This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
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2. 2
SOLO Calculus of Variations - Problems
Table of Content
Introduction
1. Brachistochrone Problems
Brachistochrone Problem 1
Brachistochrone Problem 2
2. Isoperimetric Problems
Dido Maximum Area Problem
Maximum volume with given surface area
Shape of a liquid drop on a horizontal surface
3. Problem of Minimum Surface of Revolution
4. Geodesics Problems
Geodesic in 3 Dimensional Spaces
Geodesics in Riemannian Space
Geodesics for an Implicit Equation of the Surface, G (x,y,z) = 0
5. Geometrical Optics and Fermat Principle
References
3. 3
SOLO
Introduction
The Mathematical Theory of Calculus of Variation was given in another presentation:
“Calculus of Variations”, by the same author.
Calculus of Variations tackles problems of finding Functions that Extremize a given
Cost Functional . The Solutions must satisfy Boundary Conditions and some giving
constraints. Solving those problems require finding solution to Differential Equations.
Calculus of Variations - Problems
In general few problems have analytic solutions and therefore the results can be
obtained only numerically.
In this presentation we give example f problems that could be solved analytical, in
order to facilitate the understanding of the subject.
Return to Table of Content
4. 4
SOLO
A particle slides on a frictionless wire between two fixed points A(0,0) and
B (xfc, yfc) in a constant gravity field g. The curve such that the particle takes
the least time to go from A to B is called brachistochrone (βραχιστόσ Greek for
“shortest“, χρόνοσ greek for “time).
The brachistochrone problem was posed by John Bernoulli in 1696, and
played an important part in the development of calculus of variations.
The problem was solved by Johann Bernoulli, Jacob Bernoulli, Isaac Newton,
Gottfried Leibniz and Guillaume de L’Hôpital.
Let choose a system of coordinates with the origin at point A (0,0) and the y axis in
the constant g direction
x
y
V
( )tγ
( )fcfc yxB ,fcx
fcy
N
( )0,0A
Calculus of Variations - Problems
1. Brachistochrone Problems
Brachistochrone Problem 1
5. 5
SOLO
Since the motion of the particle is in a frictionless fixed
gravitational field the total energy is conserved
( ) ygVyVygVV 2
2
1
2
1 2
0
22
0 +=→−=
x
y
V
( )tγ
( )fcfc yxB ,fcx
fcy
N
( )0,0A
Second way to get this relation is:
( ) ygVVdygdVV
sd
yd
ggV
sd
Vd
td
sd
sd
Vd
td
Vd
=−→=→====
2
0
2
2
1
sinγ
where V0 is the velocity of the particle at point A and ( ) ( )22
ydxdsd +=
td
xd
xd
yd
td
yd
td
xd
td
sd
V
222
1
+=
+
==
We have xd
ygV
xd
yd
xd
V
xd
yd
td
2
11
2
0
22
+
+
=
+
=
The cost function is
∫∫
=
+
+
=
cfcf xx
xd
xd
yd
yxFxd
ygV
xd
yd
J
00
2
0
2
,,
2
1
Calculus of Variations - Problems
1. Brachistochrone Problems
Brachistochrone Problem 1
6. 6
HISTORY OF CALCULUS OF VARIATIONS
The brachistochrone problem
In 1696 proposed the Brachistochrone (“shortest time”)
Problem:
Given two points A and B in the vertical plane, what is the curve
traced by a point acted only by gravity, which starts at A and
reaches B in the shortest time.
Johann Bernoulli
1667-1748
SOLO
7. 7
The brachistochrone problem
Jacob Bernoulli
(1654-1705)
Gottfried Wilhelm
von Leibniz
(1646-1716)
Isaac Newton
(1643-1727)
The solutions of Leibniz, Johann Bernoulli, Jacob Bernoulli
and Newton were published on May 1697 publication of
Acta Eruditorum. L’Hôpital solution was published only in 1988.
Guillaume François
Antoine de L’Hôpital
(1661-1704)
SOLO
HISTORY OF CALCULUS OF VARIATIONS
8. 8
SOLO
A particle slides on a frictionless wire between two fixed points A(0,0) and B (xfc, yfc) in a
constant gravity field g. The curve such that the particle takes the least time to go from A
to B is called brachistochrone.
x
y
V
( )tγ
( )fcfc yxB ,fcx
fcy
N
( )0,0A
∫∫
=
+
+
=
cfcf xx
xd
xd
yd
yxFxd
ygV
xd
yd
J
00
2
0
2
,,
2
1
We derived the cost function:
xd
yd
y
ygV
y
xd
yd
yxF =
+
+
=
:
2
1
:,,
2
0
2
wher
e
F doesn’t depend explicitly on the free variable x, therefore if we replace
and we use the result obtained for F not depending explicitly on x, we obtain
( ) ( )xtyx ,, →
( ) ( ) const
ygVy
y
ygV
y
yyFyyyF y ==
++
−
+
+
=− α
212
1
,,
2
0
2
2
2
0
2
const
ygVy
==
++
α
21
1
2
0
2
or
Calculus of Variations - Problems
1. Brachistochrone Problems
Brachistochrone Problem 1
9. 9
SOLO
x
y
V
( )tγ
( )fcfc yxB ,fcx
fcy
N
( )0,0A
const
ygVy
==
++
α
21
1
2
0
2
Let define a parameter τ such that
τcos
1
1
2
=
+
xd
yd
and const
ygV
ygV
xd
yd
==
+
=
+
+
α
τ
2
cos
21
1
2
02
0
2
From which ( ) ( )ττ
αα
τ
2cos12cos1
4
1
2
cos
2 22
22
0
+=+==+ r
ggg
V
y
Tacking the derivative of this equation with respect to τ we obtain τ
τ
2sin2r
d
yd
−=
Calculus of Variations - Problems
1. Brachistochrone Problems
Brachistochrone Problem 1
11. 11
HISTORY OF CALCULUS OF VARIATIONS
The brachistochrone problem
( )
( )
−−=
−+=
g
V
ry
rxx
2
cos1
sin
2
0
0
θ
θθ
Cycloid Equation
∫∫∫∫
=
+
+
===
cfcfcf xxxt
xd
xd
yd
yxFxd
ygV
xd
yd
V
sd
tdJ
00
2
0
2
00
,,
2
1
Minimization Problem
Solution of the Brachistochrone Problem:
SOLO
Johann Bernoulli
1667-1748
12. 12
SOLO
( )
( )θ
θθ
cos1
2
sin
2
0
0
−=+
−+=
r
g
V
y
rxx
θsinr
θcosr
θr
x
y
0x
0V
g
V
2
2
0
r
r
A
B
),( yx
θ
Calculus of Variations - Problems
We have
( )
( )
td
d
r
td
d
r
td
d
d
xd
d
xd
V
r
d
yd
r
d
xd
θθθ
θ
θ
θθ
θ
θ
θ
θ
=−=
+
=
=
−=
2
sin2cos12
sin
cos1
22
( ) rgrgrgygVV
=
=−=+=
2
sin2
2
sin4cos122 22
0
θθ
θ
const
r
g
td
d
==
θ
( )0
*
0
θθθ
θ
θ
−=== ∫∫ fAB
g
r
d
g
r
tdt
f
From those two equations we obtain
and the minimum time to travel between A and B will be
1. Brachistochrone Problems
Brachistochrone Problem 1
Return to Table of Content
13. 13
SOLO Calculus of Variations - Problems
Brachistochrone Problem 2
(Bryson Sec 2.7, Problem 6, pg. 81, Sec. 3.11, Example 1, pg. 119, Sec. 4.3,
Example 1, pg. 142)
A particle slides on a frictionless wire between two points A (0,0) and B (xfc, yfc) in a
constant gravity field.
The particle has an initial velocity V0 at point A.
What is the shape of the wire:
• that will produce a minimum time path between the two points with the
constraint that y ≤ x tanθ + h, where θ and h are constant.
• That provides a maximum xf for given tf and yf.
