduced order observersSolo Hermelin
 
6 radar range-doppler-angular loops
6 radar range-doppler-angular loops6 radar range-doppler-angular loops
6 radar range-doppler-angular loopsSolo Hermelin
 
Introduction to elasticity, part i
Introduction to elasticity, part iIntroduction to elasticity, part i
Introduction to elasticity, part iSolo Hermelin
 
Maxwell equations and propagation in anisotropic media
Maxwell equations and propagation in anisotropic mediaMaxwell equations and propagation in anisotropic media
Maxwell equations and propagation in anisotropic mediaSolo Hermelin
 
5 cramer-rao lower bound
5 cramer-rao lower bound5 cramer-rao lower bound
5 cramer-rao lower boundSolo Hermelin
 
11 fighter aircraft avionics - part iv
11 fighter aircraft avionics - part iv11 fighter aircraft avionics - part iv
11 fighter aircraft avionics - part ivSolo Hermelin
 
Keplerian trajectories
Keplerian trajectoriesKeplerian trajectories
Keplerian trajectoriesSolo Hermelin
 
Sliding Mode Observers
Sliding Mode ObserversSliding Mode Observers
Sliding Mode ObserversSolo Hermelin
 
Equation of motion of a variable mass system2
Equation of motion of a variable mass system2Equation of motion of a variable mass system2
Equation of motion of a variable mass system2Solo Hermelin
 
6 computing gunsight, hud and hms
6 computing gunsight, hud and hms6 computing gunsight, hud and hms
6 computing gunsight, hud and hmsSolo Hermelin
 
Fiber optics ray theory
Fiber optics ray theoryFiber optics ray theory
Fiber optics ray theorySolo Hermelin
 
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - iiSolo Hermelin
 
13 fixed wing fighter aircraft- flight performance - i
13 fixed wing fighter aircraft- flight performance - i13 fixed wing fighter aircraft- flight performance - i
13 fixed wing fighter aircraft- flight performance - iSolo Hermelin
 
Stabilization of linear time invariant systems, Factorization Approach
Stabilization of linear time invariant systems, Factorization ApproachStabilization of linear time invariant systems, Factorization Approach
Stabilization of linear time invariant systems, Factorization ApproachSolo Hermelin
 
10 fighter aircraft avionics - part iii
10 fighter aircraft avionics - part iii10 fighter aircraft avionics - part iii
10 fighter aircraft avionics - part iiiSolo Hermelin
 

Viewers also liked (20)

Kuhn tucker y lagrange
Kuhn tucker y lagrangeKuhn tucker y lagrange
Kuhn tucker y lagrange
 
De Alembert’s Principle and Generalized Force, a technical discourse on Class...
De Alembert’s Principle and Generalized Force, a technical discourse on Class...De Alembert’s Principle and Generalized Force, a technical discourse on Class...
De Alembert’s Principle and Generalized Force, a technical discourse on Class...
 
2 estimators
2 estimators2 estimators
2 estimators
 
Reduced order observers
Reduced order observersReduced order observers
Reduced order observers
 
6 radar range-doppler-angular loops
6 radar range-doppler-angular loops6 radar range-doppler-angular loops
6 radar range-doppler-angular loops
 
Introduction to elasticity, part i
Introduction to elasticity, part iIntroduction to elasticity, part i
Introduction to elasticity, part i
 
Optical aberrations
Optical aberrationsOptical aberrations
Optical aberrations
 
Maxwell equations and propagation in anisotropic media
Maxwell equations and propagation in anisotropic mediaMaxwell equations and propagation in anisotropic media
Maxwell equations and propagation in anisotropic media
 
5 cramer-rao lower bound
5 cramer-rao lower bound5 cramer-rao lower bound
5 cramer-rao lower bound
 
11 fighter aircraft avionics - part iv
11 fighter aircraft avionics - part iv11 fighter aircraft avionics - part iv
11 fighter aircraft avionics - part iv
 
Keplerian trajectories
Keplerian trajectoriesKeplerian trajectories
Keplerian trajectories
 
Sliding Mode Observers
Sliding Mode ObserversSliding Mode Observers
Sliding Mode Observers
 
Equation of motion of a variable mass system2
Equation of motion of a variable mass system2Equation of motion of a variable mass system2
Equation of motion of a variable mass system2
 
1 tracking systems1
1 tracking systems11 tracking systems1
1 tracking systems1
 
6 computing gunsight, hud and hms
6 computing gunsight, hud and hms6 computing gunsight, hud and hms
6 computing gunsight, hud and hms
 
Fiber optics ray theory
Fiber optics ray theoryFiber optics ray theory
Fiber optics ray theory
 
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
 
13 fixed wing fighter aircraft- flight performance - i
13 fixed wing fighter aircraft- flight performance - i13 fixed wing fighter aircraft- flight performance - i
13 fixed wing fighter aircraft- flight performance - i
 
Stabilization of linear time invariant systems, Factorization Approach
Stabilization of linear time invariant systems, Factorization ApproachStabilization of linear time invariant systems, Factorization Approach
Stabilization of linear time invariant systems, Factorization Approach
 
10 fighter aircraft avionics - part iii
10 fighter aircraft avionics - part iii10 fighter aircraft avionics - part iii
10 fighter aircraft avionics - part iii
 

Similar to 2 classical field theories

Hamilton application
Hamilton applicationHamilton application
Hamilton applicationSamad Akbar
 
ep ppt of it .pptx
ep ppt of it .pptxep ppt of it .pptx
ep ppt of it .pptxbsabjdsv
 
SLAM of Multi-Robot System Considering Its Network Topology
SLAM of Multi-Robot System Considering Its Network TopologySLAM of Multi-Robot System Considering Its Network Topology
SLAM of Multi-Robot System Considering Its Network Topologytoukaigi
 
final poster
final posterfinal poster
final posterNeal Woo
 
Chemical dynamics and rare events in soft matter physics
Chemical dynamics and rare events in soft matter physicsChemical dynamics and rare events in soft matter physics
Chemical dynamics and rare events in soft matter physicsBoris Fackovec
 
MMsemester project
MMsemester projectMMsemester project
MMsemester projectPreeti Sahu
 
Material modelling for design
Material modelling for design Material modelling for design
Material modelling for design Er Shambhu Chauhan
 
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
 
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
 

Similar to 2 classical field theories (20)

Analytic dynamics
Analytic dynamicsAnalytic dynamics
Analytic dynamics
 
Dynamics
DynamicsDynamics
Dynamics
 
Hamilton application
Hamilton applicationHamilton application
Hamilton application
 
ep ppt of it .pptx
ep ppt of it .pptxep ppt of it .pptx
ep ppt of it .pptx
 
Instantons in 1D QM
Instantons in 1D QMInstantons in 1D QM
Instantons in 1D QM
 
SLAM of Multi-Robot System Considering Its Network Topology
SLAM of Multi-Robot System Considering Its Network TopologySLAM of Multi-Robot System Considering Its Network Topology
SLAM of Multi-Robot System Considering Its Network Topology
 
MD_course.ppt
MD_course.pptMD_course.ppt
MD_course.ppt
 
final poster
final posterfinal poster
final poster
 
Online Signals and Systems Assignment Help
Online Signals and Systems Assignment HelpOnline Signals and Systems Assignment Help
Online Signals and Systems Assignment Help
 
Chemical dynamics and rare events in soft matter physics
Chemical dynamics and rare events in soft matter physicsChemical dynamics and rare events in soft matter physics
Chemical dynamics and rare events in soft matter physics
 
Bazzucchi-Campolmi-Zatti
Bazzucchi-Campolmi-ZattiBazzucchi-Campolmi-Zatti
Bazzucchi-Campolmi-Zatti
 
Quantum Assignment Help
Quantum Assignment HelpQuantum Assignment Help
Quantum Assignment Help
 
Quantum state
Quantum stateQuantum state
Quantum state
 
Dynamics of wind & marine turbines
Dynamics of wind & marine turbinesDynamics of wind & marine turbines
Dynamics of wind & marine turbines
 
MMsemester project
MMsemester projectMMsemester project
MMsemester project
 
Adaptive dynamic programming algorithm for uncertain nonlinear switched systems
Adaptive dynamic programming algorithm for uncertain nonlinear switched systemsAdaptive dynamic programming algorithm for uncertain nonlinear switched systems
Adaptive dynamic programming algorithm for uncertain nonlinear switched systems
 
Material modelling for design
Material modelling for design Material modelling for design
Material modelling for design
 
10 slides
10 slides10 slides
10 slides
 
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...
 
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...
 

More from Solo Hermelin

5 introduction to quantum mechanics
5 introduction to quantum mechanics5 introduction to quantum mechanics
5 introduction to quantum mechanicsSolo Hermelin
 
Slide Mode Control (S.M.C.)
Slide Mode Control (S.M.C.)Slide Mode Control (S.M.C.)
Slide Mode Control (S.M.C.)Solo Hermelin
 
Inner outer and spectral factorizations
Inner outer and spectral factorizationsInner outer and spectral factorizations
Inner outer and spectral factorizationsSolo Hermelin
 
Anti ballistic missiles ii
Anti ballistic missiles iiAnti ballistic missiles ii
Anti ballistic missiles iiSolo Hermelin
 
Anti ballistic missiles i
Anti ballistic missiles iAnti ballistic missiles i
Anti ballistic missiles iSolo Hermelin
 
12 performance of an aircraft with parabolic polar
12 performance of an aircraft with parabolic polar12 performance of an aircraft with parabolic polar
12 performance of an aircraft with parabolic polarSolo Hermelin
 
9 fighter aircraft avionics-part ii
9 fighter aircraft avionics-part ii9 fighter aircraft avionics-part ii
9 fighter aircraft avionics-part iiSolo Hermelin
 
8 fighter aircraft avionics-part i
8 fighter aircraft avionics-part i8 fighter aircraft avionics-part i
8 fighter aircraft avionics-part iSolo Hermelin
 
4 navigation systems
4 navigation systems4 navigation systems
4 navigation systemsSolo Hermelin
 
2 aircraft flight instruments
2 aircraft flight instruments2 aircraft flight instruments
2 aircraft flight instrumentsSolo Hermelin
 
3 modern aircraft cutaway
3 modern aircraft cutaway3 modern aircraft cutaway
3 modern aircraft cutawaySolo Hermelin
 
2Anti-aircraft Warhead
2Anti-aircraft Warhead2Anti-aircraft Warhead
2Anti-aircraft WarheadSolo Hermelin
 
1 susceptibility vulnerability
1 susceptibility vulnerability1 susceptibility vulnerability
1 susceptibility vulnerabilitySolo Hermelin
 
Calculus of variation problems
Calculus of variation problemsCalculus of variation problems
Calculus of variation problemsSolo Hermelin
 
Calculus of variations
Calculus of variationsCalculus of variations
Calculus of variationsSolo Hermelin
 
Aerodynamics part iii
Aerodynamics part iiiAerodynamics part iii
Aerodynamics part iiiSolo Hermelin
 
Aerodynamics part ii
Aerodynamics part iiAerodynamics part ii
Aerodynamics part iiSolo Hermelin
 

More from Solo Hermelin (20)

5 introduction to quantum mechanics
5 introduction to quantum mechanics5 introduction to quantum mechanics
5 introduction to quantum mechanics
 
Slide Mode Control (S.M.C.)
Slide Mode Control (S.M.C.)Slide Mode Control (S.M.C.)
Slide Mode Control (S.M.C.)
 
Inner outer and spectral factorizations
Inner outer and spectral factorizationsInner outer and spectral factorizations
Inner outer and spectral factorizations
 
Anti ballistic missiles ii
Anti ballistic missiles iiAnti ballistic missiles ii
Anti ballistic missiles ii
 
Anti ballistic missiles i
Anti ballistic missiles iAnti ballistic missiles i
Anti ballistic missiles i
 
12 performance of an aircraft with parabolic polar
12 performance of an aircraft with parabolic polar12 performance of an aircraft with parabolic polar
12 performance of an aircraft with parabolic polar
 
9 fighter aircraft avionics-part ii
9 fighter aircraft avionics-part ii9 fighter aircraft avionics-part ii
9 fighter aircraft avionics-part ii
 
8 fighter aircraft avionics-part i
8 fighter aircraft avionics-part i8 fighter aircraft avionics-part i
8 fighter aircraft avionics-part i
 
4 navigation systems
4 navigation systems4 navigation systems
4 navigation systems
 
3 earth atmosphere
3 earth atmosphere3 earth atmosphere
3 earth atmosphere
 
2 aircraft flight instruments
2 aircraft flight instruments2 aircraft flight instruments
2 aircraft flight instruments
 
3 modern aircraft cutaway
3 modern aircraft cutaway3 modern aircraft cutaway
3 modern aircraft cutaway
 
2Anti-aircraft Warhead
2Anti-aircraft Warhead2Anti-aircraft Warhead
2Anti-aircraft Warhead
 
1 susceptibility vulnerability
1 susceptibility vulnerability1 susceptibility vulnerability
1 susceptibility vulnerability
 
15 sky cars
15 sky cars15 sky cars
15 sky cars
 
Calculus of variation problems
Calculus of variation problemsCalculus of variation problems
Calculus of variation problems
 
Calculus of variations
Calculus of variationsCalculus of variations
Calculus of variations
 
7 air-to-air combat
7 air-to-air combat7 air-to-air combat
7 air-to-air combat
 
Aerodynamics part iii
Aerodynamics part iiiAerodynamics part iii
Aerodynamics part iii
 
Aerodynamics part ii
Aerodynamics part iiAerodynamics part ii
Aerodynamics part ii
 

Recently uploaded

Artificial Intelligence In Microbiology by Dr. Prince C P
Artificial Intelligence In Microbiology by Dr. Prince C PArtificial Intelligence In Microbiology by Dr. Prince C P
Artificial Intelligence In Microbiology by Dr. Prince C PPRINCE C P
 
Recombination DNA Technology (Nucleic Acid Hybridization )
Recombination DNA Technology (Nucleic Acid Hybridization )Recombination DNA Technology (Nucleic Acid Hybridization )
Recombination DNA Technology (Nucleic Acid Hybridization )aarthirajkumar25
 
Boyles law module in the grade 10 science
Boyles law module in the grade 10 scienceBoyles law module in the grade 10 science
Boyles law module in the grade 10 sciencefloriejanemacaya1
 
Biological Classification BioHack (3).pdf
Biological Classification BioHack (3).pdfBiological Classification BioHack (3).pdf
Biological Classification BioHack (3).pdfmuntazimhurra
 
Analytical Profile of Coleus Forskohlii | Forskolin .pptx
Analytical Profile of Coleus Forskohlii | Forskolin .pptxAnalytical Profile of Coleus Forskohlii | Forskolin .pptx
Analytical Profile of Coleus Forskohlii | Forskolin .pptxSwapnil Therkar
 
Behavioral Disorder: Schizophrenia & it's Case Study.pdf
Behavioral Disorder: Schizophrenia & it's Case Study.pdfBehavioral Disorder: Schizophrenia & it's Case Study.pdf
Behavioral Disorder: Schizophrenia & it's Case Study.pdfSELF-EXPLANATORY
 
STERILITY TESTING OF PHARMACEUTICALS ppt by DR.C.P.PRINCE
STERILITY TESTING OF PHARMACEUTICALS ppt by DR.C.P.PRINCESTERILITY TESTING OF PHARMACEUTICALS ppt by DR.C.P.PRINCE
STERILITY TESTING OF PHARMACEUTICALS ppt by DR.C.P.PRINCEPRINCE C P
 
Lucknow 💋 Russian Call Girls Lucknow Finest Escorts Service 8923113531 Availa...
Lucknow 💋 Russian Call Girls Lucknow Finest Escorts Service 8923113531 Availa...Lucknow 💋 Russian Call Girls Lucknow Finest Escorts Service 8923113531 Availa...
Lucknow 💋 Russian Call Girls Lucknow Finest Escorts Service 8923113531 Availa...anilsa9823
 
Work, Energy and Power for class 10 ICSE Physics
Work, Energy and Power for class 10 ICSE PhysicsWork, Energy and Power for class 10 ICSE Physics
Work, Energy and Power for class 10 ICSE Physicsvishikhakeshava1
 
All-domain Anomaly Resolution Office U.S. Department of Defense (U) Case: “Eg...
All-domain Anomaly Resolution Office U.S. Department of Defense (U) Case: “Eg...All-domain Anomaly Resolution Office U.S. Department of Defense (U) Case: “Eg...
All-domain Anomaly Resolution Office U.S. Department of Defense (U) Case: “Eg...Sérgio Sacani
 
VIRUSES structure and classification ppt by Dr.Prince C P
VIRUSES structure and classification ppt by Dr.Prince C PVIRUSES structure and classification ppt by Dr.Prince C P
VIRUSES structure and classification ppt by Dr.Prince C PPRINCE C P
 
Isotopic evidence of long-lived volcanism on Io
Isotopic evidence of long-lived volcanism on IoIsotopic evidence of long-lived volcanism on Io
Isotopic evidence of long-lived volcanism on IoSérgio Sacani
 
Traditional Agroforestry System in India- Shifting Cultivation, Taungya, Home...
Traditional Agroforestry System in India- Shifting Cultivation, Taungya, Home...Traditional Agroforestry System in India- Shifting Cultivation, Taungya, Home...
Traditional Agroforestry System in India- Shifting Cultivation, Taungya, Home...jana861314
 
Disentangling the origin of chemical differences using GHOST
Disentangling the origin of chemical differences using GHOSTDisentangling the origin of chemical differences using GHOST
Disentangling the origin of chemical differences using GHOSTSérgio Sacani
 