hxy +< θtan
x
y
V
( )tγ
( )fcfc yxB ,fcx
fcy
N
1. Brachistochrone Problems
14. 14
SOLO Calculus of Variations - Problems
Brachistochrone Problem 2
Solution to Brachistochrone Problem 2
Since the motion of the particle is in a frictionless fixed gravitational field the total
energy is conserved
( ) ygVyVygVV 2
2
1
2
1 2
0
22
0 +=→−=
The equations of motions are
( ) ( ) ( )
( ) ( ) ( ) fcf
fcf
ytytyyVy
xtxtxyVx
===
===
0sin
0cos
0
0
γ
γ
We have 2 First Order Differential Equations in x (t) and y (t) and 1 control γ (t)
with 4 Boundary Conditions
To find the wire path we must define the angle γ (t) such that
ftJ
γγ
minmin =
1. Brachistochrone Problems
15. 15
SOLO Calculus of Variations - Problems
Solution to Brachistochrone Problem 2 (continue – 1)
The unconstrained problem: y ≤ x tanθc + h
( )
=−
=−
=
0
0
:,
fcf
fcf
ff
yy
xx
txψ
Let adjoin the constraints to obtain the augmented cost function:
( ) ( ) ( )( ) ( )( )[ ]∫ −+−+−+−+=
ft
t
yxfcffcff dtyyVxyVxxxxtJ
0
sincos21
γλγλνν
1. The Hamiltonian ( )( )γλγλ sincos: yxyVH +=
( ) ( ) ( )fcffcffff xxxxttx −+−+=Φ 21:, ννand
2. Euler-Lagrange Equations
( )fxfx
T
tH Φ=−= λλ
( ) ( ) ( ) 10 νλλλλ ==→=
∂
∂
−= fxfxxx ttt
x
H
( ) ( ) 2sincos νλγλγλλ =+−=
∂
∂
−= fyyxy t
yd
Vd
y
H
1. Brachistochrone Problems
16. 16
SOLO Calculus of Variations - Problems
Solution to Brachistochrone Problem 2 (continue – 2)
( )( )
x
y
yxyVH
λ
λ
γγλγλγ =→=+−= tan0cossin:
Since H is not an explicit function of time we have: ( ) ( )ftHconsttH **
==
Using the Boundary Conditions: ( ) ( ) 1*
−=Φ−= ftf ttH
Therefore instead of solving the differential equation, let use:( )tyλ
( )( ) ( ) ( ) ( ) ( ) 1
sin
sincoscotsincossincos:*
−==+=
+=+=
γ
λ
γγγλγγ
λ
λ
λγλγλ y
y
y
x
yyx yVanyVyVyVH
From which
( )yV
y
γ
λ
sin
−=
Note:
If for the end point B (xfc, yfc) only xfc, is defined and yfc is free, then
( )
( )
( ) ( ) 0sin0
sin
=→=−= f
f
f
fy t
yV
t
t
c
γ
γ
λ
End Note
1. Brachistochrone Problems
17. 17
SOLO Calculus of Variations - Problems
Solution to Brachistochrone Problem 2 (continue – 3)
Also
( )( ) ( ) ( ) ( ) ( ) 1
cos
sintancossincossincos:*
−==+=
+=+=
γ
λ
γγγλγ
λ
λ
γλγλγλ x
x
x
y
xyx yVyVyVyVH
Therefore
( )
0
0
cos
1
cos γλγ
V
const
yV
x
==−=
or ( )
x
yV
λ
γcos
−=
Note:
( )
0
*
cos
01
γ
λx
VH =−=
2
0
π
γ =If V0 = V (0)=0, to keep we must have
End Note
If we take the time derivative of the last equation we obtain:
( ) ( ) ( )
td
d
yV
yd
yVd
td
yd
yd
yVd
x
γ
λ
γ
γ
sin
sin ==
or using ( ) ygVyV 2
2
0 +=
( ) ( )
( )
( ) constgyV
yV
g
yV
yd
yVd
td
d
xxx ===== ωλλλ
γ
1. Brachistochrone Problems
18. 18
SOLO Calculus of Variations - Problems
Solution to Brachistochrone Problem 2 (continue – 4)
Let find now the shape of the wire
Let use and( ) ( ) 0cos 0 == txyV
td
xd
γ ( )
x
yV
λ
γcos
−=
( )
x
x yVg
d
xd
td
d
d
xd
td
xd
λ
γ
γλ
γ
γ
γ
2
cos
cos −====
or
( )
−+−−=
+
−=−=
→−=
∫∫
002
2
2
2
2
2
2sin2sin22
4
1
2
2cos11
cos
1
cos 00
γγγγ
λ
γ
γ
λ
γγ
λ
λ
γ
γ
γ
γ
γ
γ
x
xx
x
g
d
g
d
g
x
gd
xd
From the equations: and , we obtain:( ) ygVyV 2
2
0 += ( )
x
yV
λ
γcos
−=
( )
g
V
gg
V
g
yygV
xxx
2
2cos1
4
1
2
cos
2
1cos
2
2
0
2
2
0
2
2
2
2
2
0 −+=−
=→=+ γ
λλ
γ
λ
γ
1. Brachistochrone Problems
19. 19
SOLO Calculus of Variations - Problems
Solution to Brachistochrone Problem 2 (continue – 5)
The wire shape is given by:
( )
( )
g
V
g
y
g
x
x
x
2
2cos1
4
1
2sin2sin22
4
1
2
0
2
002
−+=
−+−−=
γ
λ
γγγγ
λ
For V0 ≠ 0 from which( ) 0
00 cos
0
cos
VyV
x
γγ
λ −=
=
−= 0
0
cosγλω
γ
V
g
g
td
d
x −===
( )
( )
g
V
g
V
y
g
V
x
2
2cos1
cos4
2sin2sin22
cos4
2
0
0
2
2
0
00
0
2
2
0
−+=
−+−−=
γ
γ
γγγγ
γ
The wire shape is given by:
1. Brachistochrone Problems
20. 20
SOLO Calculus of Variations - Problems
Solution to Brachistochrone Problem 2 (continue – 6)
By using
unknown
g
V
r 0
0
2
2
0
cos4
: γ
γ
= πγθ += 2: ( ) unknownrx 0000 sin: θθθ −−=
we obtain:
( )
( )
EquationCycloid
g
V
ry
rxx
−−=
−+=
2
cos1
sin
2
0
0
θ
θθ
To find r and x0 we must solve the following 4 equations with additional 2
unknowns θ0 and θf
( )
( )
( )
( )
( )ff
fff
r
g
V
y
rxx
g
V
g
V
rr
g
V
rx
c
c
θ
θθ
θθ
θ
θθ
cos1
2
sin
2
sin4
cos12
cos1
2
sin0
2
0
0
02
2
0
0
2
0
0
2
0
000
−=+
−+=
=
−
=→−=
−+=
For V0 ≠ 0
1. Brachistochrone Problems
21. 21
SOLO Calculus of Variations - Problems
Solution to Brachistochrone Problem 2 (continue – 7)
For V0 = 0 we found 0,0
2
000 ==→= xθ
π
γ
( )
( )
−=
−=
θ
θθ
cos1
sin
ry
rx
where 2
4
1
xg
r
λ
=
and it can be calculated using the following 2 equations with 2 unknowns r and θf
( )
( )
−=
−=
ff
fff
ry
rx
c
c
θ
θθ
cos1
sin
θf is given by the following transcedental equation
( )
( )f
ff
f
f
c
c
y
x
θ
θθ
cos1
sin
−
−
=
and ( )f
fc
y
r
θcos1−
=
Now we can compute
( )
cf
f
x
ygrg
θ
λ
cos1
2
1
2
1 −
==
( )
const
y
g
y
g
r
g
g
td
d
td
d
cc f
f
f
f
x =
=
−
=====
22
sin
cos1
2
1
2
1
2
1 θθ
λω
θγand
∫∫
===
f
c
g
y
dtdt
f
f
f
AB
θ
θ
θ
θ
ω0
*
2
2
sin
2
2
1
1. Brachistochrone Problems
22. 22
SOLO
1. Brachistochrone Problems
Calculus of Variations - Problems
Solution to Brachistochrone Problem 2 (continue – 8)
If yfc is free, we found that
( ) ( )
( ) ( ) ( ) ( ) πππγθπγγ
γ
λ 2,2,00sin0
sin
=+=→=→=→=−= ffff
f
f
fy ttt
yV
t
t
c
Since for V0 = 0 we have θ0 = 0, we must have θf = π (γf = 0)
Using this result we obtain:
( )
π
πθθ c
c
f
fff
x
rrrx =→=−= sin
( )
π
θ c
c
f
ff
x
rry 22cos1 ==−=
cf
x
xgrg
π
λ
2
1
2
1
==
const
x
g
r
g
g
td
d
td
d
cf
x ======
π
λω
θγ
2
1
2
1
2
1
∫∫ ===
π
π
θ
ω0
*
2
1
g
x
dtdt cf
AB
For V0 = 0
Return to Table of Content
23. 23
HISTORY OF CALCULUS OF VARIATIONSSOLO
“When the Tyrian princess Dido landed on the Mediterranean sea she was welcomed by a local
chieftain. He offered her all the land that she could enclose between the shoreline and a rope of
knotted cowhide. While the legend does not tell us, we may assume that Princess Dido arrived at
the correct solution by stretching the rope into the shape of a circular arc and thereby maximized
the area of the land upon which she was to found Carthage. This story of the founding of
Carthage is apocryphal. Nonetheless it is probably the first account of a problem of the kind that
inspired an entire mathematical discipline, the calculus of variations and its extensions such as
the theory of optimal control.” (George Leitmann “The Calculus of Variations and Optimal
Control – An Introduction” Plenum Press, 1981)
1. Dido Maximum Area Problem
1. Isoperimetric Problems
24. 24
Given a rope of length P connected to each end of straight line of length 2 a < P find the shape
of the rope necessary to enclose the maximum area between the rope and the straight line.
( ) ( ) ( ) ∫∫∫∫∫ −−−
=+=
+=+==
a
a
a
a
a
a
dxdxdx
xd
yd
ydxdsdP θθ sectan11
2
2
22
subject o the isoperimetric constraint:
where: θtan=
xd
yd
SOLO
Rope of length P
( )xθ
x
y
a+a−
y
dx
Calculus of Variations - Problems
( ) ∫∫ −
==
a
a
xy
dxyAdJ maxmaxmaxThe problem can be formulated as:
Solution
1. Dido Maximum Area Problem
Isoperimetric Problems
25. 25
SOLO Calculus of Variations - Problems
Therefore we have the following differential equations as constraint
( ) ( ) 0&0tan ==−= ayay
xd
yd
θ
( ) ( ) Papap
xd
pd
==−= &0
cos
1
θ
1. Define the Hamiltonian:
θ
λθλ
cos
1
tan: pyyH ++=
and the end constraint function ( ) ( )Ppypyx py −+=Φ νν:,,
2. Euler-Lagrange and Boundary Conditions:
( ) ( ) ( )xax
y
a
y
H
xd
d
yyyy
y
−+=→=
∂
Φ∂
=−=
∂
∂
−= νλνλ
λ
1
( ) ( ) pypy
p
x
p
a
p
H
xd
d
νλνλ
λ
=→=
∂
Φ∂
==
∂
∂
−= 0
p
y
p
y
py
xaH
ν
ν
λ
λ
θ
θ
θ
λ
θ
λ
θ
−+
−=−=→+=
∂
∂
= *
22
sin
cos
sin
cos
1
0
3. The extremal of the Hamiltonian is given for
*
*
**
*
**
cos
cos
1
tansin
cos
1
tan θλ
θ
θθλ
θ
θ
λ
λ
λ pp
p
y
p yyyH +=
+−+=
++=
Rope of length P
( )xθ
x
y
a+a−
y
dx
Solution (continue – 1)
1. Dido Maximum Area Problem
Isoperimetric Problems
26. 26
SOLO Calculus of Variations - Problems
Since H is not an explicit function of x and
( ) constpyxH
x
H
xd
Hd
py =→=
∂
∂
= λλθ ,,,,,0
Therefore ( ) ( )
**
0
**
0
**
0
coscos
cos0cos0
ff
fpp axHaxH
θθθθ
θλθλ
±=→=→
+===+=−=
From
p
y xa
ν
ν
θ
−+
−=*
sin
we have ( ) ( )
p
y
p
y
ax
a
ax
ν
ν
θ
ν
ν
θ
±
−==±=
+
−=−= **
sin
2
sin
This equation is defined only when ± → -, and in this case
p
y
p
y a
ν
ν
ν
ν
=
+
−
2
( ) ( ) 0
**
θπθπθ −==−=−= axax
and ay −=ν
p
x
ν
θ =→ *
sin
We have and
2
*
1cos
−=
p
x
ν
θ 2
*
1
tan
−
==
p
p
x
x
xd
yd
ν
ν
θ
Rope of length P
( )xθ
x
y
a+a−
y
dx
Solution (continue – 2)
1. Dido Maximum Area Problem
Isoperimetric Problems
27. 27
SOLO Calculus of Variations - Problems
We obtained 2
*
1
tan
−
==
p
p
x
x
xd
yd
ν
ν
θ
Let integrate this equation
( )
22
2
2
0
11
12
1
−+
−−=
−
−
−=−− ∫− p
p
p
p
x
a
p
p
p
ax
x
x
d
ayy
ν
ν
ν
ν
ν
ν
ν
or 2222
xay pp −−=−− νν
Rope of length P
( )ax +=θ
x
y
a+a−
( )ax −=θ
pR ν=
22
ap −ν
0θ
and finally
22
2
22
pp xay νν =+
−−
This the equation of a circular arc
with radius R = vp and center at
− 22
,0 apν
Solution (continue – 3)
1. Dido Maximum Area Problem
Isoperimetric Problems
28. 28
SOLO Calculus of Variations - Problems
1. Dido Maximum Area Problem
Isoperimetric Problems
Rope of length P
( )ax +=θ
x
y
a+a−
( )ax −=θ
pR ν=
22
ap −ν
0θ
The radius R of the circular arc is found from the perimeter constraint
=
=
−
== −
=
−=
−
−−
∫∫ p
p
ax
axp
p
a
a
p
a
a
ax
x
xdxd
P
ν
ν
ν
ν
ν
θ
11
2*
sin2sin
1
cos
or
=
p
p
P
a
ν
ν
2
sin
From the equations
*
sinθν px =
*
0
*
22
coscos11 θνθν
ν
ν
ν
ν pp
p
p
p
p
ax
y −=
−−
−=
The maximum Area enclosed
by the Rope is
( )
−= 00
2
2sin
2
1
* θθν pA
( ) ( ) ( )∫∫∫∫ −−−−
−
+
=−==
0
0
0
0
0
0
**
0
2*
*
2***
0
**
sincos
2
2cos1
coscoscos
θ
θ
θ
θπ
θ
θπ
θθνθ
θ
νθθνθνθν dddxdyA ppppp
a
a
( )
( ) ( ) ( )
−=
−+=
−
+
=
−
00
2
000
2
0
2
2sin
2
1
2sin2sin
2
1
sincos
2
2sin
2
1 0
θθνθθθνθθ
θθ
ν
θ
θπ
ppp
The maximum Area
Solution (continue – 4)
Return to Table of Content
29. 29
SOLO Calculus of Variations - Problems
Given an area of canvas, A, to built a tent, find the shape of the canvas necessary to cover a
circular floor area of radius a ( π a2
< A ) with maximum volume inside the tent.
x
y
z
r
dr
Solution
( ) ( )∫∫ ==
aa
drrrzdrrrzV
00
max22maxmax ππ
where z (r) is the height of the canvas surface at
radius r.
Define ( ) ( ) ( ) rdrd
rd
zd
zdrd
rd
rzd
θ
θ
cos
1
1tan
2
22
=
+=+→=
and ( ) ( ) ∫∫∫ =
+=+=
aa
rd
r
rd
rd
zd
zdrdrA
00
2
22
cos
2122
θ
πππ
Isoperimetric Problems
2. Maximum volume with given surface area
30. 30
SOLO Calculus of Variations - Problems
x
y
z
r
dr
Solution (continue – 1)
Summarize
∫=
a
drrzJ
0
2π
( ) ( )
( ) ( ) 0&0tan
&00
cos
2
==
===
azdefinednotz
rd
zd
AaAA
r
rd
Ad
θ
θ
π
where
1. Define Hamiltonian θλ
θ
πλπ tan
cos
22: zA
r
rzH ++=
2. Euler-Lagrange Equations and Boundary Conditions
const
A
H
rd
d
A
A
==
∂
∂
−= λ
λ
0
( ) ( )( ) 2
0sin002 rdefinednotzcer
z
H
rd
d
zz
z
πλλπ
λ
−=→=−=
∂
∂
−=
22
22
2
1cos
222
sin
cos
1
cos
sin
20
−=→==−=→++=
∂
∂
=
AAAA
z
zA
rr
r
r
r
r
H
λ
θ
λλπ
π
λπ
λ
θ
θ
λ
θ
θ
πλ
θ
3. The extremal of the Hamiltonian is given for
Isoperimetric Problems
2. Maximum volume with given surface area
31. 31
SOLO Calculus of Variations - Problems
x
y
z
r
dr
Solution (continue – 2)
The following relations are developed
2
2
1
2
cos
sin
tan
−
===
A
A
r
r
rd
zd
λ
λ
θ
θ
θ
from which ∫∫
−
−=
−
=
−
=−
a
A
A
A
A
A
a
A
A r
r
r
d
rd
r
r
tzz
0
2
2
2
0
2
0 1
2
12
2
1
2
2
2
1
2
λ
λ
λ
λ
λ
λ
λ
that can be written as ( ) ( )222
0 22 AA rzz λλ =++−
Since 222
yxr +=
we obtain ( ) ( )2222
0 22 AA yxzz λλ =+++−
Isoperimetric Problems
2. Maximum volume with given surface area
32. 32
SOLO Calculus of Variations - Problems
Solution (continue – 3)
we obtained ( ) ( )2222
0 22 AA yxzz λλ =+++−
This is the equation of a spherical surface with
radius 2 λ A and center on the vertical axis passing
trough the center of the circular floor and located at
.Azz λ20 −=
We want to find the two unknowns z0 and λA
(1) by using the given A, the area of the canvas
( )
( ) ( )
−−=
−−=
−−=
−
==
=
=
∫∫
22
2
2
0
2
2
0
2
2
2
0
2222
2
118
2
18
2
1
2
22
cos
2
a
a
r
r
r
d
rd
r
A
AAA
A
A
ar
r
A
A
a
A
A
A
a
λλλπ
λ
πλ
λ
πλ
λ
λ
λπ
θ
π
This is a transcedental equation in 2 λ A .
Isoperimetric Problems
2. Maximum volume with given surface area
33. 33
SOLO Calculus of Variations - Problems
Isoperimetric Problems
2. Maximum volume with given surface area
Solution (continue – 4)
(2) by using the condition z (r=a) = 0
( ) ( ) ( ) 22
0
222
0 2222 azaz AAAA −±=→=++− λλλλ
We can see that for 2 λ A = a
2
2 aA π=
020 =+− Az λ
the spherical surface has the radius a and its center
on the center floor.
For A > 2 π a2
we have
( ) 22
0 22 az AA −=− λλ
the center of the spherical canvas surface is above the floor
For A< 2 π a2
we have
( ) 22
0 22 az AA −−=− λλ the center of the spherical canvas surface is below the floor
Return to Table of Content
34. 34
SOLO Calculus of Variations - Problems
A liquid drop on a horizontal surface assume an axially symmetric shape that minimize
the sum of the liquid’s potential energy in the earth’s gravitational field plus the liquid’s
surface energy.