Call Girls in Mayapuri Delhi 💯Call Us 🔝9953322196🔝 💯Escort.
Call Girls in Mayapuri Delhi 💯Call Us 🔝9953322196🔝 💯Escort.Call Girls in Mayapuri Delhi 💯Call Us 🔝9953322196🔝 💯Escort.
Call Girls in Mayapuri Delhi 💯Call Us 🔝9953322196🔝 💯Escort.aasikanpl
 
Physiochemical properties of nanomaterials and its nanotoxicity.pptx
Physiochemical properties of nanomaterials and its nanotoxicity.pptxPhysiochemical properties of nanomaterials and its nanotoxicity.pptx
Physiochemical properties of nanomaterials and its nanotoxicity.pptxAArockiyaNisha
 
Recombinant DNA technology (Immunological screening)
Recombinant DNA technology (Immunological screening)Recombinant DNA technology (Immunological screening)
Recombinant DNA technology (Immunological screening)PraveenaKalaiselvan1
 
Discovery of an Accretion Streamer and a Slow Wide-angle Outflow around FUOri...
Discovery of an Accretion Streamer and a Slow Wide-angle Outflow around FUOri...Discovery of an Accretion Streamer and a Slow Wide-angle Outflow around FUOri...
Discovery of an Accretion Streamer and a Slow Wide-angle Outflow around FUOri...Sérgio Sacani
 
Nanoparticles synthesis and characterization​ ​
Nanoparticles synthesis and characterization​ ​Nanoparticles synthesis and characterization​ ​
Nanoparticles synthesis and characterization​ ​kaibalyasahoo82800
 
Grafana in space: Monitoring Japan's SLIM moon lander in real time
Grafana in space: Monitoring Japan's SLIM moon lander in real timeGrafana in space: Monitoring Japan's SLIM moon lander in real time
Grafana in space: Monitoring Japan's SLIM moon lander in real timeSatoshi NAKAHIRA
 

Recently uploaded (20)

Artificial Intelligence In Microbiology by Dr. Prince C P
Artificial Intelligence In Microbiology by Dr. Prince C PArtificial Intelligence In Microbiology by Dr. Prince C P
Artificial Intelligence In Microbiology by Dr. Prince C P
 
Recombination DNA Technology (Nucleic Acid Hybridization )
Recombination DNA Technology (Nucleic Acid Hybridization )Recombination DNA Technology (Nucleic Acid Hybridization )
Recombination DNA Technology (Nucleic Acid Hybridization )
 
Boyles law module in the grade 10 science
Boyles law module in the grade 10 scienceBoyles law module in the grade 10 science
Boyles law module in the grade 10 science
 
Biological Classification BioHack (3).pdf
Biological Classification BioHack (3).pdfBiological Classification BioHack (3).pdf
Biological Classification BioHack (3).pdf
 
Analytical Profile of Coleus Forskohlii | Forskolin .pptx
Analytical Profile of Coleus Forskohlii | Forskolin .pptxAnalytical Profile of Coleus Forskohlii | Forskolin .pptx
Analytical Profile of Coleus Forskohlii | Forskolin .pptx
 
Behavioral Disorder: Schizophrenia & it's Case Study.pdf
Behavioral Disorder: Schizophrenia & it's Case Study.pdfBehavioral Disorder: Schizophrenia & it's Case Study.pdf
Behavioral Disorder: Schizophrenia & it's Case Study.pdf
 
STERILITY TESTING OF PHARMACEUTICALS ppt by DR.C.P.PRINCE
STERILITY TESTING OF PHARMACEUTICALS ppt by DR.C.P.PRINCESTERILITY TESTING OF PHARMACEUTICALS ppt by DR.C.P.PRINCE
STERILITY TESTING OF PHARMACEUTICALS ppt by DR.C.P.PRINCE
 
Lucknow 💋 Russian Call Girls Lucknow Finest Escorts Service 8923113531 Availa...
Lucknow 💋 Russian Call Girls Lucknow Finest Escorts Service 8923113531 Availa...Lucknow 💋 Russian Call Girls Lucknow Finest Escorts Service 8923113531 Availa...
Lucknow 💋 Russian Call Girls Lucknow Finest Escorts Service 8923113531 Availa...
 
Work, Energy and Power for class 10 ICSE Physics
Work, Energy and Power for class 10 ICSE PhysicsWork, Energy and Power for class 10 ICSE Physics
Work, Energy and Power for class 10 ICSE Physics
 
All-domain Anomaly Resolution Office U.S. Department of Defense (U) Case: “Eg...
All-domain Anomaly Resolution Office U.S. Department of Defense (U) Case: “Eg...All-domain Anomaly Resolution Office U.S. Department of Defense (U) Case: “Eg...
All-domain Anomaly Resolution Office U.S. Department of Defense (U) Case: “Eg...
 
VIRUSES structure and classification ppt by Dr.Prince C P
VIRUSES structure and classification ppt by Dr.Prince C PVIRUSES structure and classification ppt by Dr.Prince C P
VIRUSES structure and classification ppt by Dr.Prince C P
 
Isotopic evidence of long-lived volcanism on Io
Isotopic evidence of long-lived volcanism on IoIsotopic evidence of long-lived volcanism on Io
Isotopic evidence of long-lived volcanism on Io
 
Traditional Agroforestry System in India- Shifting Cultivation, Taungya, Home...
Traditional Agroforestry System in India- Shifting Cultivation, Taungya, Home...Traditional Agroforestry System in India- Shifting Cultivation, Taungya, Home...
Traditional Agroforestry System in India- Shifting Cultivation, Taungya, Home...
 
Disentangling the origin of chemical differences using GHOST
Disentangling the origin of chemical differences using GHOSTDisentangling the origin of chemical differences using GHOST
Disentangling the origin of chemical differences using GHOST
 
Call Girls in Mayapuri Delhi 💯Call Us 🔝9953322196🔝 💯Escort.
Call Girls in Mayapuri Delhi 💯Call Us 🔝9953322196🔝 💯Escort.Call Girls in Mayapuri Delhi 💯Call Us 🔝9953322196🔝 💯Escort.
Call Girls in Mayapuri Delhi 💯Call Us 🔝9953322196🔝 💯Escort.
 
Physiochemical properties of nanomaterials and its nanotoxicity.pptx
Physiochemical properties of nanomaterials and its nanotoxicity.pptxPhysiochemical properties of nanomaterials and its nanotoxicity.pptx
Physiochemical properties of nanomaterials and its nanotoxicity.pptx
 
Recombinant DNA technology (Immunological screening)
Recombinant DNA technology (Immunological screening)Recombinant DNA technology (Immunological screening)
Recombinant DNA technology (Immunological screening)
 
Discovery of an Accretion Streamer and a Slow Wide-angle Outflow around FUOri...
Discovery of an Accretion Streamer and a Slow Wide-angle Outflow around FUOri...Discovery of an Accretion Streamer and a Slow Wide-angle Outflow around FUOri...
Discovery of an Accretion Streamer and a Slow Wide-angle Outflow around FUOri...
 
Nanoparticles synthesis and characterization​ ​
Nanoparticles synthesis and characterization​ ​Nanoparticles synthesis and characterization​ ​
Nanoparticles synthesis and characterization​ ​
 
Grafana in space: Monitoring Japan's SLIM moon lander in real time
Grafana in space: Monitoring Japan's SLIM moon lander in real timeGrafana in space: Monitoring Japan's SLIM moon lander in real time
Grafana in space: Monitoring Japan's SLIM moon lander in real time
 