Solution:
Because of the axially symmetry let consider
a) a disc, at a height z ,that touches the external surface and has a radius r(z) , and
width dz that has the potential energy
( )( )zdzrzgmdzgUd vol
2
πρ−=−=
from which we obtain
( ) rd
rd
zd
rzgzdzrzgU
a
vol ∫∫
−=−=
0
22
ρπρπ
b) the change in potential energy is given by the change in the axially symmetric
surface due to surface tension
( ) ( ) ( )( ) rd
rd
zd
rrzdrdrsdrUd surface
2
22
1222
+=+== πσπσπσ
from which we obtain ∫
+=
a
surface rd
rd
zd
rU
0
2
12 σπ
x
y
r
ds
σσ
dr
σ σ
σ
dz
a
)(zrz
Isoperimetric Problems
3. Shape of a liquid drop on a horizontal surface
35. 35
SOLO Calculus of Variations - Problems
Solution (continue – 1):
x
y
r
ds
σσ
dr
σ σ
σ
dz
a
)(zrz
The sum of the liquid’s potential energy in the
earth’s gravitational field plus the liquid’s
surface energy that is minimized is given by:
∫∫
++−=+=
aa
surfacevol rd
rd
zd
rrd
rd
zd
rzgUUU
0
2
0
2
12 σπρπ
The volume of the liquid drop is fixed rd
rd
zd
rV
a
∫
=
0
2
π
Let define u
rd
zd
== θtan
Summarizing ∫∫ ++−=
aa
rdurrdurzgJ
0
2
0
2
12 σπρπ
a is not defined and we have the constraints
( ) ( )
( ) ( ) VaVVur
rd
Vd
azdefinednotzu
rd
zd
===
==
&00
0&0
2
π
Isoperimetric Problems
3. Shape of a liquid drop on a horizontal surface
36. 36
SOLO Calculus of Variations - Problems
Solution (continue – 2):
x
y
r
ds
σσ
dr
σ σ
σ
dz
a
)(zrz1. The Hamiltonian is given by
uruururgzH Vz
222
12 πλλσπρπ ++++−=
2. Euler-Lagrange Equations and Boundary Conditions
const
V
H
rd
d
V
V
==
∂
∂
−= λ
λ
0
( ) ( )( )definednotzceugr
z
H
rd
d
z
z
0sin002
==
∂
∂
−= λρπ
λ
2
2
2
1
20 r
u
u
rrgz
u
H
Vz πλλσπρπ ++
+
+−=
∂
∂
=
3. The extremal of the Hamiltonian is given for
Let take the derivative of the last equation relative to r:
( )
rrgu
rd
ud
u
u
u
r
u
u
rgzrgu Vπλρπσπσπρπρπ 2
11
1
2
1
220 2
22
2
22
2
++
+
−
+
+
+
+−−=
( ) σ
λ
σ
ρ V
z
g
u
u
rrd
ud
u
−=−
+
+
+
232 1
1
1
1
or
Isoperimetric Problems
3. Shape of a liquid drop on a horizontal surface
37. 37
SOLO Calculus of Variations - Problems
Isoperimetric Problems
3. Shape of a liquid drop on a horizontal surface
Solution (continue – 3):
x
y
r
ds
σσ
dr
σ σ
σ
dz
a
)(zrz
( ) σ
λ
σ
ρ V
z
g
u
u
rrd
ud
u
−=−
+
+
+
232 1
1
1
1
For u << 1 and using we obtainu
rd
zd
=
σ
λ
σ
ρ V
z
g
rd
zd
rrd
zd
−=−+
1
2
2
which has the solution
( )
+
+= r
g
KBr
g
IA
g
rz V
σ
ρ
σ
ρ
ρ
λ
00
where I0 and K0 are Bessel functions of order zero with imaginary argument, and
A, B and λV to be defined.
Return to Table of Content
38. 38
SOLO
Problem of Minimum Surface of Revolution
Given two points A (a,ya) and B (b, yb) a≠b in the plane. Find the curve that joints these
two points with a continuous derivative, in such a way that the surface generated by the
rotation of this curve about the x axis has the smallest possible area.
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y Minimum Surface of Revolution
The surface generated by the rotation of y (x) curve about the x – axis can be
calculated using
( ) ( ) xd
xd
yd
yydxdysdydS
2
22
1222
+=+== πππ
Therefore
( )∫
+==
b
a
xd
xd
yd
xySJ
2
12: π
2
1,,
+=
xd
yd
y
xd
yd
yxF
We can see that F
is not an explicit
function of x.
Calculus of Variations - Problems
39. 39
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Minimum Surface of Revolution
Solution:
Calculus of Variations - Problems
1. Euler-Lagrange Equation 0=− yy F
xd
d
F
2
1,,
+=
xd
yd
y
xd
yd
yxF
2
12 yFy
+= π
2
1
2
y
y
yFy
+
= π
( ) ( )
+
+
+
=
+
−
+
+
= 3
22
2
3
2
2
2
2
11
2
11
2
y
yy
y
y
y
yyy
y
yyy
F
xd
d
y
ππ
with
Substituting in the Euler-Lagrange Equation gives
( ) ( )
( )
0
1
2
1
1
2
11
1
2
11
12
2
32
2
322322
2
2
=
+
+
−+
=
+
−
+
=
+
−
+
−+=−
y
y
xd
d
y
or
y
yyy
y
yy
yy
yy
y
y
yF
xd
d
F yy
π
π
ππ
40. 40
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Minimum Surface of Revolution
Solution (continue – 1):
Calculus of Variations - Problems
( )
0
1
2
1
1
2
2
32
2
=
+
+
−+
=−
y
y
xd
d
y
or
y
yyy
F
xd
d
F yy
π
π
E-L. Equation
01 2
=−+ yyy (1) nonlinear second order differential equation
0
1 2
=
+ y
y
xd
d
)2(
Second Way to solve Euler-Lagrange Equation
Since F is not an explicit function of x, we can write:
( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )
−=−−+=− yyF
xd
d
yyFyyyF
xd
d
yyyFyyyFyyyFyyyFyyyF
xd
d
yyyyyyy
,,,,,,,,
In our case ( ) ( ) C
y
y
yyyyyFyyyF y ππ 2
1
12,,
2
2
2
=
+
−+=−
2
1 yCy +=or We obtained the same result
We obtain two
equivalent equations
constC
y
y
==
+ 2
1
41. 41
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Solution (continue – 2):
Calculus of Variations - Problems
Third Way to obtain Euler-Lagrange Equation
Start from the expression of d S
and change the free variable from x to y
( ) ( ) yd
yd
xd
yydxdysdydS
2
22
1222
+=+== πππ
Define ( )
yd
xd
xxyxxyF =←+= :12:,,
~ 2
π
Euler-Lagrange Equation ( ) ( ) 0,,
~
,,
~
=− xxyF
yd
d
xxyF xx
Since is not a function of x( ) 2
12:,,
~
xyxxyF += π ( ) 0,,
~
=xxyFx
Therefore ( ) ( )
1
2
1
1
2
1
2
,,
~
0,,
~
2
2
2
+
=
+
=
+
==→=
xd
yd
y
x
y
x
xy
xxyFconstxxyF
yd
d
xx
πππ
We recovered equation 2
1 yCy +=
42. 42
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Solution (continue – 3):
Calculus of Variations - Problems
Solving the Euler-Lagrange Equation
1
2
−
=
C
y
yRewrite the E-L Equation as
Separating variables, we obtain C
xd
C
y
C
yd
=
−
1
2
Integration of this equation, gives
−
+=− 1ln
2
1
C
y
C
y
CCx
from which 1exp
2
1
−
+=
−
C
y
C
y
C
Cx
take the square
1exp211212122exp 1
222
1
−
−
=−
−
+=−
+−
=
−
C
Cx
C
y
C
y
C
y
C
y
C
y
C
y
C
y
C
Cx
From this equation we can compute
2
expexp 11
−
−+
−
=
C
Cx
C
Cx
C
y
43. 43
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Solution (continue – 4):
Calculus of Variations - Problems
Solving the Euler-Lagrange Equation (continue – 1)
( )
−
=
C
Cx
Cxy 1
coshWe obtained
The solution is a curve called a catenary (catena = chain in Latin) and the Surface of
Revolution which is generated is called a catenoid of revolution.
The two parameters C and C1 are defined by the two End Conditions:
−
=
−
=
C
Cb
Cy
C
Ca
Cy
b
a
1
1
cosh
cosh
Since we have only two parameters C and C1 and three independent variables
ya, yb, b-a , that define the end conditions, a solution not always exists.
44. 44
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Solution (continue – 5):
Calculus of Variations - Problems
Solving the Euler-Lagrange Equation (continue – 2)
To see when a solution exists , without loss of generality choose
aybBCCa /&/&0 1 === λ
−
=
−
=
C
Cb
Cy
C
Ca
Cy
b
a
1
1
cosh
cosh
We have λcosh
ay
C =
( )
λ
λλ
cosh
coshcosh −
=
B
y
y
a
b
Since
( ) ( ) tt
tt
t ∀≥
−+
=
2
expexp
cosh
1
cosh
≤
λ
λ
therefore 0
cosh
sinhcosh
cosh
1
cosh
max 2
=
−
=
→≤
λ
λλλ
λ
λ
λλ
λ
λ d
d
The minimum is obtained for a positive λ0 that is a solution of . 0sinhcosh 000 =− λλλ
( )
λ
λ
λ
λ
λ
λλ
λ
λλ
coshcoshcosh
cosh
cosh
coshcosh
−≥−=
−
≥
−
= BB
BB
y
y
a
bwe have
We obtained
45. 45
SOLO
Problem of Minimum Surface of Revolution
Solution (continue – 6):
Calculus of Variations - Problems
Solving the Euler-Lagrange Equation
(continue – 3)
We can see that if we don't have a solution to the minimum problem.
0
0
cosh λ
λ
−<
aa
b
y
b
y
y
λ
λ
λ
λ
coshcosh
−=−≥
aa
b
y
b
B
y
y
A solution to the
minimum surface
problem exists if:
0sinhcosh 000 =− λλλ
where λ > λ0 and
λ0 is given by
Let define ( ) a
a
b y
y
b
y
−=
0
0
0
cosh
:
λ
λ
λ
( ) existscatenaryayyIf bb 0λ≥
( ) existstdoesncatenaryayyIf bb '0λ<
( )ayaA ,
x
y
( )bybB ,
( ) ( )22
ydxdsd +=
y x
y
( )bybB ,
( )ayaA ,
Catenary exists
Catenary does
not exists
46. 46
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Calculus of Variations - Problems
Legendre's Condition for a Weak Local Minimum
Let compute
( ) ( )
0
1
2
11
1
2
1
2 3
2
3
2
2
22
≥
+
=
+
−
+
=
+∂
∂
=
∂
∂
=
y
y
y
y
y
y
y
y
y
y
F
y
F yyy
π
ππ
We can see that the Legendre's Necessary Condition for a minimum is satisfied.
47. 47
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Calculus of Variations - Problems
Weierstrass' Necessary Condition for a Strong Local Minimum
Let compute the Weierstrass' Excess Function E
( ) ( ) ( ) ( ) ( )
( )YyyY
y
y
y
yYyy
Yy
y
y
yYyYyyyFyYyyFYyFE y
−++
+
=
+
−++
−+=
+
−−+−+=−−−=
22
22
22
2
2
22
11
1
2
1
1
12
1
112,,,
π
π
π
By adding to both sides of the inequality , we obtain22
Yy 01 22
>++ yY
( ) ( ) 22222222
111 YyyYYyyY >++=+++
By taking the positive square root of this inequality, we obtain YyyY >++ 22
11
Therefore ( ) 011
1
2 22
2
>−++
+
= YyyY
y
y
E
π
We can see that the Weierstrass' Necessary Condition for a strong minimum is satisfied.
48. 48
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Calculus of Variations - Problems
Conjugate Points and Jacobi’s Equation
To have an extremal for x є [a,b] we must show that there are no conjugate points to
A (a, ya) in this interval. This can be shown by finding the non-trivial solution (u(x)≠0)
of the Jacobi’s Equation:
02
2
=
−+
−++ uP
xd
Qd
xd
ud
QQ
xd
Rd
xd
ud
R
T
T
where yyyyyy FRFQFP === :::
2
12 yFy
+= πWe found
we can see that 0:,0: ==== yyyy FQFP
( )
0
1
2
3
2
≥
+
==
y
y
FR yy
π
We also found
constC
y
y
==
+ 2
1 Using in the previous equation we obtain 0
2
1
2
2
3
2
≥=
+
==
y
C
y
C
FR yy
ππ
−
+=
−
==
C
CxCC
C
Cx
CCyC
xd
ud 1
2
1222
2cosh1
2
~
cosh
~~
2
2
3
2
2
~2
0 yC
xd
ud
const
xd
ud
y
C
xd
ud
R
xd
ud
R
xd
d
xd
ud
xd
Rd
xd
ud
R =→==→=
=+
π
Substitute those results in the Jacobi’s Equation
49. 49
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Calculus of Variations - Problems
Conjugate Points and Jacobi’s Equation
−
+=
C
CxCC
xd
ud 1
2
2cosh1
2
~
( ) ( ) ∫
−
+=−
x
a
xd
C
CxCC
auxu 1
2
2cosh1
2
~
( ) ( )
−
−
−
+−=
−
++=
=
C
Ca
C
CxC
ax
CC
C
CxC
x
CC
auxu
x
ax
11
2
1
2
0
2sinh2sinh
22
~
2sinh
22
~
( ) bxa
C
Cax
C
ax
Cax
CC
xu ≤<>
−+
−
+−= 0
2
coshsinh
2
~
1
2
Therefore there are non-trivial solution (u(x)≠0) for a < x ≤ a and there are no
Conjugate Points in this interval.
50. 50
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Calculus of Variations - Problems
Canonical Equations
( ) 2
12:,, yyyyxF += π
Define 2
1
2
:
y
yy
Fp y
+
==
π
from which
( ) 222
22
2
222
2
222222
4
4
1
4
41
py
y
y
py
p
yyypy
−
=+→
−
=→=+
π
π
π
π
Substituting this in equation gives( )yyxF ,, ( ) ( ) 222
22
4
4
,,:,,
~
py
y
yyxFpyxF
yFp
−
== =
π
π
1. The Hamiltonian is
( ) ( ) 222
222
2
222
22
4
44
4
,,:,, py
py
p
py
y
ypyyxFpyxH −−=
−
+
−
−=+−= π
ππ
π
2. The canonical equations are
( )
( )
222
2
222
4
4,,
4
,,
py
y
y
pyxH
xd
pd
py
p
p
pyxH
xd
yd
−
=
∂
∂
−=
−
=
∂
∂
=
π
π
π
51. 51
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Calculus of Variations - Problems
Canonical Equations
The canonical equations are
( )
( )
222
2
222
4
4,,
4
,,
py
y
y
pyxH
xd
pd
py
p
p
pyxH
xd
yd
−
=
∂
∂
−=
−
=
∂
∂
=
π
π
π
If we divide the first equation by the second, we obtain y
p
pd
yd
2
4 π
=
or pdpydy =2
4 π
Integrating this equation gives
22222
4.4 Cconstpy ππ ==−
The reason that we choose the constant as const .= 4 π2
C2
is to recover previous results , as
we shall see. If we substitute this expression in the Hamiltonian expression we obtain:
( ) CpypyxH ππ 24,, 222
−=−−=
The fact that the Hamiltonian is constant on the extremal trajectory is a consequence
of the fact that H is not an explicit function of x.
The canonical equations become
y
Cxd
pd
p
Cxd
yd
π
π
2
2
1
=
=
52. 52
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Calculus of Variations - Problems
Canonical Equations
The canonical equations are
y
Cxd
pd
p
Cxd
yd
π
π
2
2
1
=
=
Let Differentiate first equation with respect to x and use the second, to obtain
y
Cxd
pd
Cxd
yd
22
2
1
2
1
==
π
The general solution of this Ordinary Differential Equation is:
( )
−
=
C
Cx
Cxy 1
cosh
We recovered the “catenary” equation.
53. 53
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Calculus of Variations - Problems
Hamilton-Jacoby Equation
( ) ( ) ( ) 22
2
2
1
2
1
12,,,,,
y
y
y
y
yyyyyxFyyyxFyxS yx
+
=
+
−+=−=
π
π
( ) ( ) p
y
yy
yyxFyxS yy =
+
==
2
1
2
,,,
π
Using in the first equation, we obtainC
y
y
=
+ 2
1
( ) C
y
y
yxSx π
π
2
1
2
,
2
=
+
=
from which we obtain 1
2
2
−
=
C
y
y
Substituting this equation in the Sy (x,y) equation, we obtain:
( ) 12
12
1
2
,
2
2
2
−
=
−
=
+
=
C
y
C
C
y
C
y
y
y
yy
yxSy π
π
π
( ) ( ) 022,,, =−=+ CCSyxHyxS yx ππWe have Hamilton-Jacoby Equation
We find that the Hamiltonian is ( ) CpypyxH ππ 24,, 222
−=−−=
54. 54
SOLO
Problem of Minimum Surface of Revolution
x
y
( )bybB ,
( )ayaA ,
( ) ( )22
ydxdsd +=
y
Calculus of Variations - Problems
Hamilton-Jacoby Equation
−
+=+= yd
C
y
xdCydSxdSSd yx 12
2
π
Integrating this, to obtain:
0
22
0
2
1ln
2
1
2
1
212 S
C
y
C
yC
C
y
yxCSyd
C
y
xdCS +
−
+−−
+=+
−
+= ∫∫ ππ
Using the expressions for Sx (x,y) and Sy (x,y), we obtain
Return to Table of Content
55. 55
SOLO
4. Geodesics Problems
Calculus of Variations - Problems
1. Geodesic in 3 Dimensional Spaces
Suppose we have a surface specified by two parameters
u, v and the vector .( )vur ,
The shortest path lying on the surface and connecting to
points of the surface is called a Geodesic.