2 classical field theories

  • 1. 04/08/15 1 Classical Field Theories SOLO HERMELIN Updated: 10.10.2013 16.11.2014 8.04.2015 http://www.solohermelin.com
  • 2. 04/08/15 2 SOLO Table of Content Classical Field Theories Generalized Coordinates 1. Newton’s Laws of Motion Analytic Dynamics Work and Energy Basic Definitions Constraints The Stationary Value of a Function and of a Definite Integral The Principle of Virtual Work 2. D’Alembert Principle 3. Hamilton’s Principle 4. Lagrange’s Equations of Motion 5. Hamilton’s Equations Introduction to Lagrangian and Hamiltonian Formulation Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  Second Method (Carathéodory) Equivalent Integrals Hamilton-Jacobi Theory Theorem of Noether for Single Integral
  • 3. 04/08/15 3 SOLO Table of Content (continue – 1) Classical Field Theories Fermat Principle in Optics Four-Dimensional Formulation of the Theory of Relativity Electromagnetic Field Introduction to Lagrangian and Hamiltonian Formulation for Continuous Systems and Fields Transition from a Discrete to Continuous Systems Extremal of the Functional ( ) ( )∫ ∫       ∂ ∂ ∂ ∂ = 2 1 ,,,,, t t V j k j kjk tdVd tx xtxtCI ψψ ψL Hamiltonian Formalism Theorem of Noether for Multiple Integrals Elasticity Vibrating String Vibrating Membrane Vibrating Beam Plate Theories Structural Model of the Solid Body The Inverse Square Law of Forces
  • 4. 04/08/15 4 SOLO Table of Content (continue – 2) Classical Field Theories Variational Principles of Hydrodynamics Part I – Lagrange Interpretation Part II: Euler’s Representation Part III: Simpler Eulerian Variational Principle Part IV: Problem of the Motion of Fluid Subjected to Gravity Forces and Surface Tension Forces Dynamics of Acoustic Field in Gases References Equation of Motion of a Variable Mass System – Lagrangian Approach
  • 5. 5 Analytic DynamicsSOLO Newton’s Laws of Motion “The Mathematical Principles of Natural Philosophy” 1687 First Law Every body continues in its state of rest or of uniform motion in straight line unless it is compelled to change that state by forces impressed upon it. Second Law The rate of change of momentum is proportional to the force impressed and in the same direction as that force. Third Law To every action there is always opposed an equal reaction. → =→= constvF  0 ( )vm td d p td d F  == 2112 FF  −= vmp  = td pd F  = 12F  1 2 21F  Return to Table of Content
  • 6. 6 Analytic DynamicsSOLO Work and Energy The work W of a force acting on a particle m that moves as a result of this along a curve s from to is defined by: F  1r  2r  ∫∫ ⋅      =⋅= •∆ 2 1 2 1 12 r r r r rdrm dt d rdFW      r  1r  2r  rd  rdr  + 1 2 F  m s rd  is the displacement on a real path. ••∆ ⋅= rrmT  2 1 The kinetic energy T is defined as: 1212 2 1 2 1 2 1 2 TTrrd m dtrr dt d mrdrm dt d W r r r r r r −=      ⋅=⋅      =⋅      = ∫∫∫ ⋅ ⋅ ••••⋅        For a constant mass m
  • 7. 7 Analytic DynamicsSOLO Work and Energy (continue) When the force depends on the position alone, i.e. , and the quantity is a perfect differential ( )rFF  = rdF  ⋅ ( ) ( )rdVrdrF  −=⋅ The force field is said to be conservative and the function is known as the Potential Energy. In this case: ( )rV  ( ) ( ) ( ) 212112 2 1 2 1 VVrVrVrdVrdFW r r r r −=−=−=⋅= ∫∫ ∆      The work does not depend on the path from to . It is clear that in a conservative field, the integral of over a closed path is zero. 12W 1r  2r  rdF  ⋅ ( ) ( ) 01221 21 1 2 2 1 =−+−=⋅+⋅=⋅ ∫∫∫ VVVVrdFrdFrdF path r r path r rC         Using Stoke’s Theorem it means that∫∫∫ ⋅×∇=⋅ SC sdFrdF  0=×∇= FFrot  Therefore is the gradient of some scalar functionF  ( ) rdrVdVrdF  ⋅−∇=−=⋅ ( )rVF  −∇=
  • 8. 8 Analytic DynamicsSOLO Work and Energy (continue) and • →∆→∆ ⋅−=⋅−= ∆ ∆ = rF dt rd F t V dt dV tt  00 limlim But also for a constant mass system •••••••••••• ⋅=⋅=      ⋅+⋅=      ⋅= rFrrmrrrr m rrm dt d dt dT  22 1 Therefore for a constant mass in a conservative field ( ) .0 constEnergyTotalVTVT dt d ==+⇒=+ Return to Table of Content
  • 9. 9 Analytic DynamicsSOLO 1.4 Basic Definitions Given a system of N particles defined by their coordinates: ( ) ( ) ( ) ( ) Nkktzjtyitxzyxrr kkkkkkkk ,,2,1,,   =++== where are the unit vectors defining any Inertial Coordinate Systemkji  ,, The real displacement of the particle mk : ( ) ( ) ( ) Nkktdzjtdyitdxrd kkkk ,,2,1   =++= is the infinitesimal change in the coordinates along real path caused by all the forces acting on the particle mk . The virtual displacements (Δxk , Δyk, Δzk, Δt) are infinitesimal changes in the coordinates; they are not real changes because they are not caused by real forces. The virtual displacements define a virtual path that coincides with the real one at the end points.
  • 10. 10 Analytic DynamicsSOLO Basic Definitions (continue) ( )trk  ( )1trk  ( )2trk  krd  1 2 F  km ),,,( dttdzzdyydxxPrdr kkkkkkkk ++++=+  ),,,( tzyxP kkk ),,,( ttzzyyxxP kkkkkk ∆+∆+∆+∆+ ),,,( tzzyyxxP kkkkkk ∆+∆+∆+ tvrd kk ∆=  i  j  k  True (Dynamical or Newton) Path Virtual Path Return to Table of Content
  • 11. 11 Analytic DynamicsSOLO 1.5 Constraints If the N particles are free the system has n = 3 N degrees of freedom.( ) Nkzyxr kkkk ,,2,1,,   = The constraints on the system can be of the following types: (1) Equality Constraints: The general form (the Pffafian form) ( ) ( ) ( )[ ] ( ) mldttradztradytradxtra l t N k k l zkk l ykk l xk ,,2,10,,,, 1   ==+++∑= or { } maaarankmldtarda l zk l yk l xk l t N k k l k ===+⋅∑= ,,,,2,10 1   We can classify the constraints as follows: (a) Time Dependency (a1) Catastatic mlal t ,,2,10 == (a2) Acatastatic mlal t ,,2,10 =≠ (1) Equality Constraints (2) Inequality Constraints
  • 12. 12 Analytic DynamicsSOLO Constraints (continue) Equality Constraints: The general form (the Pffafian form) (continue) { } maaarankmldtarda l zk l yk l xk l t N k k l k ===+⋅∑= ,,,,2,10 1   (b) Integrability (b1) Holonomic if the Pffafian forms are integrable; i.e.: mldt t f zd z f yd y f xd x f df N k l k k l k k l k k l l ,,2,1 1 = ∂ ∂ +      ∂ ∂ + ∂ ∂ + ∂ ∂ = ∑= or ( ) mltzyxzyxf NNNl ,,2,10,,,,,,, 111  == (b2) Non-holonomic if the Pffafian forms are not integrable (b2.1) Scleronomic: (b2.2) Rheonomic: ml l t f ,,2,1 0 = = ∂ ∂ or ( ) mlzyxzyxf NNNl ,,2,10,,,,,, 111  == ml l t f ,,2,1 0 = ≠ ∂ ∂
  • 13. 13 Analytic DynamicsSOLO Constraints (continue) (2) Inequality Constraints: (a) Stationary Boundaries (time independent): (b) Non-stationary Boundaries (time dependent): ( ) mlzyxzyxf NNNl ,,2,10,,,,,, 111  =≥ ( ) mltzyxzyxf NNNl ,,2,10,,,,,,, 111  =≥
  • 14. 14 Analytic DynamicsSOLO Constraints (continue) Displacements Consistent with the Constraints: The real displacement consistent with the General Equality Constraints (Pffafian form) is: The virtual displacement consistent with the General Equality Constraints (Pffafian form) is: dtkdzjdyidxrd kkkk ,  ++= [ ] mldtardadtadzadyadxa l t N k k l k l t N k k l zkk l ykk l xk ,,2,10 11   ==+⋅=+++ ∑∑ == tkzjyixr kkkk ∆∆+∆+∆=∆ ,  [ ] mltaratazayaxa l t N k k l k l t N k k l zkk l ykk l xk ,,2,10 11   ==∆+∆⋅=∆+∆+∆+∆ ∑∑ == Dividing the Pffafian equation by dt and taking the limit, we obtain: mlraa N k k l k l t ,,2,1 1   =⋅−= ∑= ⋅ Now replace in the virtual displacement equationl ta mltrra N k kk l k ,,2,10 1   ==      ∆−∆⋅∑= ⋅ Define the δ variation as: td d t∆−∆= ∆ δ
  • 15. 15 Analytic DynamicsSOLO Constraints (continue) Displacements Consistent with the Constraints (continue): Define the δ variation as: td d t∆−∆= ∆ δ ( )trk  kr  δ km ),,,( dttdzzdyydxxPrdr kkkkkkkk ++++=+  ),,,( tzyxP kkk ),,,( ttzzyyxxP kkkkkk ∆+∆+∆+∆+ ),,,( tzzyyxxP kkkkkk ∆+∆+∆+ dtrrd kk ⋅ =  i  j  k  True (Dynamical or Newton) Path Virtual Path kr  ∆ trr kk ∆=∆ ⋅ Then: kkk r td d trr  ∆−∆= ∆ δ From the Figure we can see that δ variation corresponds to a virtual displacement in which the time t is held fixed and the coordinates varied to the constraints imposed on the system. mlra N k k l k ,,2,10 1   ==⋅∑= δ For the Holonomic Constraints: ( ) mltzyxzyxf NNNl ,,2,10,,,,,,, 111  == mlrf N k klk ,,2,10 1   ==⋅∇∑= δ Return to Table of Content
  • 16. 16 Analytic DynamicsSOLO 1.6 Generalized Coordinates The motion of a mechanical system of N particles is completely defined by n = 3N coordinates . Quite frequently we may find it more advantageous to express the motion of the system in terms of a different set of coordinates, say . If we take in consideration the m constraints we can reduce the coordinates to n = 3N-m generalized coordinates. ( ) ( ) ( ) ( )Nktztytx kkk ,,2,1,, = ( )T nqqqq ,,, 21   = ( ) ( ) ( ) ( ) ( ) Nkktqzjtqyitqxtqqqrtqr kkknkk ,,2,1,,,,,,,, 21     =++== Nkkdzjdyidxdt t r dq q r rd kkk k j n j j k k ,,2,1 1    =++= ∂ ∂ + ∂ ∂ = ∑= Nk t r q q r td rd rv k j n j j kk kk ,,2,1 1      = ∂ ∂ + ∂ ∂ === ∑= ⋅ In the same way Nkkzjyixt t r q q r r kkk k j n j j k k ,,2,1 1    =∆+∆+∆=∆ ∂ ∂ +∆ ∂ ∂ =∆ ∑= and Nkt t r tq q r t t r q q r trrr k j n j j kk j n j j k kkk ,,2,1 11      ==∆ ∂ ∂ −∆ ∂ ∂ −∆ ∂ ∂ +∆ ∂ ∂ =∆−∆= ∑∑ == ⋅ δ
  • 17. 17 Analytic DynamicsSOLO Generalized Coordinates (continue) ( ) Nkq q r tqq q r r n j j j k jj n j j k k ,,2,1 11      = ∂ ∂ =∆−∆ ∂ ∂ = ∑∑ == δδ where tqqq jjj ∆−∆= ∆ δ The Generalized Equality Constraints in Generalized Coordinates will be: mldt t r aadq q r a dt t r aadq q r adtarda N k kl k l ti n i i k N k l k N k N k kl k l ti n i i kl k l t N k k l k ,,2,10 11 1 1 111          ==      ∂ ∂ ⋅++      ∂ ∂ ⋅= =      ∂ ∂ ⋅++ ∂ ∂ ⋅=+⋅ ∑∑ ∑ ∑ ∑∑∑ == = = === If we define ∑ ∑∑= = ∆ = ∆ ∂ ∂ ⋅+= ∂ ∂ ⋅= N k N k kl k l t l t n i i kl k l i t r aac q r ac 1 11 &     we obtain mldtcdqc l ti n i l i ,,2,10 1 ==+∑= and the virtual displacements compatible with the constraints are mlqc i n i l i ,,2,10 1 ==∑= δ
  • 18. 18 Analytic DynamicsSOLO Generalized Coordinates (continue) The number of degrees of freedom of the system is n = 3N-m. However, when the system is nonholonomic, it is possible to solve the m constraint equations for the corresponding coordinates so that we are forced to work with a number of coordinates exceeding the degrees of freedom of the system. This is permissible provided the surplus number of coordinates matches the number of constraint equations. Although in the case of a holonomic system it may be possible to solve for the excess coordinates, thus eliminating them, this is not always necessary or desirable. If surplus coordinates are used, the corresponding constraint equations must be retained. Return to Table of Content
  • 19. 19 Analytic DynamicsSOLO 1.7 The Stationary Value of a Function and of a Definite Integral In problems of dynamics is often sufficient to find the stationary value of functions instead of the extremum (minimum or maximum). Definition: A function is said to have a stationary value at a certain point if the rate of change in every direction of the point is zero. Examples: (1) ( ) ni u f du u f dfuuuf i n i i i n ,,2,100,,, 1 21  == ∂ ∂ →= ∂ ∂ =→ ∑= By solving those n equations we obtain for which f is stationary ( )nuuu ,,, 21 
  • 20. 20 Analytic DynamicsSOLO The Stationary Value of a Function and of a Definite Integral (continue) Examples (continue): (2) ( )nuuuf ,,, 21  with the constraints { } marankmldua l k N k k l k ===∑= ,,2,10 1  Lagrange’s multipliers solution gives: 0 1 1 =      + ∂ ∂ = ∑ ∑= = i n i m l l il i dua u f df λ By choosing the m Lagrange’s multipliers λl to annihilate the coefficients of the m dependent differentials dui we have equationsmn mldua nia u f n l i l i m l l il i +        == ==+ ∂ ∂ ∑ ∑ = = ,,2,10 ,,2,10 1 1  λ
  • 21. 21 Analytic DynamicsSOLO The Stationary Value of a Function and of a Definite Integral (continue) Examples (continue): (3) The functional ( ) ( ) ∫       = 2 1 ,, x x dx xd xyd xyxFI We want to find such that I is stationary, when the end points and are given. ( )xy ( )1xy ( )2xy ( )xy ( ) ( ) ( ) ( )xxyxyxy ηεδ +=+ ( )11, yx ( )22 , yx x y The variation of is( )xy ( ) ( ) ( ) ( ) ( ) ( ) ( ) 021 ==+=+= xxxxyxyxyxy ηηηεδ and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫∫       ++=      = 2 1 2 1 ,,,, x x x x dx xd xd xd xyd xxyxFdx xd xyd xyxFI η εηεε ( ) ( ) +++== == 2 0 2 2 0 2 1 0 ε ε ε ε εε εε d d Id d d Id II
  • 22. 22 Analytic DynamicsSOLO The Stationary Value of a Function and of a Definite Integral (continue) Examples (continue): Continue: The functional ( ) ( ) ∫       = 2 1 ,, x x dx xd xyd xyxFI The necessary condition for a stationary value is ( ) ( ) ( ) ( ) ( )[ ] 0 0 12 0 2 1 2 1 =−       ∂ ∂ +                     ∂ ∂ − ∂ ∂ =                     ∂ ∂ + ∂ ∂ = ∫ ∫ =    xx xd yd F dxx xd yd F xd d y F dx xd xd xd yd F x y F d Id x x nintegratio partsby x x ηηη η η ε ε Since this must be true for every continuous function η(x) we have 210 xxx xd yd F xd d y F ≤≤=                     ∂ ∂ − ∂ ∂ Euler-Lagrange Differential Equation By solving this differential equation, y(x),for which I is stationary is found. JOSEPH-LOUIS LAGRANGE 1736-1813 LEONHARD EULER 1707-1783
  • 23. 23 Leonhard Euler (1707-1783) generalized the brothers Bernoulli methods in “Me tho dus inve nie ndi line as curvas m axim i m inim ive pro prie tate g aude nte s sive so lutio pro ble m atis iso pe rim e trici latissim o se nsu acce pti” (“Me tho d fo r finding plane curve s that sho w so m e pro pe rty o f m axim a and m inim a”) published in 1744. Euler solved the Ge o de sic Pro ble m , i.e. the curves of minimum length constrained to lie on a given surface. Joseph-Louis Lagrange (1736-1813) gave the first analytic methods of Calculus of Variations in "Essay on a new method of determining the maxima and minima of indefinite integral formulas" published in 1760. Euler-Lagrange Equation: SOLO CALCULUS OF VARIATIONS Return to Table of Content
  • 24. 24 Analytic DynamicsSOLO 1.8 The Principle of Virtual Work This is a statement of the Static Equation of a mechanical system. If the system of N particles is in dynamic equilibrium the resultant force on each particle is zero; i.e.: 0=iR  0 1 =⋅= ∑= N i ii rRW  δδ Because of this, for a virtual displacement the Virtual Work of the system is ir  δ If the system is subjected to the constraints: { } maaarankmldtarda l zk l yk l xk l t N k k l k ===+⋅∑= ,,,,2,10 1   Then we denote the external forces on particle i by and the constraint’s forces by . The resultant force on i is: iF  iF'  0' =+= iii FFR 
  • 25. 25 Analytic DynamicsSOLO The Principle of Virtual Work (continue) We want to find the Virtual Work of the constraint forces. There are two kind of constraints: (1) The particle i is constrained to move on a definite surface. We assume that the motion is without friction and therefore the constraint forces must be normal to the surface. The virtual variation compatible with the constraint must be on the surface, therefore . iF  ir  δ 0' =⋅ ii rF  δ ir  δ iF'  (2) The particle i is acting on the particle j and the distance between them is l(t). . iF'  i j jF'  ir  jr  ( )tl
  • 26. 26 Analytic DynamicsSOLO The Principle of Virtual Work (continue) ( ) ( ) ( )tlrrrr jiji 2 =−⋅−  ( ) ( ) tllrrrr jiji ∆=∆−∆⋅−  ( ) llrrrr jiji  =      −⋅− ⋅⋅ ( ) ( ) ( ) ji rr jiji jjiiji rrrrrr trrtrrrr ji    δδδδ =→=−⋅−→ →=            ∆−∆−      ∆−∆⋅− ≠ ⋅⋅ 0 0 ji FF ''  −= If we compute the virtual variation and differential and we multiply the second equation by and add to the first we obtaint∆− Because is a real (not a generalized) force we can use Newton’s Third Law: i.e.:iF'  and the virtual work of the constraint forces of this system is: ( ) 0''''' =⋅−+⋅=⋅+⋅= rFrFrFrFW iijjii  δδδδδ We can generalized this by saying that: 0' 1 =⋅∑= N i ii rF  δ The work done by the constraint forces in virtual displacements compatible with the constraints (without dissipation) is zero.
  • 27. 27 Analytic DynamicsSOLO The Principle of Virtual Work (continue) From equation we obtain:0' =+= iii FFR  ∑∑∑∑ ==== ⋅=⋅+⋅=⋅= N i ii N i ii N i ii N i ii rFrFrFrR 1 0 111 '0    δδδδ or 0 1 =⋅= ∑= N i ii rFW  δδ The Principle of Virtual Work The work done by the applied forces in infinitesimal virtual displacements compatible with the constraints (without dissipation) is zero
  • 28. 28 Analytic DynamicsSOLO The Principle of Virtual Work (continue) { } mjNimaaarank mjra j zi j yi j xi N k k j k ,,2,1&,,1,, ,,2,10 1    === ==⋅∑= δ We found that the General Equality Constraint the virtual displacement compatible with the constraint must be: ir  δ Let adjoin the m constraint equations by the m Lagrange’s multipliers and add to the virtual work equation: jλ 0 1 11 11 =⋅        +=      ⋅+⋅= ∑ ∑∑ ∑∑ = == == N i i m j j iji m j N i i j ij N i ii raFrarFW  δλδλδδ There are 3N virtual displacements from which m are dependent of the constraint relations and 3N-m are independent. We will choose the m Lagrange’s multipliers to annihilate the coefficients of the m dependent variables: jλ    −+= = =+ ∑= iationsmNtindependenNmi mtheofbecausemi aF j m j j iji var33,,1 ,,2,1 0 1   λ λ
  • 29. 29 Analytic DynamicsSOLO The Principle of Virtual Work (continue) From we obtain:0' =+= iii FFR  ∑= = m j j iji aF 1 '  λ where are chosen such thatjλ mjforaF m j j iji ,,2,10 1   ==+ ∑= λ Since , we obtain:k n k k i i q q r r δδ ∑= ∂ ∂ = 1   0 1 1 11 1 111 1 =                       ∂ ∂ +      ∂ ∂ = = ∂ ∂ ⋅        +=⋅        += ∑ ∑ ∑∑ ∑ ∑∑∑ ∑ = = == = === = n k k m j N i k ij ij N i k i i N i n k k k i m j j iji N i i m j j iji q q r a q r F q q r aFraFW δλ δλδλδ      We define: nk q r FQ N i k i ik ,,2,1 1   = ∂ ∂ =∑= ∆ nkc q r aQ m j j kj m j N i k ij ijk ,,2,1' 11 1    ==      ∂ ∂ = ∑∑ ∑ == = ∆ λλ nk q r ac N i k ij i j k ,,2,1 1    = ∂ ∂ =∑= ∆ Generalized Forces Generalized Constraint Forces Return to Table of Content
  • 30. 30 Analytic DynamicsSOLO 2. D’Alembert Principle Jean Le Rond d’ Alembert 1717-1783 Newton’s Second Law for a particle of mass and a linear momentum vector can be written as im iii vmp  = D’Alembert Principle: 0' =−+ • iii pFF  D’Alembert Principle enables us to trait dynamical problems as if they were statical. Let extend the Principle of Virtual Work to dynamic systems: 0' 1 =⋅      −+∑= •N i iiii rpFF  δ Assuming that the constraints are without friction the virtual work of the constraint force is zero . Then we have Generalized D’Alembert Principle: 0 1 =⋅      −∑= •N i iii rpF  δ 0' 1 =⋅∑= N i ii rF  δ The Generalized D’Alembert Principle The total Virtual Work performed by the effective forces through infinitesimal Virtual Displacement, compatibile with the system constraints are zero. 0=− • ii pF  is the effective force. “Traité de Dynamique” 1743 Return to Table of Content where and are the applied and constraint forces, respectively.iF  iF' 
  • 31. 31 Analytic DynamicsSOLO 3. Hamilton’s Principle William Rowan Hamilton 1805-1865 Let write the D’Alembert Principle: in integral form0 1 =⋅      −∑= •N i iii rpF  δ 0 2 1 1 =⋅      −∫∑= • t t N i iii dtrpF  δ But ( )∑∑∑ === • ⋅+      ⋅−=⋅− N i iii N i iii N i iii r td d vmrvm td d rvm 111  δδδ Let find ( )ir td d  δ iiiii r td d ttvrr        ∆−∆=∆−∆=δ are the virtual displacements compatible with the constraints mjra N i i j i ,,2,10 1   ==⋅∑= δ ( )tri  ir  δ tvi ∆  ( )tPi( )tP i' ( )ttP i ∆+' ir  ∆ Virtual Path True Path (P) Newtonian or Dynamic Path The Constraint Space at t mjra N i i j i ,,10 1   ==⋅∑= δ ( ) ( ) ( ) ( ) ( ) ( )     =∆=∆ =∆=∆ == 0 0 0 21 21 21 tttt trtr trtr ii ii   δδ 1t 2t
  • 32. 32 Analytic DynamicsSOLO Hamilton’s Principle (continue) Since td rd vv i Pi i   == ( ) ( ) ( ) t td d vr td d vt td d r td d v t td d r td d td rd tdtd rdrd ttd rrd vvv iiiii i i iiii ttPii i ∆−∆+≈      ∆−      ∆+≈ ∆+ ∆+ = = ∆+ ∆+ = ∆+ ∆+ ==∆+ ∆+      1 1 ' ( ) ( ) tar td d tatvr td d t td d vr td d v iiiiiiii ∆+=∆+∆−∆=∆−∆=∆  δ Therefore ( ) ( ) ecommutativare td d and r td d vv td d ttavr td d iiiiii δ δδδ → →==      ∆−∆=∆−∆= 
  • 33. 33 Analytic DynamicsSOLO Hamilton’s Principle (continue) Now we can develop the expression: tavmvvmrvm td d ram N i iii N i iii N i iii N i iii ∆⋅−      ∆⋅+      ⋅−=⋅− ∑∑∑∑ ==== 1111  δδ But the Kinetic Energy T of the system is: ∑= ⋅= N i iii vvmT 12 1  ∑= ∆⋅=∆ N i iii vvmT 1  ∑∑∑ === ⋅ ⋅=⋅=⋅= N i iii N i iii N i iii vFmavmvvmT 111  Therefore Trvm td d tTTrvm td d ram N i iii N i iii N i iii δδ δδ +      ⋅−= =∆−∆+      ⋅−=⋅− ∑ ∑∑ = • == 1 11  
  • 34. 34 Analytic DynamicsSOLO Hamilton’s Principle (continue) From the integral form of D’Alembert Principle we have: ( ) ∫ ∑∫ ∑∑ ∫ ∑∫ ∑ ∫∑       ⋅+=      ⋅++⋅−= =      ⋅++      ⋅−= =⋅+−= === == = 2 1 2 1 2 1 2 1 2 1 2 1 111 0 11 1 0 t t N i ii t t N i ii N i t tiii t t N i ii t t N i iii t t N i iiii dtrFTdtrFTrvm dtrFTdtrvm td d dtrFam      δδδδδ δδδ δ We obtained ( ) 0 2 1 2 1 1 =+=      ⋅+ ∫∫ ∑= dtWTdtrFT t t t t N i ii δδδ  Extended Hamilton’s Principle
  • 35. 35 Analytic DynamicsSOLO Hamilton’s Principle (continue) If we develop and we can writetTTT ∆−∆= δ tvrr iii ∆−∆=  δ 0 2 1 2 1 111 =      ∆      ⋅+−      ∆⋅+∆=      ⋅+ ∫ ∑∑∫ ∑ === dttvFTrFTdtrFT t t N i ii N i ii t t N i ii   δδ and because ∑= ⋅= N i ii vFT 1   02 2 1 1 =      ∆−      ∆⋅+∆∫ ∑= dttTrFT t t N i ii  The pair and is arbitrary but compatible with the constraints:ir  ∆ t∆ mjtara j t N i i j i ,,2,10 1   ==∆+∆⋅∑=
  • 36. 36 Analytic DynamicsSOLO Hamilton’s Principle (continue) For a Conservative System VF ii −∇=  VrVrFW N i ii N i ii δδδδ −=⋅∇−=⋅= ∑∑ == 11  We have ( ) ( ) 0 2 1 2 1 2 1 ==−=+ ∫∫∫ t t t t t t dtLdtVTdtWT δδδ where WTVTL +=−= ∆       =−∇=−== ∆ ∫ NiVFVTLdtL ii t t ,,2,1;0 2 1   δ Hamilton’s Principle for Conservative Systems Hamilton’s Principle for Conservative Systems: The actual path of a conservative system in the configuration space renders the value of the integral stationary with respect to all arbitrary variations (compatible with the constraints) of the path between the two instants and provided that the path variations vanish at those two points. ∫= 2 1 t t dtLI 1t 2t Return to Table of Content
  • 37. 37 Analytic DynamicsSOLO 4. Lagrange’s Equations of Motion Joseph Louis Lagrange 1736-1813 “Mecanique Analitique” 1788 The Extended Hamilton’s Principle states: 0 2 1 1 =      ⋅+∫ ∑= dtrFT t t N i ii  δδ where are the virtual displacements compatible with the constraints: ir  δ mjqcq q r ara n k k k i n k k N i k ij i N i i j i ,,2,10 11 11   ===      ∂ ∂ =⋅ ∑∑ ∑∑ == == δδδ T the kinetic energy of the system is given by: ∑ ∑∑∑ = ⋅ === ⋅⋅       =      ∂ ∂ + ∂ ∂ ⋅      ∂ ∂ + ∂ ∂ =⋅= N j n i j i i j n i j i i j j N j jjj tqqT t r q q r t r q q r mrrmT 1 111 ,, 2 1 2 1        where is the vector of generalized coordinates.( )nqqqq ,,, 21   =       −∆ ∂ ∂ +      ∆ ∂ ∂ +∆ ∂ ∂ +      =      −      ∆+∆+∆+=∆ ⋅ = ⋅⋅⋅⋅ ∑ tqqTt t T q q T q q T tqqTtqqTttqqqqTT n i i i i i ,,,,,,,, 1     t T q q T q q T T n i i i i i ∂ ∂ +      ∂ ∂ + ∂ ∂ = ∑=1    ( ) ( ) ∑∑ ==       ∂ ∂ + ∂ ∂ =      ∆−∆ ∂ ∂ +∆−∆ ∂ ∂ =∆−∆= n i i i i i n i ii i ii i q q T q q T tqq q T tqq q T tTTT 11      δδδ
  • 38. 38 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue) ( ) ( ) ∑∑ ==       ∂ ∂ + ∂ ∂ =      ∆−∆ ∂ ∂ +∆−∆ ∂ ∂ =∆−∆= n i i i i i n i ii i ii i q q T q q T tqq q T tqq q T tTTT 11      δδδ But because δ and are commutative and:td d ( )ii q dt d q δδ = ∑=       ∂ ∂ + ∂ ∂ = n i i i i i q dt d q T q q T T 1 δδδ  This is an expected result because the variation δ keeps the time t constant. We found that , therefore∑= ∂ ∂ = n i i i j j q q r r 1 δδ   ∫∑∫∑ ∑∫ ∑ ∑∫ ∑ == == == =        ∂ ∂ ⋅=        ∂ ∂ ⋅=        ⋅ 2 1 2 1 2 1 2 1 11 11 11 t t n i ii t t n i i N j i j j t t N j n i i i j j t t N j jj dtqQdtq q r Fdtq q r FdtrF δδδδ     where ForcesdGeneralizeni q r FQ N j i j ji ,,2,1 1    = ∂ ∂ ⋅=∑= ∆ Now ( ) ∫∑∑ ∫∑∫ ∑ == ==       − ∂ ∂ −      ∂ ∂ − ∂ ∂ =       + ∂ ∂ + ∂ ∂ =        ⋅+= 2 1 2 1 2 1 2 1 11 0 .int 11 0 t t n i iii i i i n i t ti i partsby t t n i iii i i i t t N j jj dtqQq q T q q T td d q q T dtqQq q T q td d q T dtrFT δδδδ δδδδδ    
  • 39. 39 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue) 0 2 1 1 =      − ∂ ∂ −      ∂ ∂ ∫∑= t t i n i i ii dtqQ q T q T td d δ  where the virtual displacements must be consistent with the constraints . Let adjoin the previous equations by the constraints multiplied by the Lagrange’s multipliers iqδ mkqc n i i k i ,,2,10 1 ==∑= δ ( )mkk ,,2,1 =λ 0 1 11 1 =      =      ∑ ∑∑ ∑ = == = n i i m k k ik m k n i i k ik qcqc δλδλ to obtain 0 2 1 1 1 =      −− ∂ ∂ −      ∂ ∂ ∫∑ ∑= = t t i n i m k k iki ii dtqcQ q T q T td d δλ  While the virtual displacements are still not independent, we can chose the Lagrangian’s multipliers so as to render the bracketed coefficients of equal to zero. The remaining being independent can be chosen arbitrarily, which leads to the conclusion that the coefficients of are zero. It follows iqδ iqδ ( )mkk ,,2,1 =λ ( )nmiqi ,,2,1 +=δ ( )miqi ,,2,1 =δ nicQ q T q T dt d m k k iki ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λ
  • 40. 40 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue) nicQ q T q T dt d m k k iki ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λ We have here n equations with n+m unknowns . To find all the unknowns we must add the m equations defined by the constraints, to obtain ( ) ( ) mn tqtq λλ ,,,,, 11  nicQ q T q T dt d m k k iki ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λLagrange’s Equations: mkcqc k t n i i k i ,,2,10 1  ==+∑= Let define Generalized Constraint Forces: nicQ m k k iki ,,2,1' 1 == ∑= λ
  • 41. 41 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue) If the system is acted upon by some forces which are derivable from a potential function and some forces which are not, we can write:( ) ( )nn qqqVrrrV ,,,,,, 2121     −=− n jF  n jjj FVF  +−∇= ( ) ∑∑ ∑∑ ∑∑ == == == =              ∂ ∂ ⋅+ ∂ ∂ ⋅∇−=      ∂ ∂ ⋅+∇−=⋅ n i ii n i i N j i jn j i j j N j n i i i jn jj N j jj qQq q r F q r Vq q r FVrF 11 11 11 δδδδ     But where∑= ∂ ∂ ⋅∇= ∂ ∂ N j i j j i q r V q V 1  k z V j y V i x V V jjj j  ∂ ∂ + ∂ ∂ + ∂ ∂ =∇ Therefore: niQ q V q r F q r VQ in i N j i jn j N j i j ji ,,2,1 11     =+ ∂ ∂ −= ∂ ∂ ⋅+ ∂ ∂ ⋅∇−= ∑∑ == Generalized External Forces: Generalized External Nonconservative Forces: ni q r FQ N j i jn jin ,,2,1 1    = ∂ ∂ ⋅=∑= ∆
  • 42. 42 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue) The Lagrange’s Equations nicQ q T q T dt d m k k iki ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λ Define: ( ) ( ) ( )qVtqqTtqqL  −= ∆ ,,,, Because we assume that , we have( ) i q qV i ∀= ∂ ∂ 0   Lagrange’s Equations: nicQ q L q L dt d m k k ikin ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λ mkcqc k t n i i k i ,,2,10 1  ==+∑= We proved       =−∇=−== ∆ ∫ NiVFVTLdtL ii t t ,,2,1;0 2 1   δ Hamilton’s Principle for Conservative Systems Lagrange’s Equations for a Conservative System without Constraints: ( )0,,,,2,10 =−=−∇=== ∂ ∂ −      ∂ ∂ k iii ii cVTLVFni q L q L dt d    If they are no constraints, from the Lagrange’s Equations, or from Euler- Lagrange Equation for a stationary solution of , we obtain:∫= 2 1 t t dtLI
  • 43. Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue) The Lagrangian gives unique Euler-Lagrange Equations, but the inverse is not true, there more than one Lagrangian that gives the same Euler-Lagrange Equations. Example 1: ( ) ( ) ( ) td tqqd tqqLtqqL ,, ,,:,,1   Φ += ( ) ( ) ( ) ( )   ( )∫∫∫∫ = ∂ Φ∂ + ∂ Φ∂ +=Φ+= 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ,,,,,,,,:,, 00 1 t t t t t t t t t t t t t t tdtqqLq q q q tdtqqLtqqtdtqqLtdtqqL     δδδδδδδ Example 2: This is not the most general case ( ) ( ) [ ]2 2 2 1 2 2 2 1221211 2 1 :,,&:,, qqqqtqqLqqqqtqqL −−+=−=          =−= ∂ ∂ −      ∂ ∂ =−= ∂ ∂ −      ∂ ∂ 0 0 11 2 1 2 1 22 1 1 1 1 qq q L q L dt d qq q L q L dt d            =−= ∂ ∂ −      ∂ ∂ =−= ∂ ∂ −      ∂ ∂ 0 0 22 2 2 2 2 11 1 2 1 2 qq q L q L dt d qq q L q L dt d     Return to Table of Content
  • 44. 44 Analytic DynamicsSOLO 5. Hamilton’s Equations The Lagrange’s Equations nicQ q T q T dt d m k k iki ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λ can be rewritten as: nicQ q T tq T q qq T q qq T q T dt d m k k iki i n i i j ji j jii ,,2,1 11 222       =++ ∂ ∂ =         ∂∂ ∂ + ∂∂ ∂ + ∂∂ ∂ =      ∂ ∂ ∑∑ == λ therefore consist of a set of n simultaneous second-order differential equations. They must be solved tacking in consideration the m constraint equations. mkcqc k t n i i k i ,,2,10 1  ==+∑= A procedure for the replacement of the n second-order partial-differential equations by 2n first-order ordinary-differential equations consists of formulating the problem in terms of 2n Hamilton’s Equations. We define first: General Momentum: ni q T p i i ,,2,1   = ∂ ∂ = ∆ We want to find the transformation from the set of variables to the set by the Legendre’s Dual Transformation. ( )tqq ,,  ( )tpq ,, 
  • 45. 45 Analytic DynamicsSOLO Hamilton’s Equations (continue) Legendre’s Dual Transformation. Adrien-Marie Legendre 1752-1833 Let consider a function of n variables , m variables and time t.ix iy ( )tyyxxF mn ,,,,,, 11  and introduce a new set of variables defined by the transformation:iu ni x F u i i ,,2,1 = ∂ ∂ = ∆ We can see that:                             ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ +                             ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ ∂ =             m mnn m n nn n n dy dy dy yx F yx F yx F yx F dx dx dx x F xx F xx F x F du du du          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 by the new variables . We can see that the new n variables are independent if the Hessian Matrix is nonsingular. ( )nidxi ,,2,1 = ni njji xx F ,,1 ,,1 2   = =         ∂∂ ∂