A
B
( )vur ,
vdrv
udru
rd
The Shortest Path on a Surface
The arc length differential is
td
td
vd
v
r
v
r
td
vd
td
ud
v
r
u
r
td
ud
u
r
u
r
td
td
vd
v
r
td
ud
u
r
td
vd
v
r
td
ud
u
r
td
td
rd
td
rd
td
td
rd
ds
2/122
2/12/1
2
∂
∂
⋅
∂
∂
+
∂
∂
⋅
∂
∂
+
∂
∂
⋅
∂
∂
=
∂
∂
+
∂
∂
⋅
∂
∂
+
∂
∂
=
⋅==
The length of the path between the two points A and B is:
∫
+
+
==
B
A
r
r
td
td
vd
G
td
vd
td
ud
F
td
ud
ESJ
2/122
2:
where ( ) ( ) ( )
∂
∂
⋅
∂
∂
=
∂
∂
⋅
∂
∂
=
∂
∂
⋅
∂
∂
=
v
r
v
r
vuG
v
r
u
r
vuF
u
r
u
r
vuE
:,,:,,:,
57. 57
SOLO
Geodesics Problems
Calculus of Variations - Problems
1. Geodesic in 3 Dimensional Spaces
If instead of t we use the variable s, we obtain
( ) ( )
( ) ( ) ( ) ( ) ( )
0
,,
2
,,2,
',',,,',',,,
22
'
=
+
−
+
+
=
−
dt
ds
dt
ds
ds
dv
vuF
dt
ds
ds
du
vuE
dt
ds
sd
d
dt
ds
dt
ds
ds
dv
vuG
dt
ds
ds
dv
dt
ds
ds
du
vuF
dt
ds
ds
du
vuE
vuvutF
td
d
vuvutF
uuu
uu
A
B
( )vur ,
vdrv
udru
rd
( ) ( )
( ) ( ) ( ) ( ) ( )
0
,,
2
,,2,
',',,,',',,,
22
'
=
+
−
+
+
=
−
dt
ds
dt
ds
ds
dv
vuG
dt
ds
ds
du
vuF
dt
ds
sd
d
dt
ds
dt
ds
ds
dv
vuG
dt
ds
ds
dv
dt
ds
ds
du
vuF
dt
ds
ds
du
vuE
vuvutF
td
d
vuvutF
vvv
vv
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) 0,,,,2,
2
1
0,,,,2,
2
1
22
22
=
+−
+
+
=
+−
+
+
ds
dv
vuG
ds
du
vuF
sd
d
ds
dv
vuG
ds
dv
ds
du
vuF
ds
du
vuE
ds
dv
vuF
ds
du
vuE
sd
d
ds
dv
vuG
ds
dv
ds
du
vuF
ds
du
vuE
vvv
uuu
or
58. 58
SOLO
Geodesics Problems
Calculus of Variations - Problems
1. Geodesic in 3 Dimensional Spaces
Example: Spherical Surface
A spherical surface is defined by
or ( ) ( )kjiRr ˆcosˆsinsinˆcossin, θφθφθθφ ++=
θ
φφ
φθ
θθ
22
2
sin
0
R
rr
G
rr
F
R
rr
E
=
∂
∂
⋅
∂
∂
=
=
∂
∂
⋅
∂
∂
=
=
∂
∂
⋅
∂
∂
=
( ) ( ) ( ) ( ) ( ) ( ) θ
θ
φ
θφθθφφθθ d
d
d
RddRdGddFdEsd
2
222222
sin1sin2
+=+=++=
θθθφθφθφ cossin20 2
RGGFFEE ======
Let write the Euler-Lagrange Equations for this particular problem
02
2
1
02
2
1
22
22
=
+−
+
+
=
+−
+
+
ds
dv
G
ds
du
F
sd
d
ds
dv
G
ds
dv
ds
du
F
ds
du
E
ds
dv
F
ds
du
E
sd
d
ds
dv
G
ds
dv
ds
du
F
ds
du
E
vvv
uuu
φθθ dRdvdRdu sin, ==
0sin
0cossin
22
2
2
3
2
3
=
−
=−
ds
d
R
sd
d
sd
d
R
sd
d
R
φ
θ
θφ
θθ
φθφθφ cos,sinsin,cossin RzRyRx ===
59. 59
SOLO
Geodesics Problems
Calculus of Variations - Problems
1. Geodesic in 3 Dimensional Spaces
Example: Spherical Surface (continue – 1)
0sin
0cossin
22
2
2
3
2
3
=
−
=−
ds
d
R
sd
d
sd
d
R
sd
d
R
φ
θ
θφ
θθ
From last equation and we obtainθ
θ
φ
θ d
d
d
Rsd
2
2
sin1
+=
constAR
d
d
d
d
R
ds
d
d
d
R
ds
d
R ==
+
==
2
2
2
2222
sin1
sin
sinsin
θ
φ
θ
θ
φ
θ
θ
θ
φ
θ
φ
θ
2
222
2
4
2
2
2
sinsin
sin1
sin
+=
⇒=
+
θ
φ
θ
θ
φ
θ
θ
φ
θ
θ
φ
θ
d
d
AA
d
d
A
d
d
d
d
This equation can be rewritten as
θ
θ
θθθ
φ
2
2
2
22
sin
1sin
sinsin A
A
A
A
d
d
−
±
=
−
±
=
60. 60
SOLO
Geodesics Problems
Calculus of Variations - Problems
1. Geodesic in 3 Dimensional Spaces
Example: Spherical Surface (continue – 2)
We obtain
θ
θ
θ
φ
2
2
2
sin
1sin
A
A
d
d
−
±
=
To solve this equation let write the constant A as the sinus of the constant A = sin α
and define a new independent variable w as w = ctg θ.
θθ
θ
2
2
2
sin
1
1,
sin
=+−= w
d
wdTherefore
( )
( ) α
α
α
α
θ
θ
α
θ
αθ
θ
φφ
2222
2
2
2
2 tan1
tan
sin11
sin
sin
sin
sin
1sin
sin
wwdw
d
d
d
dw
d
−
=
+−
=−
−
±
==
−⇒±
and
ffffffff RzRyRxB
RzRyRxA
φθφθφ
φθφθφ
cos,sinsin,cossin:
cos,sinsin,cossin: 00000000
===
===
The Geodesic must pass through the points A and B given by:
.
tan1
tan
22
constdw
w
d =
−
= α
α
α
φThe Geodesic O,D.E. is
61. 61
SOLO
Geodesics Problems
Calculus of Variations - Problems
1. Geodesic in 3 Dimensional Spaces
Example: Spherical Surface (continue – 2)
We obtained the O. D. E.
The solution is defined by 4 parameters ϕ0, θ0, ϕf. θf that define the initial and
final points A and B.
This satisfies the point A (ϕ0, θ0 ) constraints.
.
tan1
tan
22
constdw
w
d =
−
= α
α
α
φ
( ) ( )
−
+=−+= −−
=
−−
0
11
00
11
0
tan
tan
cos
tan
tan
costancostancos
θ
α
θ
α
φααφφ
θctgw
ww
The O.D.E. can be easily integrated
from which we obtain ( )
θ
α
φφ
tan
tan
cos 0 =−
To simplify the expressions, without affect the solution generality, let choose the direction of
x and y axes, relative to points A and B such that
0
tan
tan
cos
0
1
=
−
θ
α
( ) ff θφφα tancostan 0−=and
62. 62
SOLO
Geodesics Problems
Calculus of Variations - Problems
1. Geodesic in 3 Dimensional Spaces
Example: Spherical Surface (continue –3)
Great Circle
( )
θ
α
φφ
tan
tan
cos 0 =−
Multiply this equation by R sinϕ to obtain
αθφφθφφθ tancossinsinsincoscossin 00 RRR =+
Using , in the previous equation,
we obtain
φθφθφ cos,sinsin,cossin RzRyRx ===
αφφ tansincos 00 zyx =+
This is the equation of a plane passing through x=y=z=0 (center of the sphere).
Thus the geodesic of a spherical surface lies in a
plane passing through the center of the sphere,
therefore a Great Circle.
Return to Table of Content
63. 63
SOLO
Geodesics Problems
Calculus of Variations - Problems
2. Geodesics in Riemannian Space
Riemann Spaces
Let assume a n dimensional space Vn in which we have a m (m , n) dimensional
subspace Vm (Vm submersed in Vn).nm VV ⊂
Let take a point M defined by the vector as function of m independent parameters
x1
, x2
,…,xm
:
r
( ) ( ) ( ) nkexxxexxxxxxr km
kk
mkm
,,2,1,,,,,,,,, 212121
=== ξξ
→
1N
M
C
→
2r
→
1r
2V 3V
→
r
Τ
dt
dr
→
→
1r
→
2r
→
1N
constx =1
Τ
and define the tangent hyper-surface T, defined by the m vectors tangents to C at the
point M:
mi
x
r
r ii ,,2,1,:
=
∂
∂
=
We assume that those vectors are linear independent since the parameters
x1
, x2
,…,xm
are independent.
64. 64
SOLO
Geodesics Problems
Calculus of Variations - Problems
2. Geodesics in Riemannian Space
Riemann Spaces →
1N
M
C
→
2r
→
1r
2V 3V
→
r
Τ
dt
dr
→
→
1r
→
2r
→
1N
constx =1
Τ
m
xxx ,,, 21
m
xxx ,,, 21
Change of Coordinates from to
( ) ( )mm
xxxrxxxr ,,,,,, 2121
=
At each point since the parameters are independent
the Jacobian is nonzero:
( )
( ) 0det
,,,
,,,
1
21
1
2
1
1
1
21
21
≠
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
∂
∂
m
mm
m
m
m
x
x
x
x
x
x
x
x
x
x
x
x
xxx
xxx
We have: j
j
j
j
i
i
i
i
xd
x
r
xd
x
x
x
r
xd
x
r
rd
∂
∂
=
∂
∂
∂
∂
=
∂
∂
=
ji
j
iij
i
i
j
i
j
j
j
mm
i
i
j
m
m
i
i
j
i
i
j
jj
j
i
i
r
x
x
rr
x
x
x
r
x
x
x
r
r
x
x
x
x
xd
x
x
x
x
xd
x
x
xdxd
x
x
xd
∂
∂
=
∂
∂
=
∂
∂
∂
∂
=
∂
∂
=
=
∂
∂
∂
∂
⇒
∂
∂
∂
∂
=
∂
∂
=
∂
∂
=
&:
& δ
65. 65
SOLO
Geodesics Problems
Calculus of Variations - Problems
2. Geodesics in Riemannian Space
Riemann Spaces →
1N
M
C
→
2r
→
1r
2V 3V
→
r
Τ
dt
dr
→
→
1r
→
2r
→
1N
constx =1
Τ
mj
x
r
r
j
j ,...,2,1: =
∂
∂
=
mi
x
r
r
i
i ,...,2,1: =
∂
∂
=
The vectors are also in T since are
given by linear combinations of
Metric in T over the base .
mi
x
r
r
i
i ,...,2,1: =
∂
∂
=
Let assume that a Scalar Product is defined in T and define the Metric in T over the base
asmi
x
r
r
i
i ,...,2,1: =
∂
∂
=
( ) ( ) jiij
ji
jiij grr
x
r
x
r
rrg ==
∂
∂
∂
∂
==
,,,:
Dual Base to .
mi
x
r
r
i
i ,...,2,1: =
∂
∂
=
Since are linear independent; i.e.:miri ,...,2,1=
{ }
( ) ( ){ } { } 0det,...,10,,0
,...,100
≠=⇔=∀=⇔===→
→=∀=⇔=
ij
i
ij
i
ji
i
ji
i
i
i
i
ggmigrrrr
mir
αααα
αα
66. 66
SOLO
Geodesics Problems
Calculus of Variations - Problems
2. Geodesics in Riemannian Space
Riemann Spaces →
1N
M
C
→
2r
→
1r
2V 3V
→
r
Τ
dt
dr
→
→
1r
→
2r
→
1N
constx =1
Τ
the mxm matrix is nonsingular and we can
define the matrix:
{ }ijgG =
{ } { } { }{ } mxmij
ij
ij
ij
IgggGg =→==
−−
∆
11
i
jkj
ik
gg δ=or
Let define now: mi
x
r
grgr
k
ik
k
iki
,...,2,1: ==
∂
∂
==
We can see that ( ) ( ) ( ) i
jkj
ik
jk
ik
jk
ik
j
i
ggrrgrrgrr δ====
,,,
mir i
,...,2,1=
miri ,...,2,1=
Therefore are dual to .
Moreover
( ) ( ) ( ) ijj
k
ikj
k
ikj
k
ikji
ggrrgrrgrr ==== δ
,,,
gik
is the Metric of the dual base .mir i
,...,2,1=
67. 67
SOLO
Geodesics Problems
Calculus of Variations - Problems
2. Geodesics in Riemannian Space
In a Riemann space the length of a space curve, defined by it’s
coordinates xi (t), as a function of a parameter t, is given by
∫
=
1
0
2/1t
t
dt
td
xd
td
xd
gs βα
βα
where we used the tensor index summation rule
∑∑= =
⇔
m m
td
xd
td
xd
g
td
xd
td
xd
g
1 1α β
βα
βα
βα
βα
mi
x
r
r
i
i ,...,2,1: =
∂
∂
=
( ) ( ) jiij
ji
jiij grr
x
r
x
r
rrg ==
∂
∂
∂
∂
==
,,,:
Define m
td
sd
td
xd
td
xd
g
td
xd
xtF ,,1,:,,
2/1
==
=
βαβα
βα
To find the geodesics (the curve of the minimum length) let apply the Euler-Lagrange
Equations
mi
x
F
x
F
td
d
ii
,,10
==
∂
∂
−
∂
∂
→
1N
M
C
→
2r
→
1r
2V 3V
→
r
Τ
dt
dr
→
→
1r
→
2r
→
1N
constx =1
Τ
68. 68
SOLO
Geodesics Problems
Calculus of Variations - Problems
2. Geodesics in Riemannian Space
m
td
sd
td
xd
td
xd
g
td
xd
xtF ,,1,:,,
2/1
==
=
βαβα
βα
1. To find the geodesics (the curve of the minimum
length) let apply the Euler-Lagrange Equations
mi
x
F
x
F
td
d
ii
,,10
==
∂
∂
−
∂
∂
→
1N
M
C
→
2r
→
1r
2V 3V
→
r
Τ
dt
dr
→
→
1r
→
2r
→
1N
constx =1
Τ
( )
mi
td
xd
g
td
xd
g
td
sd
dtdstd
xd
td
xd
x
g
td
xd
td
xd
x
g
td
xd
g
td
xd
g
dtds
dtds
td
xd
g
td
xd
g
dt
d
x
F
dt
d
ii
ii
ii
ii
i
,,1
/2
1
/2
1
/2
2
2
22
2
2
2
=
+−
∂
∂
+
∂
∂
++=
+
=
∂
∂
β
β
α
α
βα
α
ββα
β
αβ
β
α
α
β
β
α
α
mi
td
xd
td
xd
x
g
dtdsx
F
ii
,,1
/2
1
=
∂
∂
=
∂
∂ βαβα
We have
69. 69
SOLO
Geodesics Problems
Calculus of Variations - Problems
2. Geodesics in Riemannian Space
→
1N
M
C
→
2r
→
1r
2V 3V
→
r
Τ
dt
dr
→
→
1r
→
2r
→
1N
constx =1
Τ
If we choose 0,1 2
2
==→=
td
sd
dt
ds
st
mi
sd
xd
sd
xd
x
g
x
F
ii
,,1
2
1
=
∂
∂
=
∂
∂ βαβα
mi
sd
xd
sd
xd
x
g
x
g
sd
xd
g
sd
xd
sd
xd
x
g
sd
xd
sd
xd
x
g
sd
xd
g
sd
xd
g
x
F
ds
d
ii
i
ii
ii
i
,,1
2
1
2
1
2
2
2
2
2
2
=
∂
∂
+
∂
∂
+=
∂
∂
+
∂
∂
++=
∂
∂
βα
α
β
β
αα
α
βα
α
ββα
β
αβ
β
α
α
Therefore we obtain: mi
sd
xd
sd
xd
x
g
x
g
x
g
sd
xd
g
i
ii
i ,,1
2
1
2
2
=
∂
∂
−
∂
∂
+
∂
∂
+
βαβα
α
β
β
αα
α
Multiplying by gri
and summing on i, we obtain:
mi
sd
xd
sd
xd
x
g
x
g
x
gg
sd
xd
i
ii
ir
r
,,1
22
2
=
∂
∂
−
∂
∂
+
∂
∂
+
βαβα
α
β
β
α
We define
∂
∂
−
∂
∂
+
∂
∂
=Γ
i
ii
ir
r
x
g
x
g
x
gg βα
α
β
β
α
βα
2
: Christoffel Symbol of Second Kind
mi
sd
xd
sd
xd
sd
xd rr
,,12
2
=Γ+ βα
βα Differential Equations of the Geodesics
Return to Table of Content
70. 70
SOLO
Geodesics Problems
Calculus of Variations - Problems
3. Geodesics for an Implicit Equation of the Surface
Suppose that we want to find the geodesics on a surface defined by an implicit function:
( ) 0,, =zyxG
We want to find the extremal of ∫
+
+
=
ft
t
dt
td
zd
td
yd
td
xd
s
0
222
Let adjoin the first equation to the second using the Lagrange multiplier μ (t).
( ) ( )∫
+
+
+
=
ft
t
dtzyxGt
td
zd
td
yd
td
xd
s
0
,,
222
µ
Define ( ) ( )zyxGt
td
zd
td
yd
td
xd
td
zd
td
yd
td
xd
zyxtF ,,:,,,,,,
222
µ+
+
+
=
dt
ds
td
zd
td
yd
td
xd
f =
+
+
=
222
:and
71. 71
SOLO
Geodesics Problems
Calculus of Variations - Problems
3. Geodesics for an Implicit Equation of the Surface
The Euler-Lagrange Equations are:
( )
( )
( ) 0
1
0
1
0
1
=
∂
∂
−
=
∂
∂
−
=
∂
∂
−
z
G
t
td
zd
ftd
d
y
G
t
td
yd
ftd
d
x
G
t
td
xd
ftd
d
µ
µ
µ
( ) ( )zyxGt
td
zd
td
yd
td
xd
f
td
zd
td
yd
td
xd
zyxtF ,,,,:,,,,,, µ+
=
dt
ds
td
zd
td
yd
td
xd
f =
+
+
=
222
:
μ (t) can be eliminated to obtain ( )t
z
G
td
zd
ftd
d
y
G
td
yd
ftd
d
x
G
td
xd
ftd
d
µ=
∂
∂
=
∂
∂
=
∂
∂
111
Using f = ds/dt we can write the last expression as
( )t
z
G
td
zd
sd
td
sd
d
td
sd
y
G
td
yd
sd
td
sd
d
td
sd
x
G
td
xd
sd
td
sd
d
td
sd
µ=
∂
∂
=
∂
∂
=
∂
∂
( )
td
sd
t
z
G
sd
zd
sd
d
y
G
sd
yd
sd
d
x
G
sd
xd
sd
d
µ
=
∂
∂
=
∂
∂
=
∂
∂
or
72. 72
SOLO
Geodesics Problems
Calculus of Variations - Problems
3. Geodesics for an Implicit Equation of the Surface
( )
td
sd
t
z
G
sd
zd
sd
d
y
G
sd
yd
sd
d
x
G
sd
xd
sd
d
µ
=
∂
∂
=
∂
∂
=
∂
∂
We found
From Differential Geometry we know that for a three dimensional curve
( ) ( ) ( )kszjsyisxr ˆˆˆ ++=
we have ( ) ( ) ( ) k
sd
szd
j
sd
syd
i
sd
sxd
sd
rd
t ˆˆˆ: ++==
the unit vector tangent to the curve
( ) ( ) ( ) nk
sd
szd
sd
d
j
sd
syd
sd
d
i
sd
sxd
sd
d
sd
rd
sd
d
sd
td
ˆˆˆˆ κ=
+
+
=
=
is the unit vector that defines the principal normal to the curve and κ is the
magnitude of the curvature. Since is a unit vector
nˆ
tˆ
0
ˆ
ˆˆˆ =⋅=⋅
sd
td
tnt
We also have k
z
G
j
y
G
i
x
G
G ˆˆˆ
∂
∂
+
∂
∂
+
∂
∂
=∇
We can see that the principal normal to any point of the geodesic curve is parallel to
the normal to the surface.