  • 46. 46 Analytic DynamicsSOLO Hamilton’s Equations (continue) Legendre’s Dual Transformation (continue-1) Let define a new function G of the variables , and t.iu iy ( )tyyuuGFxuG mn n i ii ,,,,,, 11 1 =−=∑= ∆ Then: ( ) dt t F dy y F dx x F udux dt t F dy y F dx x F dxuduxdG m j j j n i i i iii n i m j j j i i n i iiii ∂ ∂ − ∂ ∂ −                   ∂ ∂ −+= = ∂ ∂ − ∂ ∂ − ∂ ∂ −+= ∑∑ ∑ ∑∑ == = == 11 0 1 11  But because: ( )tyyuuGG mn ,,,,,, 11 = dt t G dy y G du u G dG n i m j j j i i ∂ ∂ + ∂ ∂ + ∂ ∂ = ∑ ∑= =1 1 Because all the variations are independent we have: t F t G mj y F y G ni u G x jji i ∂ ∂ −= ∂ ∂ = ∂ ∂ −= ∂ ∂ = ∂ ∂ = ;,,1;,,1 
  • 47. 47 Analytic DynamicsSOLO Hamilton’s Equations (continue) Legendre’s Dual Transformation (continue-2) Now we can define the Dual Legendre’s Transformation from ( )tyyxxF mn ,,,,,, 11  ( ) FxutyyuuG n i iimn −= ∑=1 11 ,,,,,, to by using ni x F u i i ,,2,1 = ∂ ∂ = ni u G x i i ,,2,1 = ∂ ∂ = End of Legendre’s Dual Transformation
  • 48. 48 Analytic DynamicsSOLO Hamilton’s Equations (continue) Following the same pattern to find the transformation from the set of variables to the set , we introduce the Hamiltonian:( )tqq ,,  ( )tpq ,,  ( )tqqTqpH n i ii ,, 1   −=∑= ∆ where ni q T p i i ,,2,1   = ∂ ∂ = Then ( )tpqHH ,,  = dt t H dp p H dq q H dt t T dq q T dpq dt t T dq q T qd q T qdpdpqdH n i i i i i n i i i ii n i i i i i iiii ∂ ∂ +      ∂ ∂ + ∂ ∂ = ∂ ∂ −      ∂ ∂ −= = ∂ ∂ −      ∂ ∂ − ∂ ∂ −+= ∑∑ ∑ == = 11 1     and 0 1 =      ∂ ∂ + ∂ ∂ +            − ∂ ∂ +      ∂ ∂ + ∂ ∂ ∑= dt t T t H dpq p H dq q T q Hn i ii i i ii 
  • 49. 49 Analytic DynamicsSOLO Hamilton’s Equations (continue) If the Hessian Matrix is nonsingular, all the are independent, but not the that must be consistent with the constraints: ni njji qq T ,,1 ,,1 2    = =         ∂∂ ∂ ( )nidpi ,,2,1 = ( )nidqi ,,2,1 = mjdtcdqc j t n i i j i ,,2,10 1 ==+∑= Let adjoin the previous equations by the constraint equations multiplied by the m Lagrange’s multipliers :j'λ 0''' 11 11 1 =+        =      + ∑∑ ∑∑ ∑ == == = m j j ij n i i m j j ij m j j t n i i j ij dtcdqcdtcdqc λλλ We have 0'' 11 1 =        + ∂ ∂ + ∂ ∂ +               − ∂ ∂ +        + ∂ ∂ + ∂ ∂ ∑∑ ∑ == = dtc t T t H dpq p H dqc q T q H m j j tj n i ii i i m j j ij ii λλ 
  • 50. 50 Analytic DynamicsSOLO Hamilton’s Equations (continue) By proper choosing the m Lagrange’s multipliers λ’j ,the remainder differentials and dt are independent and therefore we have:ii dpdq , ni c t H t T c q H q T p H q m j j tj m j j ij ii i i ,,2,1 ' ' 1 1   =           − ∂ ∂ −= ∂ ∂ − ∂ ∂ −= ∂ ∂ ∂ ∂ = ∑ ∑ = = λ λ Legendre’s Dual Transformation By differentiating the General Momentum Equation and using Lagrange’s Equations we obtain: ( )∑∑ == −++ ∂ ∂ −=++ ∂ ∂ =      ∂ ∂ = m j j ijji i m j j iji ii i cQ q H cQ q T q T dt d p 11 ''''' λλλ   ni cQ q H p p H q m j j iji i i i i ,,2,1 1    =        ++ ∂ ∂ −= ∂ ∂ = ∑= λ mkcqc k t n i i k i ,,2,10 1  ==+∑= Extended Hamilton’s Equations Constrained Differential Equations
  • 51. 51 Analytic DynamicsSOLO Hamilton’s Equations (continue) For Holonomic Constraints (constraints of the form ) we can (theoretically) reduce the number of generalized coordinates to n-m and we can assume that and n represents the number of degrees of freedom of the system (this reduction is not possible for Nonholonomic Constraints). Then: ( ) mjtqqf nj .,10,,,1  == 0== j t j i cc ni Q q H p p H q i i i i i ,,2,1    =        + ∂ ∂ −= ∂ ∂ = Extended Hamilton’s Equations for Holonomic Constraints ni q V Q i i ,,2,1 = ∂ ∂ −= ( ) ( ) ( )qVtqqTtqqL  −= ∆ ,,,, ni q L q T p ii i ,,2,1   = ∂ ∂ = ∂ ∂ = Extended Hamilton’s Equations for Holonomic Constraints and a Conservative System Conservative System
  • 52. 52 Analytic DynamicsSOLO Hamilton’s Equations (continue) Define: Hamiltonian for Conservative Systems ( ) ( ) ( ) ( )qVtqqTtqqLqptqqH n i ii   −=−=∑= ∆ ,,,,,, 1 Hamilton’s Canonical Equations for Conservative Systems with Holonomic Constraints ni q H p p H q i i i i ,,2,1    =        ∂ ∂ −= ∂ ∂ = We have: Return to Table of Content
  • 53. 04/08/15 53 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Second Method (Carathéodory) Constantin Carathéodory (1873-1950) Carathéodory developed another approach to the Euler-Lagrange Equations ( ) ( )          Covariant k l l jj k jj k q q q qqtL q qqtL p ∂ ∂ ∂ ∂ = ∂ ∂ = ,,,, :Define Canonical Momentum We assume that is a one to one correspondence between the n components of pj and the n components of and from the definition of pk (by the Inverse Function Theorem) we have j q ( )k kjj pqtx ,,φ= From we see that the one to one correspondence is possible only if ( ) ( ) ( ) td tq qqtL qd qq qqtL qd qq qqtL pd k jj j jk jj j jk jj k ∂∂ ∂ + ∂∂ ∂ + ∂∂ ∂ =        ,,,,,, : 222 ( ) 0 ,, det 2 ≠      ∂∂ ∂ jk jj qq qqtL   Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
  • 54. 04/08/15 54 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Second Method (Carathéodory) ( ) ( ) k l l jj k jj k q q q qqtL q qqtL p       ∂ ∂ ∂ ∂ = ∂ ∂ = ,,,, : Canonical Momentum ( )k jj ptq ,φ= Define ( ) ( )[ ] ( )k j ji kk k k ptpptqtLpqtH ,,,,:,, φφ +−= Hamiltonian Let compute j q q L p j j k k j k k j qq p p p q q L p H kk kk      φ φ = ∂ ∂ = =+ ∂ ∂ + ∂ ∂ ∂ ∂ −= ∂ ∂ j q L p jjjjjj q L q p qq L q L q H jj ∂ ∂ −= ∂ ∂ + ∂ ∂ ∂ ∂ − ∂ ∂ −= ∂ ∂ ∂ ∂ =   φφ t L t p t q q L t L t H kk kk q q L p j j j j ∂ ∂ −= ∂ ∂ + ∂ ∂ ∂ ∂ − ∂ ∂ −= ∂ ∂ = ∂ ∂ = φ φ     Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  William Rowan Hamilton 1805-1865
  • 55. 04/08/15 55 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Second Method (Carathéodory) ( ) k jj k q qqtL p   ∂ ∂ = ,, : Canonical Momentum ( ) ( )[ ] ( )k j ji kk k k ptpptqtLpxtH ,,,,:,, φφ +−= Hamiltonian j j q p H = ∂ ∂ jj q L q H ∂ ∂ −= ∂ ∂ t L t H ∂ ∂ −= ∂ ∂ Let compute ( ) j j jjj q H td pd q L q L td d LE ∂ ∂ += ∂ ∂ −      ∂ ∂ =  Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  Return to Table of Content
  • 56. 04/08/15 56 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Equivalent Integrals Given a Scalar S = S (t,qk ) є C2 . Along a curve C: qj = qj (t) we can form the Total Derivative: j j q q S t S td Sd  ∂ ∂ + ∂ ∂ = With the aid of this Scalar ,S, we may construct an alternative Lagrangian by writing ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) td tqtSd tqtqtLcq q S t S tqtqtLctqtqtL j jjk k jjjj , ,,,,:,,* −= ∂ ∂ − ∂ ∂ −=  We obtain a new Integral ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )12 ** 2 1 2 1 ,,,, SSCIctd td Sd tqtqtLctdtqtqtLCI t t jj t t jj −−=      −== ∫∫  ( ) ( ) 21 * SSCIcCI −=− where c > 0 is a constant. Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
  • 57. 04/08/15 57 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Equivalent Integrals Alternative Lagrangian ( ) ( )( ) ( ) ( )( ) ( )( ) td tqtSd tqtqtLctqtqtL j jjjj , ,,:,,* −=  ( ) ( ) ( )( ) ( ) ( )12 ** 2 1 ,, SSCIctdtqtqtLCI t t jj −−== ∫  We obtain That is independent of the choice of the curve C joining the points P1 and P2. It follows that C will be an extremal of the integral I* , if and only if, it is an extremal of the integral I. Accordingly I and I* are called Equivalent Integrals. ( ) ( ) 21 * SSCIcCI −=− Therefore the Lagrangian that gives the Equations of Motions is not necessarily L = T – V but it can be any function L* = c (T – V) – dS/dt. Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
  • 58. 04/08/15 58 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Equivalent Integrals Alternative Lagrangian that gives the same equation of Motion doesn’t have necessarily to differ by a total time derivative. For example Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  ( ) ( ) ( ) ( )[ ]2 2 2 1 2 2 2 1 2121 2 1 qqqqL qqqqL b a −−+= −=   0&0 0&0 22 22 11 11 11 22 22 11 =+=      ∂ ∂ −      ∂ ∂ =+=      ∂ ∂ −      ∂ ∂ =+=      ∂ ∂ −      ∂ ∂ =+=      ∂ ∂ −      ∂ ∂ qq q L q L td d qq q L q L td d qq q L q L td d qq q L q L td d bbbb aaaa         ( ) ( )[ ]2 21 2 21 2 1 qqqqLL ab −−−=−  Return to Table of Content
  • 59. A Geodesic Field is defined as a Field of Contravariant Vectors , given at each point of a finite Region G in the n+1 Space (t, q1 , q2 ,…,qn ) by a set of n class C2 functions ψj (t,qk ) that are such that, for a suitably chosen function S = S (t,qk ), the following conditions are satisfied while One other condition (Jacobi) that the curves that define a Geodesic Field, is that they don’t intersect for t1 < t < t2. ( )kjj qtq ,ψ= ( ) ( ) ( )kjjkkkk qtqwhenever td Sd qqtLqqtL ,0,,:,,* ψ==−=  ( ) otherwiseqqtL kk 0,,* > ( )kjj qtq ,ψ= 04/08/15 59 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Geodesic Field See “Calculus of Variation” for a detailed exposition. Note: Since we see that it is equivalent to “Action”.( ) ( )∫= t t kk tdqqtLtS 1 ,,  Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
  • 60. 04/08/15 60 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Geodesic Field (continue – 1) For any other curve K in G that joins the points P1 and P2 we have ( ) ( ) FieldGeodesictqqtdqqtL jj t t kk ⊂=Γ=∫ :0,, 2 1 0 *   ( ) ( ) FieldGeodesictqqKtdqqtL jj t t kk ⊄=>∫ > :0,, 2 1 0 *   Therefore the Integral I* (Γ) is a local minimum, and since this is an Equivalent Integral to I (Γ), this is also a local minimum in G. ( )kjj qtq ,ψ= is a System of n First-Order Differential Equations, that may be integrated to cover the region G. Let Γ: qj = qj (t) be a member of this family and Let P1 and P2 the points on Γ, corresponding to t1 and t2, It follows that Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
  • 61. 04/08/15 61 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Basic Properties of Geodesic Field Since ( ) ( ) ( ) FieldGeodesictqqq q S t S qqtLqqtL jjl l kkkk ⊂=Γ= ∂ ∂ − ∂ ∂ −= :0,,:,,*  ( ) ( ) ( )kjj jj p q L jj kk j kk qtqwhenever q S p qd S q qqtL q qqtL jj , ,,,, 0 * ψ= ∂ ∂ −= ∂ − ∂ ∂ = ∂ ∂ = = ∂ ∂       Therefore ( ) FieldGeodesictqq q S p jj jj ⊂=Γ ∂ ∂ = : ( ) ( )kj k kjj qtpqtq ,,, ψφ == This defines the Covariant Vector Field pj = pj (t, qk ) as a function of position in G. Assuming the one-to-one correspondence between pj and (j,k=1,2,…,n), we have k q We get ( ) l l q S p q l l kk p t S q q S t S qqtL ll ll φ φ + ∂ ∂ = ∂ ∂ + ∂ ∂ = ∂ ∂ = =     ,, ( ) ( ) ( )       ∂ ∂ =+−= = ∂ ∂ = k k pqtq q S p l l kkkk q S qtHpqqtLqqtH k kkk kk ,,,,,, ,,φ φ   0,, =      ∂ ∂ + ∂ ∂ k k q S qtH t S and Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
  • 62. 04/08/15 62 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory William Rowan Hamilton 1805-1865 Carl Gustav Jacob Jacobi (1804-1851) 0,, =      ∂ ∂ + ∂ ∂ k k q S qtH t S Hamilton-Jacobi Equation Hamilton-Jacobi is a First-Order Nonlinear Partial Differential Equation for the Scalar Function S = S (t,qk ), that defines the Geodesic Field through from which a Unique Vector Field is obtained jj q S p ∂ ∂ = ( ) ( )kj k kjj qtpqtq ,,, ψφ == Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
  • 63. 04/08/15 63 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory William Rowan Hamilton 1805-1865 Carl Gustav Jacob Jacobi (1804-1851) 0,, =      ∂ ∂ + ∂ ∂ k k q S qtH t S Start with jj q S p ∂ ∂ = ( ) l ljjj k j q qq S xt S q qtS td d td pd  ∂∂ ∂ + ∂∂ ∂ =      ∂ ∂ = 22 , Partially Differentiate the Hamilton Jacobi Equation with respect to qj ( ) ( ) 0 ,,,, 222 = ∂∂ ∂ + ∂ ∂ + ∂∂ ∂ = ∂ ∂ ∂ ∂ + ∂ ∂ + ∂∂ ∂ ∂ ∂ = = ∂ ∂ l ljjj q S p x p Hj l l k k j k k j q qq S q H tq S q p p pqtH q pqtH tq S ll l l   we obtain j j q H td pd ∂ ∂ −= Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
  • 64. 04/08/15 64 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory William Rowan Hamilton 1805-1865 Carl Gustav Jacob Jacobi (1804-1851) we obtain        = ∂ ∂ −= = ∂ ∂ = nj q H td pd nj p H td qd j j j j ,,2,1 ,,2,1   ( ) 0= ∂ ∂ += ∂ ∂ −      ∂ ∂ = = ∂ ∂ ∂ ∂ = ∂ ∂ j j td pd q L q H q Ljjj q H td pd q L q L td d LE j j jj   A System of 2n First-Order Ordinary Differential Equations which the curve Γ must satisfy. This is equivalent to Euler-Lagrange n Second-Order Partial Differential Equations, since ( ) k jj k q qqtL p   ∂ ∂ = ,, : Canonical Momentum ( ) ( )[ ] ( )k kj ji ikk k k pqtppqtqtLpqtH ,,,,,,:,, φφ +−= Hamiltonian Hamilton’s Equations Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI 
  • 65. 04/08/15 65 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory William Rowan Hamilton 1805-1865 Carl Gustav Jacob Jacobi (1804-1851) Let return to the condition ( ) ( ) ( ) ( ) 0 ,, ,,,,* ≥ ∂ ∂ − ∂ ∂ −= k k jj jjjj q q qtS t qtS qqtLqqtL  where the equality holds for ( ) ( )kj k kjj qtpqtq ,,, ψφ == We want to eliminate S from this inequality. For this use ( ) ( )[ ] ( )l lk kl ljj p q S j j j pqtppqtqtL q S qtH t qtS jj ,,,,,,:,, , φφ +−=      ∂ ∂ = ∂ ∂ − = ∂ ∂ ( ) ( )[ ] ( ) ( ) 0 , ,,,,,,,, ≥ ∂ ∂ −+− k k j l lk kl ljjjj q q qtS pqtppqtqtLqqtL  φφ ( )jjj j qqtE ,,, φ : Weierstrass Excess Function Weierstrass Sufficiency Condition for a Local Minimum ( ) ( ) ( ) ( ) 0,,,,:,,, ≥− ∂ ∂ +−= kk k jjjjjjj j q q L qtLqqtLqqtE  φφφ and ( ) ( ) k j k jj k q qtS q qqtL p ∂ ∂ = ∂ ∂ = ,,, :   Karl Theodor Wilhelm Weierstrass (1815-1897) Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  Return to Table of Content
  • 66. 04/08/15 66 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtxtxtLCI  Example: Recovering Newton Equation VF −∇= - Force on Mass m, due to External Potential V rrmT  ⋅= 2 1 - Kinetic Energy of Mass m ( )rV  - Potential Energy of the External Force rm r L p    = ∂ ∂ = - Canonical Momentum - HamiltonianEVTVrrmrrmVrrmxpLH =+=+⋅=⋅++⋅−=+−= : 2 1 2 1 :   VrrmVTL −⋅=−=  2 1 : - Lagrangian
  • 67. 04/08/15 67 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtxtxtLCI  Example: Recovering Newton Equation (continue – 1) S r S rm r L p ∇= ∂ ∂ == ∂ ∂ =    - Canonical Momentum - HamiltonianVTVrrmH +=+⋅=  2 1 Hamilton-Jacobi Equation where S is given by VSS m VrrmH +∇⋅∇=+⋅= 2 1 2 1  0 2 1 =+∇⋅∇+ ∂ ∂ VSS mt S Hamilton-Jacobi Equation ( ) 0,, =∇+ ∂ ∂ SrtH t S 
  • 68. 04/08/15 68 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtxtxtLCI  Example: Recovering Newton Equation (continue – 2) 0 2 1 =+∇⋅∇+ ∂ ∂ VSS mt S Hamilton-Jacobi Equation Let take the Gradient of this equation ( ) ( ) Vrr r m p t VSS mt S rmpS ∇+⋅ ∂ ∂ + ∂ ∂ =∇+∇⋅∇∇+ ∂ ∂ ∇= ==∇      0 22 1 0 FVp t p td d  =−∇= ∂ ∂ = Newton Equation ( ) p t rm r rp t p td d      ∂ ∂ = ∂ ∂ ⋅+ ∂ ∂ = 0
  • 69. 04/08/15 69 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtxtxtLCI  Example: Classical Harmonic Oscillator In Equilibrium Displaced from Equilibrium xmxkF =−= - Force of Spring on Mass m 2 2 1 xmT = - Kinetic Energy of Mass m 2 2 1 xkV = - Potential Energy of the Spring xm x L p   = ∂ ∂ = - Canonical Momentum - HamiltonianEVTxkxmxxmxkxmxpLH =+=+=⋅++−=+−= : 2 1 2 1 2 1 2 1 : 2222  22 2 1 2 1 : xkxmVTL −=−=  - Lagrangian 02 2 >= ∂ ∂ m x L 
  • 70. 04/08/15 70 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtxtxtLCI  Example: Classical Harmonic Oscillator (continue – 1) In Equilibrium Displaced from Equilibrium- Hamiltonian( ) 2 2 2 1 2 1 :, xk m p pxH += x m k td pd mtd xd m p p H td xd xk x H td pd −==⇒        = ∂ ∂ = −= ∂ ∂ −= 1 2 2 Initial Conditions: ( ) ( )     == == 00 0 t td xd Atx ( ) ( ) ( ) m k tAmtxmtp tAtx =    −== = : sin cos ω ωω ω  : Solution On the minimizing curve Γ ( ) . 2 1 cos 2 1 sin 1 2 1 2 1 2 1 :, 2222222 2 constAktAkt m k Am m xk m p pxH ==+      =+= ωω Hamilton’s Equations:
  • 71. 04/08/15 71 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Hamilton-Jacobi Theory Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtxtxtLCI  Example: Classical Harmonic Oscillator (continue – 2) In Equilibrium Displaced from Equilibrium ( ) ( ) td Sd tAktAktA m k m m xk m p VTpxL =−=−        =−=−= ωωω 2cos 2 1 cos 2 1 sin 1 2 1 2 1 2 1 , 22 2 2 2 ( ) t t tA k ttA k tA k S ω ω ω ω ωω ω ω ω cos sin cos 2 1 cossin 2 1 2sin 4 1 222 −=−=−= ( ) t t x k xtS ω ω ω cos sin 2 1 , 2 −= ( ) HAk t xk t xtS tAx −=−=−= ∂ ∂ = 2 cos 2 2 2 1 cos 1 2 1, ω ω ( ) ptA k t t x k x xtS tAmp m k tAx ωω ω ω ω ω ωω ω ω sincos sin cos sin, −= = = =−=−= ∂ ∂ ω ω m m k mkm mk kk ==== / Return to Table of Content
  • 72. 04/08/15 72 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  Invariance Properties of the Fundamental Integral Theorem of Noether for Single Integral Amalie Emmy Noether (1882 –1935) Consider the Functional ( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  Symmetry of Lagrangian is a Geometric Property in which quantities remain unchanged under coordinate transformation. Invariance is an Algebraic or Analytic Property in which “Integrals of Motion” are constant along System Trajectories.. Consider also a continuous family of coordinate transformation qj (t) → qj (t,ε) parameterized by a single quantity ε. We assume that the Lagrangian is invariant to this parameter which implies a Symmetry of the System. Thus ( ) ( )( ) ( ) ( )( )0,,0,,,,,, === εεεε tqtqtLtqtqtL jjjj  ( ) ( )( ) nj q q Lq q Lq q Lq q L tqtqtL d d j j j j n j j j j j jj ,,100,,,, 1       == ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ⇒=         ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∑ = αααα εε α
  • 73. 73 SOLO Classical Field Theories Introduction to Lagrangian and Hamiltonian Formulation Extremal of the Functional .( ) ( ) ( )( )∫= 2 1 ,, t t jj tdtqtqtLCI  Invariance Properties of the Fundamental Integral Theorem of Noether for Single Integral (continue – 1) Amalie Emmy Noether (1882 –1935) nj q q Lq q L j j j j ,,10    == ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ αα Since qi(ε) is a solution of Euler-Lagrange’s Equation we have nj q L q L td d jj ,,10   == ∂ ∂ −         ∂ ∂ Substitute this in previous equation we obtain ( ) ( ) ( ) nj tq q L td dtq q Ltq q L td d j j j j j j ,,10 ,,,     ==         ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ + ∂ ∂         ∂ ∂ α α α α α α The quantity is a Constant of Motion. ( ) ( ) α α α α ∂ ∂ = ∂ ∂ ∂ ∂ ,, tq p tq q L j j j j  Noether Theorem is not of great importance in Particle Dynamics. It becomes extremely important in Field Theory and in Quantum Mechanics. nj q L p j j ,,1 = ∂ ∂ = Return to Table of Content
  • 74. 74 SOLO Foundation of Geometrical Optics Fermat’s Principle (1657) in Optics 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). The idea that the light travels in the shortest path was first put forward by Hero of Alexandria in his work “Catoptrics”, cc 100B.C.-150 A.C. Hero showed by a geometrical method that the actual path taken by a ray of light reflected from plane mirror is shorter than any other reflected path that might be drawn between the source and point of observation.
  • 75. 75 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations 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  ++=      +      +=
  • 76. 76 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 1) 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   
  • 77. 77 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 2) 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 ∂ ∂ =     
  • 78. 78 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 3) 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
  • 79. 79 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 4) 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 ∂ ∂ =     
  • 80. 80 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 5) 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 ∇=       The equation is called Eikonal Equation Eikonal (from Greek έίκων = eikon → image) . See “Geometrical Optics” Presentation
  • 81. 81 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 6) Hamilton’s Canonical Equations Define ( ) [ ] ( ) ( ) [ ] ( ) sd zd zyxn zy zzyxn z F p sd yd zyxn zy yzyxn y F p z y ,, 1 ,, : ,, 1 ,, : 2/122 2/122 = ++ = ∂ ∂ = = ++ = ∂ ∂ =       ( )( ) ( )2222222 1 zynzypp zy  +=+++ Adding the square of twose two equations gives ( ) ( ) 2 222 2 22 1       = +− =++ xd sd ppn n zy zy  from which Substituting in ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      += gives ( ) ( )222 2 ,,,, zy zy ppn n ppzyxF +− = ( ) ( ) ( ) 22 22 1,,1,,,,,, zyzyxn xd zd xd yd zyxnzyzyxF  ++=      +      +=
  • 82. 82 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 7) Hamilton’s Canonical Equations (continue – 1) From ( ) [ ] ( ) ( ) [ ] ( ) sd zd zyxn zy zzyxn z F p sd yd zyxn zy yzyxn y F p z y ,, 1 ,, : ,, 1 ,, : 2/122 2/122 = ++ = ∂ ∂ = = ++ = ∂ ∂ =       solve for ( ) ( )222 2 ,,,, zy zy ppn n ppzyxF +− =and ( ) ( )222 222 zy z zy y ppn p z ppn p y +− = +− =   Define the Hamiltonian ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sd xd zyxnppzyxn ppn p ppn p ppn n zpypppzyxFppzyxH zy zy z zy y zy zyzyzy ,,,, ,,,,:,,,, 222 222 2 222 2 222 2 −=+−−= +− + +− + +− −= ++−= 
  • 83. 83 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 8) Hamilton’s Canonical Equations (continue – 2) From We obtain the Hamilton’s Canonical Equations ( ) ( ) ( ) ( ) sd xd zyxnppzyxnppzyxH zyzy ,,,,,,,, 222 −=+−−= ( ) ( )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 +− ∂ ∂ −= ∂ ∂ −= +− ∂ ∂ −= ∂ ∂ −=
  • 84. 84 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 9) Hamilton’s Canonical Equations (continue – 3) From ( ) ( ) ( ) ( ) sd xd zyxnppzyxnppzyxH zyzy ,,,,,,,, 222 −=+−−= ( )222 zy ppn n sd xd +− = By similarity with ( ) sd yd zyxnpy ,,= define ( ) ( ) ( ) ( )222 ,,,,,,,,: zyzyx ppzyxnppzyxH sd xd zyxnp +−=−== Let differentiate px with respect to x ( ) x H xd Hd ppn x n n xd pd zy x ∂ ∂ −=−= +− ∂ ∂ = 222 Let compute ( ) ( ) x n n ppn ppn x n n sd xd xd pd sd pd zy zy xx ∂ ∂ = +− +− ∂ ∂ == 222 222
  • 85. 85 SOLO Foundation of Geometrical Optics Proof of Fermat’s Principle Using Calculus of Variations (continue – 10) Hamilton’s Canonical Equations (continue – 4) and ( ) ( ) x n n ppn ppn x n n sd xd xd pd sd pd zy zy xx ∂ ∂ = +− +− ∂ ∂ == 222 222 ( ) ( ) y n n ppn ppn y n n sd xd xd pd sd pd zy zy yy ∂ ∂ = +− +− ∂ ∂ == 222 222 ( ) ( ) z n n ppn ppn z n n sd xd xd pd sd pd zy zy zz ∂ ∂ = +− +− ∂ ∂ == 222 222 n sd pd ∇=  xp nsd xd 1 = ( ) ( ) y zy zy y p nn ppn ppn p sd xd xd yd sd yd 1 222 222 = +− +− == ( ) ( ) z zy zy z p nn ppn ppn p sd xd xd zd sd zd 1 222 222 = +− +− == p nsd rd ray   1 = We recover the result from Geometrical Optics Return to Table of Content
  • 86. 04/08/15 86 SOLO Classical Field Theories The Inverse Square Law of Forces The Two Body Central Force Problem Assume a system of two mass points m1 and m2 located at , respectively, subject to an interaction potential V, where V is any function of the range vector between particles. 21 randr  Define: ( ) ( )212211 12 / : mmrmrmR rrr c ++= −=   - range vector between particles - system center of mass 21 1 2 21 2 1 , mm rm Rr mm rm Rr cc + += + −=   We obtain: The kinetic energy of the system:         + +⋅        + ++        + −⋅        + −=⋅+⋅= 21 1 21 1 2 21 2 21 2 1222111 2 1 2 1 2 1 2 1 mm rm R mm rm Rm mm rm R mm rm RmrrmrrmT cccc       ( ) rr mm mm RRmmT M cc    ⋅ + +⋅+= 21 21 21 2 1 2 1or ( ) ( ) ,, 2 1 2 1 21 rrVrrMRRmmVTL cc −⋅+⋅+=−= The Lagrangian of the system is
  • 87. 04/08/15 87 SOLO Classical Field Theories The Inverse Square Law of Forces ( ) ( ) ( ) ( )2121 222 /: 2 1 2 1 mmmmM rVrrMrVrrMVTL += −+=−⋅=−= θ When V is a function of r only like in classical gravitation and electromagnetic fields, we can write, in polar coordinates and ignoring the term describing the motion of center of mass The Lagrange’s Equations are ,2,10 =∀= ∂ ∂ −       ∂ ∂ ∂ ∂ i q L t q L td d ii 2. r - independent variable r V rM r L rM r L ∂ ∂ −= ∂ ∂ = ∂ ∂ 2 , θ  1. θ - independent variable 0,2 = ∂ ∂ = ∂ ∂ θ θ θ L rM L   ( ) ( ) 02 == ∂ ∂ − ∂ ∂ θ θθ   rM td dLL td d ( ) 02 = ∂ ∂ +−= ∂ ∂ − ∂ ∂ r V rMrM r L r L td d θ 
  • 88. 04/08/15 88 SOLO Classical Field Theories The Inverse Square Law of Forces The Specific Angular Momentum and Angular Momentum are defined as θθ  22 :&: rMhMlrh === ( ) ( ) 02 == ∂ ∂ − ∂ ∂ θ θθ   rM td dLL td d The Equation means that Angular Momentum is constant (Conservation of Angular Momentum ) This Equation can be rewritten as θdrMdtl 2 = that implies       =⇒= θθθ d d rM l d d rM l dt d d d rM l dt d 222 2 2 a b The Area swept out by one of the body when moving around the other (see Figure) is given by ( )θdrrAd 2 1 = It follows that . 22 const h d Ad rM l td Ad === θ This is Kepler Second Law.
  • 89. 04/08/15 89 SOLO Classical Field Theories The Inverse Square Law of Forces 02 = ∂ ∂ +− r V rMrM θReturn to the equation θ2 : rMl =and We have ( ) ( ) 0 22 2 2 2 2 3 2 =      ++=      + ∂ ∂ += ∂ ∂ +− rV rM l rd d rMrV rM l r rM r V rM l rM  or ( ) ( ) ( ) ( ) 2 2 2 2 2 : 2 r l rVrV rV rd d rV rM l rd d rM eff eff µ += −=      +−= Multiplying the differential equation by we obtainr ( ) ( )rV td d rV rd d td rd rM td d rrM effeff −=−=      = 2 2 1  or   0 2 1 2 ==               + energy total energy potential eff energy kinetic E td d VrM td d   The total Energy is conservedeffVrME += 2 2 1 : 
  • 90. 04/08/15 90 SOLO Classical Field Theories The Inverse Square Law of Forces The total Energy is conservedeffVrME += 2 2 1 :  Solving for we obtainr ( )       −−±=−±= 2 2 2 22 rM l VE M VE M r eff  or       −− = 2 2 2 2 rM l VE M rd td Using we obtain θd d rM l dt d 2 = 2 222 122 1 r rd rl VM l EM d −− =θ Consider the case where V (r) = -k/r = - k u ud uu l kM l EM d 2 22 22 1 −− − =θ Integrating this expression gives 1 2 1 1 sin 28 2 2 sin 2 2 2 1 2 22 2 1 +       − =       + +− =− −− kM lE rkM l l kM l EM l kM u iθθ
  • 91. 04/08/15 91 SOLO Classical Field Theories The Inverse Square Law of Forces 1 2 1 1 sin 28 2 2 sin 2 2 2 1 2 22 2 1 +       − =       + +− =− −− kM lE rkM l l kM l EM l kM u iθθ Inverting this expression we obtain ( )         −+−= i kM lE l kM r θθsin 2 11 1 2 2 2 This is usually written as ( ){ } 2 2 2 2 1: sin1 1 kM lE e e l kM r i += −−= θθ This is a equation of Conic Section: 1.If e > 1 and E > 0, the trajectory is a Hyperbola. 2. If e = 1 and E = 0, the trajectory is a Parabola. 3. If e < 1 and E < 0, the trajectory is a Ellipse. 4. If e = 0 and E = -mk2 /(2l2 ), the trajectory is a Circle. Return to Table of Content
  • 92. 04/08/15 92 SOLO Four-Dimensional Formulation of the Theory of Relativity Special Relativity Theory We introduce a 4-dimensional space-time or four-vector x with components: ( ) ( ) ctxxxxxxxx == 003210 ,,,,: µ The differential length element is defined as: ( ) ( ) ( ) ( ) ( ) ( ) 220232221202 : xdxdxdxdxdxds  −=−−−= or ( ) νµ µν µ ν νµ µν dxdxgdxdxgds summationsEimstein convention '3 0 3 0 2 == ∑∑= = The metric corresponding to this differential length is given by g: ( )µνgg =               − − − = 1000 0100 0010 0001 therefore: ( )νµµν ≠=−==== 0,1,1 33221100 ggggg We can see that Igg =               =               − − −               − − − = 1000 0100 0010 0001 1000 0100 0010 0001 1000 0100 0010 0001
  • 93. 04/08/15 93 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 1) Therefore since we have γ µ νγ µν δ=gg µν µν gg = If a 4-vector has the contravariant components A0 ,A1 ,A2 ,A3 we have ( ) ( ) ( )32103210 ,,,,,, AAAAwhereAAAAAAA === α Using the g metric we get the same 4-vector described by the covariant components: ( )AA A A A A A A A A AgA  −=               − − − =                             − − − == , 1000 0100 0010 0001 0 3 2 1 0 3 2 1 0 β αβα The Scalar Product of two 4-vectors is: ( )( ) ( )( ) αα αβαβ αβα αα α ABgABgABABAABBABAABBAB ==⋅−=−==−=  000000 ,,,, Special Relativity Theory
  • 94. 04/08/15 94 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 1) Therefore since we have γ µ νγ µν δ=gg µν µν gg = If a 4-vector has the contravariant components A0 ,A1 ,A2 ,A3 we have ( ) ( ) ( )32103210 ,,,,,, AAAAwhereAAAAAAA === α Using the g metric we get the same 4-vector described by the covariant components: ( )AA A A A A A A A A AgA  −=               − − − =                             − − − == , 1000 0100 0010 0001 0 3 2 1 0 3 2 1 0 β αβα The Scalar Product of two 4-vectors is: ( )( ) ( )( ) αα αβαβ αβα αα α ABgABgABABAABBABAABBAB ==⋅−=−==−=  000000 ,,,, Special Relativity Theory
  • 95. 95 SOLO Special Relativity Theory Four-Dimensional Formulation of the Theory of Relativity (continue – 2) Let introduce the following definitions:       ∇ ∂ ∂ =      ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ =∂       −∇ ∂ ∂ =      ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ = ∂ ∂ =∂ ,,,,: ,,,,: 03210 03210 xxxxxx xxxxxx αα α α The 4-divergence of a 4-vector A is the invariant: ( ) ( ) A x A AA x AAA x A  ⋅∇+ ∂ ∂ =      ∇ ∂ ∂ =∂=−      −∇ ∂ ∂ =∂ 0 0 0 0 0 0 ,,,, α αα α The four-dimensional Laplacian operator (d’Alembertian) is defined as:         ∇− ∂ ∂ =      ∇ ∂ ∂       −∇ ∂ ∂ =∂∂=      −∇ ∂ ∂       ∇ ∂ ∂ =∂∂= 2 20 2 0000 ,,,,: xxxxx α αα α
  • 96. 04/08/15 96 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 3) We have: ( ) ( ) ( ) ( ) ( )         −=−=−−−= 2 2 22222232221202 1 1 dt xd c dtcxddtcdxdxdxdxds   We define - the velocity vector of the particle in the inertial frame. td xd u   = 2 1 1 :&: u uu c u β γβ − == ( ) ( ) 22222222 2 2 222 /11 τγβ dcdtcdtc c u dtcds uu ==−=      −= where ( ) ( )t dt dtd u u γ βτ =−= 2 1 is the differential of the proper time τ. Special Relativity Theory
  • 97. 04/08/15 97 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 4) 4-Coordinates Vector The 4-coordinates vector is: ( ) ( ) ( ) ( )xxxxxxxxxxxxxx  −=−−−=== ,,,,&,,,, 0321003210 α α ( )( ) ( ) ( ) 23210000 22222 ,, sdxdxdxdxdxdxdxdxdxdxdxdxdxd =−−−=⋅−=−=  α α Special Relativity Theory
  • 98. 04/08/15 98 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 5) 4-Velocity Vector ( ) ( ) td xd uUUuc d dt dt xd d dt dt dx d xd d dx U uu    ===      =      = ,,,,: 0 00 γγ ττττ α ( ) ( ) 20 0 u 1 1 :,,,:       − =−=−=      −= c UUuc d xd d dx U uuu γγγ ττ α   The 4-Velocity vector is defined as: ( ) ( ) ( ) 2 2 2 2 2 2 222 1 1 1 ,, c c u c c u ucucucUU uuuuu =      −       − =−=−= ∆ γγγγγα α  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )         −=−        = =−=        −+−        = =−+−    === ττττ ττττττ ττττ α α d Ud d dU UUUU d Ud d dU d Ud U d dU U d Ud d dU UUUU d Ud d dU UU d d UUUUUU d d UU d d d dc          ,,2,,2 22,,,, ,,,,0 0 00 0 0 0 0 00 0 0000 2 ( ) ( ) ( ) ( ) 0==== α αα αα αα α ττττ UU d d U d d UU d d UUU d d Special Relativity Theory
  • 99. 04/08/15 99 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 5) 4-Moment Vector The 4-Momentum vector of the mass m is defined as: ( )UUmUmp  ,000 == αα 2 02 2 0 000 0 1 1 :       − == ==       − === c u m mofEnergymcE c E mc c u cm cmUmp uγ pumu c u m umUm u  ==       − == 2 0 00 1 γ Therefore: ( ) ( ) ( ) ( )UUmUmpppUUmUmppp  −==−==== ,,&,, 000 0 000 0 αα αα 22 0 2 0 2 2 2 ,, cmUUmp c E p c E p c E pp ==      −=      −      = α α α α  from which we get: ( ) ( )22 0 22222 cmcpcmE +== Special Relativity Theory
  • 100. 100 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue – 6) 4-Force Vector The 4-Force vector on the mass m is defined as:       =      == ττττ µµ d pd d Ed c p c E d d p d d F   , 1 ,: From the relations: ( ) u u u d dtdt dtd γ τγ βτ =→=−= 2 1 ( ) ( ) c uF d td td rd c F d Ed c EdcmEdrdFdT u   ⋅ =⋅=→=−=⋅= γ ττ 12 0 F td pd d td d pd u  γ ττ == Special Relativity Theory 2 u 1 1 :       − = c uγ ( ) ( )       −⋅=        − ⋅ ==       ⋅=        ⋅ == F c u FF c uF d pd F F c u FF c uF d pd F uu u uu u   ,,: ,,: γγ γ τ γγ γ τ µ µ µ µ
  • 101. 101 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue –7) 4-Force Vector (continue – 1) Assuming that the rest mass doesn’t change :00 = τd dm ( ) τττ µ µµµ d dU mUm d d p d d F 00 === Using the relation: ( ) ( ) ( ) ( ) 0==== α αα αα αα α ττττ UU d d U d d UU d d UUU d d we get: 00 == τ µ µ µ µ d dU UmFU Special Relativity Theory
  • 102. 102 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue –8) 4-Force Vector (continue – 2) Assuming that the rest mass changes ,:00 ≠ τd dm ( ) µµµ µ µµ ττττ Π+=+== FU d md d Ud mUm d d p d d 0 00 Using the relation: ( ) ( ) ( ) ( ) 0==== α αα αα αα α ττττ UU d d U d d UU d d UUU d d where: 00 == τ µ µ µ µ d dU UmFU Special Relativity Theory µµ µ µ ττ U d md d Ud mF 0 0 :,: =Π= ( ) ( )         − ⋅ ==         ⋅ == = = F c uF d pd F F c uF d pd F u u constm u u constm   γ γ τ γ γ τ µ µ µ µ ,: ,: . . 0 0
  • 103. 103 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue –9) 4-Force Vector (continue – 3) Assuming that the rest mass changes ,:00 ≠ τd dm Special Relativity Theory ( ) ( ) dt xd uUUuc d dt dt xd d dt dt dx d xd d dx U uu    ===      =      = ,,,,: 0 00 γγ ττττ α ( ) ( ) ( )2 0 0 /1/1:,,,: cuUUuc d xd d dx U uuu −=−=−=      −= γγγ ττ α   The 4-Velocity vector is defined as: Let define:       Π− Φ =Π      Π Φ =Π uuuu cc γγγγ α α  ,:,,: ( ) ( ) 02 :,, Φ=Π⋅−Φ=      Π− Φ =Π  u c ucU uuuuu γγγγγα α µµµ µ µ ττ Π+=+ FU d md d Ud mU 0 0  02 0 0 0 0 Φ Π+=+ µ µ µ µ µ µ µ µ ττ UFUUU d md d Ud Um c τ α α d md cU 020 =Π=Φ The 4-Velocity Momentum vector that the particle loses per unit proper time τ.
  • 104. 104 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue –10) 4-Force Vector (continue – 4) Assuming that the rest mass changes ,:00 ≠ τd dm Special Relativity Theory       µ µ µµµ µ ττ D R U d md F d Ud m 0 0 −Π+= µ µ τ D d Ud m =0 µµ τ µµµ µµµ τ U c U d md R RFD c d md 2 0 0 020 : : Φ −Π=−Π= += Φ= Fμ – External Force Rμ – Reaction Force Dμ – Driving Force 0 0 00 2 0 20 =+= = Φ −Π= Φ   µ µ µ µ µ µ µ µ µ µ µ µ RUFUDU UU c URU c       Π+Φ+⋅=Π+=      =  F c uF c Fp c E d d p d d u , 11 , 0 γ ττ µµµ ( ) u u u d dtdt dtd γ τγ βτ =→=−= 2 1 Also       Π+Φ+⋅=      =  F c uF c p c E td d p td d , 11 , 0µ
  • 105. 105 SOLO Four-Dimensional Formulation of the Theory of Relativity (continue –11) 4-Force Vector (continue – 5) Assuming that the rest mass changes ,:00 ≠ τd dm Special Relativity Theory ( ) ( ) ( )PcQdtcdt dt cdQ u uu dtd u  δδ γ γγτδ γτ µµ ,/,/,/ 00 / =ΠΦ=ΠΦ=Π= = In special cases the rate of change of Πμ may be due to emission or absorption of Heat by the particle where     Π= Φ= tdP tdQ  δ δ 0
  • 106. 04/08/15 106 SOLO Four-Dimensional Formulation of the Theory of Relativity Special Relativity Theory Principle of Last Action for 4-Vector for a Free Particle (zero external forces) We want to define the Action Integral for 4-Vectors, similar to Fermat Principle in Optics, to recover the 4-Momentum and 4-Force Vectors. Define ∫∫ =−= 2 1 : t t b a tdLsdαA α is a constant, to be defined ( )22 /1 cudtcsd −=Using we obtain ∫∫ =−−= 2 1 2 1 22 /1: t t t t tdLtdcucαA Therefore the Lagrangian is c u ccucL 2 /1 2 22 α αα +−≈−−= Since the constant α c in the Lagrangian doesn’t affect the equations of motion, we concentrate on the second term that must be equal to Energy. We choose α to obtain the non-relativistic kinetic energy m0u2 /2 (m0 – rest mass), i.e.: cm0=α 222 0 /1 cucmL −−= ∫∫ −−=−−= 2 1 2 1 0 222 0 /1: s s t t sdcmtdcucmA ( ) ( ) uzyxqzyxq   === ,,&,,
  • 107. 04/08/15 107 SOLO Four-Dimensional Formulation of the Theory of Relativity Special Relativity Theory Canonical Momentum 222 0 /1 cucmL −−= ( )( ) um cu um czyxcm z y x q L p         = − =++−−           ∂∂ ∂∂ ∂∂ = ∂ ∂ = 22 022222 0 /1 /1 / / / ( )( ) 0/1 / / / 22222 0 =++−−           ∂∂ ∂∂ ∂∂ = ∂ ∂ czyxcm z y x q L  Particle Equation of Motion Q td pd cu um td d q L q L td d   ==        − = ∂ ∂ −      ∂ ∂ 22 0 /1 - are the External Forces acting on the particle (zero for free particle)Q  Principle of Last Action for 4-Vector for a Free Particle 22 0 /1 : cu m m − = - relativistic mass of Particle at velocity u Free Particle Lagrangian
  • 108. 04/08/15 108 SOLO Four-Dimensional Formulation of the Theory of Relativity Special Relativity Theory 222 0 /1 cucmL −−= Principle of Last Action for 4-Vector for a Free Particle The Free Particle Hamiltonian is defined as: LpH −⋅= u:  ( )( ) um cu um czyxcm z y x q L p         = − =++−−           ∂∂ ∂∂ ∂∂ = ∂ ∂ = 22 022222 0 /1 /1 / / / where: therefore: 2 22 2 0222 022 0 /1 /1u /1 u: cm cu cm cucm cu um LpH = − =−+⋅ − =−⋅=    We van see that the Hamiltonian H derived from the Lagrangian L is equal to the Total Energy E of the Free Particle Ecm cu cm LpH == − =−⋅= 2 22 2 0 /1 u:  Free Particle Lagrangian
  • 109. 04/08/15 109 SOLO Four-Dimensional Formulation of the Theory of Relativity Special Relativity Theory α αUUcucmcucmL 22 0 222 0 /1/1 −−=−−= Principle of Last Action for 4-Vector for a Free Particle (Covariant Treatment) Let use: UαUα = c2 to write ∫∫∫ −=−=−−= 2 1 2 1 2 1 00 222 0 /1: τ τ α α τ τ ττ dUUcmdccmtdcucm t t A We defined ( ) u u u d dtdt dtd γ τγ βτ =→=−= 2 1 2 u 1 1 :       − = c uγ therefore the Action Integral is To use this for a variational calculation we must add the constraint 2 cUU =α α 0= τ α α d Ud Uor equivalently This can be added using Lagrange’s multipliers technique, but here we use a different approach. ττ ττ τ βα αβα α α αα α β αβα dcxdxdgxdxdd d xd d xd dUU xdgxd ==== =
  • 110. 04/08/15 110 SOLO Four-Dimensional Formulation of the Theory of Relativity Special Relativity Theory Principle of Last Action for 4-Vector for a Free Particle (Covariant Treatment) ∫∫∫ −=−=−−= 2 1 2 1 2 1 00 222 0 /1: τ τ α α τ τ ττ dUUcmdccmtdcucm t t A Action Integral is where s = s (τ) is any monotonically increasing function of τ. ττ βααβ βα αβα α dcsd sd xd sd xd gxdxdgdUU === ∫−= 2 1 0: s s sd sd xd sd xd gcm βααβ A sd xd sd xd gcmL βααβ 0: −= ( ) ( )γδ αβ gg =               − − − = 1000 0100 0010 0001 sd xd sd xd sd xd cm sd xd sd xd g sd xd g sd xd g cm sd xd L x L jj β β α βααβ αββα αα 2 2 2 ,0 00 −= + −=       ∂ ∂ = ∂ ∂ Euler-Lagrange Equation             −=               −= ∂ ∂ −                   ∂ ∂ = sd d c sd xd d d sd d cm sd xd sd xd sd xd sd d cm x L sd xd L sd d sd d ττ τ α β β α αα  000 02 2 0 = τ α d xd m 4-Vector Free Particle Equation of MotionReturn to Table of Content
  • 111. 04/08/15 111 SOLO Classic Electrodynamics Start with Microscopic Maxwell’s Equations in Gaussian Coordinates: eE ρπ4=⋅∇  Gauss’ Law (Electric) j ct E c B    π41 = ∂ ∂ −×∇ Ampère’ Law (with Maxwell’s Extension) 0 1    = ∂ ∂ +×∇ t B c EFaraday’ Law of Induction Gauss’ Law (Magnetic) 0=⋅∇ B  Lorentz Force Equation: Electric Field Intensity [statV . cm-1 ] E  Magnetic Induction [statV . sec . cm-2 = gauss] B  ρe Charge density [statC . cm-3 ] j  Current Density [statA . cm-3 ] eF  Electromagnetic force [dynes] u  Charge velocity [cm . sec] 1 V = 1/3x10-2 statV 1 C = 3x109 statC 1 A= 3x109 statA       ×+= B c u EqFe  Classical Electromagnetic Theory q electric charge [statC]
  • 113. 04/08/15 113 SOLO 0 111      =        ∂ ∂ +×∇=×∇ ∂ ∂ +×∇= ∂ ∂ +×∇ t A c EA tc E t B c E Therefore we can define a Potential φ, such that We can define a vector such that0=⋅∇ B  0≡×∇⋅∇⇐×∇= AAB  A  0 1 ≡∇×∇⇐−∇= ∂ ∂ + ϕϕ t A c E   and φ are not uniquely defined since will give the same results for . Such a transformation that doesn’t change the results is called a “Gauge Transformation” on the Potentials. ( )BE  , A  t f c andfgradAA ∂ ∂ −=+= 1 11 ϕϕ  ϕ∇− ∂ ∂ −= ×∇= t A c E AB    1 We obtained Classical Electromagnetic Theory Classic Electrodynamics Electrodynamic Potentials and φA 
  • 114. 04/08/15 114 SOLO Classical Electromagnetic Theory j ct E c B    π41 = ∂ ∂ −×∇From we obtain ϕ∇− ∂ ∂ −= ×∇= t A c E AB    1 ( ) j c A t A ctc A tct A c A      πϕϕ 41111 2 2 2 22 2 2 =        ∇− ∂ ∂ +      ∂ ∂ +⋅∇∇= ∂ ∂ ∇− ∂ ∂ +×∇×∇ From we obtaineE ρπ4=⋅∇  eA tc ρπϕ 4 1 2 =∇−⋅∇ ∂ ∂ −  Since and φ are not uniquely defined, let add the following relation: A  0 1 = ∂ ∂ +⋅∇ tc A ϕ We obtain Lorenz Condition e tc j c A t A c ρπϕ ϕ π 4 1 41 2 2 2 2 2 2 2 2 =∇− ∂ ∂ =∇− ∂ ∂   Waveform Equations for and φ.A  Ludwig Valentin Lorenz 1829-1891 Classic Electrodynamics
  • 115. 04/08/15 115 SOLO 0 1 = ∂ ∂ +⋅∇ tc A ϕ Lorenz Condition e tc j c A t A c ρπϕ ϕ π 4 1 41 2 2 2 2 2 2 2 2 =∇− ∂ ∂ =∇− ∂ ∂   Waveform Equations for and φ.A  If we define the new Potentials through the relations t f c andfAA ∂ ∂ −=∇+= 1 11 ϕϕ  To satisfy the Lorenz Condition 0 111 2 2 2 2 0 1 1 = ∂ ∂ −∇+ ∂ ∂ +⋅∇= ∂ ∂ +⋅∇ t f c f tc A tc A   ϕϕ 0 1 2 2 2 2 = ∂ ∂ −∇ t f c fTherefore f must satisfy Classical Electromagnetic Theory Classic Electrodynamics
  • 116. 116 SOLO Energy and Momentum ( ) EJB c u Eup e Buu Ju e ee    ⋅=      ×+⋅= =×⋅ = 0 ρ ρ or ( ) ( ) ( ) ( ) ( )[ ] ( )BE c t B BE t E E t E BEEB c E t E c B c EJp t B c E BEBEEB J ct E c B e e                 ×⋅∇−        ∂ ∂ ⋅+⋅ ∂ ∂ −= ⋅ ∂ ∂ −×⋅∇−×∇⋅= ⋅        ∂ ∂ −×∇=⋅= ∂ ∂ −=∇× ×⋅∇=∇×⋅−∇×⋅ + ∂ ∂ =∇× ππ ππ π π 44 1 4 1 4 1 4 1 41 ( ) ( ) ( ) ( )          =⋅∇ =⋅∇ + ∂ ∂ =×∇ ∂ ∂ −=×∇ 0 4 41 1 BGM EGE J ct E c BA t B c EF e e        ρπ π Classical Electromagnetic Theory ( )BE cBBEE t EJp e    ×⋅∇−        ⋅ + ⋅ ∂ ∂ −=⋅= ππ 4224 1 The power density of the Lorentz Force the charge ρe and velocity isu 
  • 117. 117 SOLO Energy and Momentum (continue -1) We identify the following quantities EJe  ⋅       ⋅ ∂ ∂ =⋅= BB t pBBw mm  ππ 8 1 , 8 1       ⋅ ∂ ∂ =⋅= EE t pEEw ee  ππ 8 1 , 8 1 ( )BE c pR  ×⋅∇= π4 - Magnetic energy and power densities, respectively (Energy transferred from the field to the particles) - Electric energy and power densities, respectively - (Energy transferred from the field to the particles) - Radiation power density (Power lost through the boundaries ( )EE tt E E    ⋅ ∂ ∂ = ∂ ∂ ⋅ 2 1 ( )BB tt B B    ⋅ ∂ ∂ = ∂ ∂ ⋅ 2 1 -Power density of the current density eJ  Classical Electromagnetic Theory ( )BE cBBEE t EJp e    ×⋅∇−        ⋅ + ⋅ ∂ ∂ −=⋅= ππ 4224 1 Note: The minus sign means transfer of power from (loss) the electromagnetic field. ( )BE c S  ×= π4 : - Poynting power flux vector
  • 118. 118 SOLO Energy and Momentum (continue – 2) Let integrate this equation over a constant volume V         = ∂ ∂ ∫∫ VV td d t Classical Electromagnetic Theory ( )BE cBBEE t EJp e    ×⋅∇−        ⋅ + ⋅ ∂ ∂ −=⋅= πππ 488               PowerRadiationEM V PowerMagnetic V PowerEletrical V Powet V e vd c BE cvd BB t vd EE t vdEJ ∫∫∫∫         × ⋅∇− ⋅ ∂ ∂ − ⋅ ∂ ∂ −=⋅ πππ 488 2
  • 119. 04/08/15 119 SOLO Electromagnetic Stress Lorentz Force Density ( ) ( ) ( ) ( )          =⋅∇ =⋅∇ + ∂ ∂ =×∇ ∂ ∂ −=×∇ 0 4 41 1 BGM EGE J ct E c BA t B c EF e e        ρπ π Electric Field Intensity [statV . cm-1 ]E  Magnetic Induction [statV . sec . cm-2 = gauss]B  ρe Charge density [statC . cm-3 ] eJ  Current Density [statA . cm-3 ] Maxwell Electromagnetic Stress Start with Maxwell Equations in Gaussian Coordinates uJ BJ c Ef ee eee   ρ ρ = ×+= 1 ef  Electromagnetic force density [dynes . cm-3 ] u  Charge velocity [cm . sec] ( ) ( ) ( ) ( ) ( ) ( ) B t E c BBBB t B E c EEEE B t E c BBEEfe              × ∂ ∂ −         ⋅∇+××∇+                     ∂ ∂ ×−×∇×−+⋅∇= × ∂ ∂ −××∇+⋅∇= 1 4 1 4 11 4 1 1 4 1 4 1 4 1 0 0 πππ πππ ( ) ( )[ ] ( ) ( )         ∂ ∂ ×+× ∂ ∂ −         ⋅∇−×∇×−⋅∇−×∇×−= t B EB t E c BBBBEEEEfe      1 4 1 4 1 4 1 0 πππ 1 V = 1/3x10-2 statV 1 C = 3x109 statC 1 A= 3x109 statA uJ BJ c Ef ee eee   ρ ρ = ×+= 1
  • 120. 04/08/15 120 SOLO Electromagnetic Stress Maxwell Electromagnetic Stress ( ) ( )[ ] ( ) ( )[ ]         ∂ ∂ ×+× ∂ ∂ −×∇×−⋅∇+×∇×−⋅∇= t B EB t E c BBBBEEEEfe     1 4 1 4 1 4 1 πππ Use to get( ) ( ) ( )BABABA B  ∇⋅−⋅∇=×∇× ( ) ( ) ( )EEEEEE E  ∇⋅−⋅∇=×∇× ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )EEIEEEEEEEEEEEE EE  ⋅∇⋅−⋅∇=⋅∇−∇⋅+⋅∇=×∇×−⋅∇ and ( ) ( )[ ] ( ) ( )[ ] ( ) ( )    ⋅−⋅∇=⋅∇−∇⋅+⋅∇=×∇×−⋅∇ EEIEEEEEEEEEEEE  2 1 2 1 therefore where the unit dyadic.           = 100 010 001 I  ( ) ( )[ ] ( ) ( )[ ] ( ) ( )      ⋅−⋅∇=⋅∇−∇⋅+⋅∇=×∇×−⋅∇ BBIBBBBBBBBBBBB  2 1 2 1 On the same
  • 121. 04/08/15 121 SOLO Electromagnetic Stress Maxwell Electromagnetic Stress ( ) ( )         × ∂ ∂ −      ⋅−⋅∇+      ⋅−⋅∇= c BE t BBIBBEEIEEfe πππ 42 1 4 1 2 1 4 1   Define ( )    ⋅−= EEIEETe  2 1 4 1 : π ( )    ⋅−= BBIBBTm  2 1 4 1 : π ( )      ⋅+⋅−+=+= BBEEIBBEETTT me  2 1 4 1 : π [ ]2− ⋅cmdynesTm  . - Magnetic Stress Tensor [ ]2− ⋅cmdynesTe  . - Electric Stress Tensor [ ]2− ⋅+≡ cmdynesTTT me  . - Electromagnetic Stress Tensor c BE G π4 :   × = [ ]213 secsec −−− ⋅⋅=⋅⋅ cmergcmdynesG  - Momentum Density Vector t S Tfe ∂ ∂ −⋅∇=   Return to Relativistic EM Stress

Editor's Notes

  1. J. V. José, E. J. Saletan, “Classical Dynamics – A Contemporary Approach”, Cambridge University Press, 1998, pp. 67-68
  2. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  3. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  4. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  5. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  6. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  7. J.V. José, E.J. Saletan, “Classical Dynamics – A Contemporary Approach”, Cambridge University, 1998, pg. 68
  8. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  9. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  10. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  11. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  12. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  13. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  14. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  15. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  16. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  17. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  18. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  19. J.V. José, E. J. Saletan, “Classical Dynamics – A Contemporary Approach”, Cambridge University, 1998, pp. 124-128
  20. Hero’s proof is described in “Optics”, M.V. Klein, T.E. Furtak. Pp. 3-5
  21. A.C. Tribble, “Princeton Guide to Advanced Physics”, Princeton University Press, 1996, pp. 37-42
  22. A.C. Tribble, “Princeton Guide to Advanced Physics”, Princeton University Press, 1996, pp. 37-42
  23. A.C. Tribble, “Princeton Guide to Advanced Physics”, Princeton University Press, 1996, pp. 37-42
  24. A.C. Tribble, “Princeton Guide to Advanced Physics”, Princeton University Press, 1996, pp. 37-42
  25. A.C. Tribble, “Princeton Guide to Advanced Physics”, Princeton University Press, 1996, pp. 37-42
  26. A.C. Tribble, “Princeton Guide to Advanced Physics”, Princeton University Press, 1996, pp. 37-42
  27. C. Møller, “The Theory of Relativity”, 2nd Ed., Clarendon Press, Oxford, 1972, pp. 102-107
  28. C. Møller, “The Theory of Relativity”, 2nd Ed., Clarendon Press, Oxford, 1972, pp. 102-107
  29. C. Møller, “The Theory of Relativity”, 2nd Ed., Clarendon Press, Oxford, 1972, pp. 102-107
  30. C. Møller, “The Theory of Relativity”, 2nd Ed., Clarendon Press, Oxford, 1972, pp. 102-107
  31. L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999
  32. L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999
  33. L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999
  34. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 583-585
  35. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 583-585
  36. http://en.wikipedia.org/wiki/Maxwell&amp;apos;s_equations L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999
  37. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  38. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558 http://en.wikipedia.org/wiki/Maxwell&amp;apos;s_equations
  39. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  40. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  41. J. Schwinger, L.L. DeRaad, Jr., K.A. Milton, Wu-yang Tsai, “Classical Electrodynamics”, Perseus Books, 1998, pp. 64-65
  42. J. Schwinger, L.L. DeRaad, Jr., K.A. Milton, Wu-yang Tsai, “Classical Electrodynamics”, Perseus Books, 1998, pp. 64-65
  43. J. Schwinger, L.L. DeRaad, Jr., K.A. Milton, Wu-yang Tsai, “Classical Electrodynamics”, Perseus Books, 1998, pp. 64-65
  44. http://en.wikipedia.org/wiki/Maxwell&amp;apos;s_equations L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999
  45. J. Schwinger, L.L. DeRaad, Jr., K.A. Milton, Wu-yang Tsai, “Classical Electrodynamics”, Perseus Books, 1998
  46. J. Schwinger, L.L. DeRaad, Jr., K.A. Milton, Wu-yang Tsai, “Classical Electrodynamics”, Perseus Books, 1998
  47. J. Schwinger, L.L. DeRaad, Jr., K.A. Milton, Wu-yang Tsai, “Classical Electrodynamics”, Perseus Books, 1998
  48. C. Möller, “The Theory of Relativity”, Clarendon Press Oxford, 2nd Ed., 1972, pp. 162-167 L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975, pp. 69 W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968 A.O. Barut, “Electrodynamics and Classical Theory of Fields and Particles”, Dover, 1964, pp.93-103 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 605-608
  49. L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975, pg. 69
  50. L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975, pg. 69
  51. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  52. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  53. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  54. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  55. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  56. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  57. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  58. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  59. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  60. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 543-548, 558-561
  61. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 543-548, 558-561
  62. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 543-548, 558-561
  63. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 543-548, 558-561
  64. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 543-548, 558-561
  65. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 543-548, 558-561
  66. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  67. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  68. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  69. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  70. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  71. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 553-558
  72. C. Möller, “The Theory of Relativity”, Clarendon Press Oxford, 2nd Ed., 1972, pp. 165-167
  73. C. Möller, “The Theory of Relativity”, Clarendon Press Oxford, 2nd Ed., 1972, pp. 165-167
  74. http://en.wikipedia.org/wiki/Maxwell&amp;apos;s_equations L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999
  75. http://en.wikipedia.org/wiki/Maxwell&amp;apos;s_equations L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999
  76. http://en.wikipedia.org/wiki/Maxwell&amp;apos;s_equations L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999
  77. http://en.wikipedia.org/wiki/Maxwell&amp;apos;s_equations L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999
  78. L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975, pg. 46 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999
  79. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 583-585
  80. J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 583-585
  81. W.K.H. Panofsky, M. Phillips, “Classical Electricity and Magnetism”, Addison Wesley, 2nd Ed., pp. 427 - 430 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 583-585
  82. W.K.H. Panofsky, M. Phillips, “Classical Electricity and Magnetism”, Addison Wesley, 2nd Ed., pp. 427 - 430 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 583-585
  83. L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975, pp. 60-62
  84. L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975, pp. 60-62
  85. L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975, pp. 60-62
  86. C. Möller, “The Theory of Relativity”, Clarendon Press Oxford, 2nd Ed., 1972, pp. 162-167 L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975, pp. 69 W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968 A.O. Barut, “Electrodynamics and Classical Theory of Fields and Particles”, Dover, 1964, pp.93-103 J.D. Jackson, “Classical Electrodynamics”, John Wiley &amp; Sons, 3th Ed., 1999, pp. 605-608
  87. L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975, pg. 69
  88. L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975, pg. 69
  89. L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975, pg. 69
  90. L.D. Landau, E.M. Lifshitz, “The Classical Theory of Fields”, 4th Revised English Edition, Pergamon Press, 1975, pg. 69
  91. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980
  92. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980
  93. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980
  94. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980 B.L. Moiseiwitsch, “Variational Principles”, Dover, 1966, 2004
  95. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980 B.L. Moiseiwitsch, “Variational Principles”, Dover, 1966, 2004
  96. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980 B.L. Moiseiwitsch, “Variational Principles”, Dover, 1966, 2004
  97. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980 B.L. Moiseiwitsch, “Variational Principles”, Dover, 1966, 2004
  98. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980 B.L. Moiseiwitsch, “Variational Principles”, Dover, 1966, 2004
  99. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980 B.L. Moiseiwitsch, “Variational Principles”, Dover, 1966, 2004
  100. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980 B.L. Moiseiwitsch, “Variational Principles”, Dover, 1966, 2004
  101. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980 B.L. Moiseiwitsch, “Variational Principles”, Dover, 1966, 2004
  102. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980 B.L. Moiseiwitsch, “Variational Principles”, Dover, 1966, 2004
  103. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980 B.L. Moiseiwitsch, “Variational Principles”, Dover, 1966, 2004
  104. H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980 B.L. Moiseiwitsch, “Variational Principles”, Dover, 1966, 2004
  105. I.M. Gelfand, S.V.Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 81-83, 176-179 A.O. Barut, “Electrodynamics and Classical Theory of Particles”, Dover, 1964, pp. 103-105 D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989, pp. 201-207 H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980, pp.590-591 T. Frankel, “The Geometry of Physics – An Introduction”, Cambridge University, 1997, pp. 523 - 529
  106. I.M. Gelfand, S.V.Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 81-83, 176-179 A.O. Barut, “Electrodynamics and Classical Theory of Particles”, Dover, 1964, pp. 103-105 D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989, pp. 201-207 H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980, pp.590-591
  107. I.M. Gelfand, S.V.Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 81-83, 176-179 A.O. Barut, “Electrodynamics and Classical Theory of Particles”, Dover, 1964, pp. 103-105 D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989, pp. 201-207 H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980, pp.590-591
  108. I.M. Gelfand, S.V.Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 81-83, 176-179 A.O. Barut, “Electrodynamics and Classical Theory of Particles”, Dover, 1964, pp. 103-105 D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989, pp. 201-207 H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980, pp.590-591
  109. I.M. Gelfand, S.V.Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 81-83, 176-179 A.O. Barut, “Electrodynamics and Classical Theory of Particles”, Dover, 1964, pp. 103-105 D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989, pp. 201-207 H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980, pp.590-591
  110. I.M. Gelfand, S.V.Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 81-83, 176-179 A.O. Barut, “Electrodynamics and Classical Theory of Particles”, Dover, 1964, pp. 103-105 D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989, pp. 201-207 H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980, pp.590-591
  111. I.M. Gelfand, S.V.Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 81-83, 176-179 A.O. Barut, “Electrodynamics and Classical Theory of Particles”, Dover, 1964, pp. 103-105 D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989, pp. 201-207 H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980, pp.590-591
  112. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  113. D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989
  114. H. Sagan, “Boundary and Eigenvalue Problems in Mathematical Physics”, John Wiley &amp; Sons, 1961, Dover, 1989, pp. 34-49 B.L. Moiseiwitsch, “Variational Principles”, John Wiley &amp; Sons, 1966, Dover, 2004, pp. 91-94 P. R. Wallace, “Mathematical Analysis of Physical Properties”, Dover, 1972, 1984, “The Vibrating String”, pp. 5-65
  115. H. Sagan, “Boundary and Eigenvalue Problems in Mathematical Physics”, John Wiley &amp; Sons, 1961, Dover, 1989, pp. 34-49 B.L. Moiseiwitsch, “Variational Principles”, John Wiley &amp; Sons, 1966, Dover, 2004, pp. 91-94 P. R. Wallace, “Mathematical Analysis of Physical Properties”, Dover, 1972, 1984, “The Vibrating String”, pp. 5-65
  116. H. Sagan, “Boundary and Eigenvalue Problems in Mathematical Physics”, John Wiley &amp; Sons, 1961, Dover, 1989, pp. 34-49 B.L. Moiseiwitsch, “Variational Principles”, John Wiley &amp; Sons, 1966, Dover, 2004, pp. 91-94 P. R. Wallace, “Mathematical Analysis of Physical Properties”, Dover, 1972, 1984, “The Vibrating String”, pp. 5-65
  117. H. Sagan, “Boundary and Eigenvalue Problems in Mathematical Physics”, John Wiley &amp; Sons, 1961, Dover, 1989, pp. 49-65 B.L. Moiseiwitsch, “Variational Principles”, John Wiley &amp; Sons, 1966, Dover, 2004, pp. 94-95
  118. H. Sagan, “Boundary and Eigenvalue Problems in Mathematical Physics”, John Wiley &amp; Sons, 1961, Dover, 1989, pp. 49-65 B.L. Moiseiwitsch, “Variational Principles”, John Wiley &amp; Sons, 1966, Dover, 2004, pp. 94-95
  119. http://scholar.lib.vt.edu/theses/available/etd-05122000-14040026/unrestricted/T5012000.pdf
  120. http://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory
  121. http://scholar.lib.vt.edu/theses/available/etd-05122000-14040026/unrestricted/T5012000.pdf
  122. http://scholar.lib.vt.edu/theses/available/etd-05122000-14040026/unrestricted/T5012000.pdf
  123. http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch20.d/AFEM.Ch20.pdf
  124. http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch20.d/AFEM.Ch20.pdf
  125. http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch20.d/AFEM.Ch20.pdf
  126. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  127. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  128. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  129. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  130. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  131. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  132. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  133. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  134. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  135. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  136. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  137. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  138. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  139. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  140. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  141. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  142. Asher Yahalom, “A Simpler Eulerian Variational Principle for Barotropic Fluids”, February 2, 2008, http://arxiv.org/pdf/physics/9906050.pdf W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  143. Asher Yahalom, “A Simpler Eulerian Variational Principle for Barotropic Fluids”, February 2, 2008, http://arxiv.org/pdf/physics/9906050.pdf W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  144. Asher Yahalom, “A Simpler Eulerian Variational Principle for Barotropic Fluids”, February 2, 2008, http://arxiv.org/pdf/physics/9906050.pdf W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  145. Asher Yahalom, “A Simpler Eulerian Variational Principle for Barotropic Fluids”, February 2, 2008, http://arxiv.org/pdf/physics/9906050.pdf W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  146. Asher Yahalom, “A Simpler Eulerian Variational Principle for Barotropic Fluids”, February 2, 2008, http://arxiv.org/pdf/physics/9906050.pdf W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  147. Asher Yahalom, “A Simpler Eulerian Variational Principle for Barotropic Fluids”, February 2, 2008, http://arxiv.org/pdf/physics/9906050.pdf W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  148. Asher Yahalom, “A Simpler Eulerian Variational Principle for Barotropic Fluids”, February 2, 2008, http://arxiv.org/pdf/physics/9906050.pdf W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  149. Asher Yahalom, “A Simpler Eulerian Variational Principle for Barotropic Fluids”, February 2, 2008, http://arxiv.org/pdf/physics/9906050.pdf W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 M. Stone, P. Goldbart, “Mathematics for Physics”, Pimander-Casaubon Publishers, Ch.1, “Calculus of Variations”, Problem 1-12, pg. 50 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  150. N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  151. N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  152. N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  153. N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  154. N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  155. N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  156. N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
  157. P. R. Wallace, ‘Mathematical Analysis of Physical Problems”, Dover, 1972, 1984, pp. 232 - 239 B. L. Moiseiwitsch, “Variational Principles”, John Wiley &amp; Sons, 1966, Dover, 2004, pp. 96 – 100 H. Goldstein, “Classical Mechanics”, Addison Wesley, 2nd Ed., 1980, Appendix E, pp. 616 - 619
  158. P. R. Wallace, ‘Mathematical Analysis of Physical Problems”, Dover, 1972, 1984, pp. 232 - 239 B. L. Moiseiwitsch, “Variational Principles”, John Wiley &amp; Sons, 1966, Dover, 2004, pp. 96 – 100 H. Goldstein, “Classical Mechanics”, Addison Wesley, 2nd Ed., 1980, Appendix E, pp. 616 - 619
  159. P. R. Wallace, ‘Mathematical Analysis of Physical Problems”, Dover, 1972, 1984, pp. 232 - 239 B. L. Moiseiwitsch, “Variational Principles”, John Wiley &amp; Sons, 1966, Dover, 2004, pp. 96 – 100 H. Goldstein, “Classical Mechanics”, Addison Wesley, 2nd Ed., 1980, Appendix E, pp. 616 - 619
  160. P. R. Wallace, ‘Mathematical Analysis of Physical Problems”, Dover, 1972, 1984, pp. 232 - 239 B. L. Moiseiwitsch, “Variational Principles”, John Wiley &amp; Sons, 1966, Dover, 2004, pp. 96 – 100 H. Goldstein, “Classical Mechanics”, Addison Wesley, 2nd Ed., 1980, Appendix E, pp. 616 - 619
  161. P. R. Wallace, ‘Mathematical Analysis of Physical Problems”, Dover, 1972, 1984, pp. 232 - 239 B. L. Moiseiwitsch, “Variational Principles”, John Wiley &amp; Sons, 1966, Dover, 2004, pp. 96 – 100 H. Goldstein, “Classical Mechanics”, Addison Wesley, 2nd Ed., 1980, Appendix E, pp. 616 - 619
  162. P. R. Wallace, ‘Mathematical Analysis of Physical Problems”, Dover, 1972, 1984, pp. 232 - 239 B. L. Moiseiwitsch, “Variational Principles”, John Wiley &amp; Sons, 1966, Dover, 2004, pp. 96 – 100 H. Goldstein, “Classical Mechanics”, Addison Wesley, 2nd Ed., 1980, Appendix E, pp. 616 - 619
  163. P. R. Wallace, ‘Mathematical Analysis of Physical Problems”, Dover, 1972, 1984, pp. 232 - 239 B. L. Moiseiwitsch, “Variational Principles”, John Wiley &amp; Sons, 1966, Dover, 2004, pp. 96 – 100 H. Goldstein, “Classical Mechanics”, Addison Wesley, 2nd Ed., 1980, Appendix E, pp. 616 - 619
  164. P. R. Wallace, ‘Mathematical Analysis of Physical Problems”, Dover, 1972, 1984, pp. 232 - 239 B. L. Moiseiwitsch, “Variational Principles”, John Wiley &amp; Sons, 1966, Dover, 2004, pp. 96 – 100 H. Goldstein, “Classical Mechanics”, Addison Wesley, 2nd Ed., 1980, Appendix E, pp. 616 - 619
  165. I.M. Gelfand, S.V.Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 81-83, 176-179 A.O. Barut, “Electrodynamics and Classical Theory of Particles”, Dover, 1964, pp. 103-105 D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989, pp. 201-207 H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980, pp.590-591
  166. I.M. Gelfand, S.V.Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 81-83, 176-179 A.O. Barut, “Electrodynamics and Classical Theory of Particles”, Dover, 1964, pp. 103-105 D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989, pp. 201-207 H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980, pp.590-591
  167. I.M. Gelfand, S.V.Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 81-83, 176-179 A.O. Barut, “Electrodynamics and Classical Theory of Particles”, Dover, 1964, pp. 103-105 D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989, pp. 201-207 H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980, pp.590-591
  168. I.M. Gelfand, S.V.Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 81-83, 176-179 A.O. Barut, “Electrodynamics and Classical Theory of Particles”, Dover, 1964, pp. 103-105 D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989, pp. 201-207 H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980, pp.590-591
  169. I.M. Gelfand, S.V.Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 81-83, 176-179 A.O. Barut, “Electrodynamics and Classical Theory of Particles”, Dover, 1964, pp. 103-105 D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989, pp. 201-207 H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980, pp.590-591
  170. I.M. Gelfand, S.V.Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 81-83, 176-179 A.O. Barut, “Electrodynamics and Classical Theory of Particles”, Dover, 1964, pp. 103-105 D. Lovelock, H. Rund, “Tensors, Differential Forms, and Variational Principles”, Dover, 1975, 1989, pp. 201-207 H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison Wesley, 1980, pp.590-591