73. 73
SOLO
Geodesics Problems
Calculus of Variations - Problems
3. Geodesics for an Implicit Equation of the Surface
Example: Geodesics on a Spherical Surface
( ) 0,, 2222
=−++= RzyxzyxGSpherical Surface:
kzjyixk
z
G
j
y
G
i
x
G
G ˆ2ˆ2ˆ2ˆˆˆ ++=
∂
∂
+
∂
∂
+
∂
∂
=∇
Let differentiate twice G (x,y,z) = 0 as a function of s
0=++
sd
zd
z
sd
yd
y
sd
xd
x ( ) 0,, =zyxG
sd
d
( ) 0,,2
2
=zyxG
sd
d02
2
2
2
2
2222
=+++
+
+
sd
zd
z
sd
yd
y
sd
xd
x
sd
zd
sd
yd
sd
xd
1
222
=
+
+
sd
zd
sd
yd
sd
xd
dt
ds
td
zd
td
yd
td
xd
f =
+
+
=
222
:
dsdt =
( )
td
sd
t
z
G
sd
zd
sd
d
y
G
sd
yd
sd
d
x
G
sd
xd
sd
d
µ
=
∂
∂
=
∂
∂
=
∂
∂
( )tk
z
sd
zd
y
sd
yd
x
sd
xd
===
222
2
2
2
2
2
2
( ) ( ) ( ) ( ) 2
2222
2
1
02121
R
tkRtkzyxtk −=→=+=+++
74. 74
SOLO
Geodesics Problems
Calculus of Variations - Problems
3. Geodesics for an Implicit Equation of the Surface
Example: Geodesics on a Spherical Surface
( ) 2
2
2
2
2
2
2
2
1
222 R
tk
z
sd
zd
y
sd
yd
x
sd
xd
−====We found
+
−
+
−
+
−
R
s
C
R
s
Cz
R
s
C
R
s
Cy
R
s
C
R
s
Cx
cossin
cossin
cossin
65
43
21
The solutions of those equation are
0
cos
sin
1
65
43
21
=
−
R
s
R
s
CCz
CCy
CCx
0
0
0
22
2
22
2
22
2
=+
=+
=+
R
z
sd
zd
R
y
sd
yd
R
x
sd
xd
75. 75
SOLO
Geodesics Problems
Calculus of Variations - Problems
3. Geodesics for an Implicit Equation of the Surface
Example: Geodesics on a Spherical Surface
0
cos
sin
1
65
43
21
=
−
R
s
R
s
CCz
CCy
CCx
We found
The solution exists only if
0det
65
43
21
=
CCz
CCy
CCx
0=++ CzByAx
Thus the Geodesic on a Spherical Surface lies in a plane passing through the
center of the sphere, therefore a Great Circle.
Great Circle
Return to Table of Content
76. 76
SOLO
Geometrical Optics and Fermat Principle
The Principle of Fermat (principle of the shortest optical path) asserts that the optical
length
of an actual ray between any two points is shorter than the optical ray of any other
curve that joints these two points and which is in a certain neighborhood of it.
An other formulation of the Fermat’s Principle requires only Stationarity (instead of
minimal length).
∫
2
1
P
P
dsn
An other form of the Fermat’s Principle is:
Principle of Least Time
The path following by a ray in going from one point in
space to another is the path that makes the time of transit of
the associated wave stationary (usually a minimum).
Calculus of Variations - Problems
77. 77
SOLO
We have:
( ) ( ) ( )∫∫∫∫ =
+
+===
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 ++=
+
+=
constS =
constdSS =+
sˆ
∫
2
1
P
P
dsn
1
P
2
P
Paths of Rays Between Two Points
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
78. 78
SOLO
Necessary Conditions for Stationarity (Euler-Lagrange Equations)
( ) ( ) ( ) 22
22
1,,1,,,,,, zyzyxn
xd
zd
xd
yd
zyxnzyzyxF ++=
+
+=
0=
∂
∂
−
∂
∂
y
F
y
F
dx
d
( )
[ ] 2/122
1
,,
zy
yzyxn
y
F
++
=
∂
∂ [ ] ( )
y
zyxn
zy
y
F
∂
∂
++=
∂
∂ ,,
1 2/122
( )
[ ]
[ ] 01
1
,, 2/122
2/122
=
∂
∂
++−
++ y
n
zy
zy
yzyxn
xd
d
0=
∂
∂
−
∂
∂
z
F
z
F
dx
d
[ ] [ ]
0
11
2/1222/122
=
∂
∂
−
++++ y
n
zy
yn
xdzy
d
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
79. 79
SOLO
Necessary Conditions for Stationarity (continue - 1)
We have
[ ]
0
1
2/122
=
∂
∂
−
++ y
n
zy
yn
sd
d
y
n
sd
yd
n
sd
d
∂
∂
=
In the same way
[ ]
0
1
2/122
=
∂
∂
−
++ z
n
zy
zn
sd
d
z
n
sd
zd
n
sd
d
∂
∂
=
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
80. 80
SOLO
Necessary Conditions for Stationarity (continue - 2)
Using ( ) ( ) ( ) xd
xd
zd
xd
yd
zdydxdds
22
222
1
+
+=++=
we obtain 1
222
=
+
+
sd
zd
sd
yd
sd
xd
Differentiate this equation with respect to s and multiply by n
sd
d
0=
+
+
sd
zd
sd
d
n
sd
zd
sd
yd
sd
d
n
sd
yd
sd
xd
sd
d
n
sd
xd
sd
nd
sd
zd
sd
nd
sd
yd
sd
nd
sd
xd
sd
nd
=
+
+
222
sd
nd
and
sd
nd
sd
zd
n
sd
d
sd
zd
sd
yd
n
sd
d
sd
yd
sd
xd
n
sd
d
sd
xd
=
+
+
add those two equations
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
81. 81
SOLO
Necessary Conditions for Stationarity (continue - 3)
sd
nd
sd
zd
n
sd
d
sd
zd
sd
yd
n
sd
d
sd
yd
sd
xd
n
sd
d
sd
xd
=
+
+
Multiply this by and use the fact that to obtain
xd
sd
cd
ad
cd
bd
bd
ad
=
xd
nd
sd
zd
n
sd
d
xd
zd
sd
yd
n
sd
d
xd
yd
sd
xd
n
sd
d
=
+
+
Substitute and in this equation to obtain
y
n
sd
yd
n
sd
d
∂
∂
=
z
n
sd
zd
n
sd
d
∂
∂
=
xd
zd
z
n
xd
yd
y
n
xd
nd
sd
xd
n
sd
d
∂
∂
−
∂
∂
−=
Since n is a function of x, y, z
x
n
xd
zd
z
n
xd
yd
y
n
xd
nd
zd
z
n
yd
y
n
xd
x
n
nd
∂
∂
=
∂
∂
−
∂
∂
−→
∂
∂
+
∂
∂
+
∂
∂
=
and the previous equation becomes
x
n
sd
xd
n
sd
d
∂
∂
=
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
82. 82
SOLO
Necessary Conditions for Stationarity (continue - 4)
We obtained the Euler-Lagrange Equations:
x
n
sd
xd
n
sd
d
∂
∂
=
y
n
sd
yd
n
sd
d
∂
∂
=
z
n
sd
zd
n
sd
d
∂
∂
=
k
sd
zd
j
sd
yd
i
sd
xd
sd
rd
kzjyixr
ˆˆˆ
ˆˆˆ
++=
++=
Define the unit vectors in the x, y, z directionskji ˆ,ˆ,ˆ
The Euler-Lagrange Equations can be written as:
n
sd
rd
n
sd
d
∇=
This is the Eikonal Equation from Geometrical Optics.
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
83. 83
SOLO
Transversality Conditions for Geometrical Optics and Fermat’s Principle
( ) ( ) ( ) 22
22
1,,1,,,,,, zyzyxn
xd
zd
xd
yd
zyxnzyzyxF ++=
+
+=
For the Geometrical Optics we obtained:
Assume that the initial and final boundaries are defined by the surfaces A (x0, y0, z0) and
B (xf, yf, zf) respectively. The transversality conditions at the boundaries i=0,f are defined by
( ) ( ) ( )[ ]
( ) ( ) 0,,,,,,,,
,,,,,,,,,,,,
=++
−−
iziy
izy
dzzyzyxFdyzyzyxF
dxzyzyxFzzyzyxFyzyzyxF
( ) [ ]
[ ] [ ]
[ ]
( )
sd
xd
zyxn
zy
n
zy
zn
z
zy
yn
yzynFzFyzyzyxF zy
,,
1
11
1,,,,
2/122
2/1222/122
2/122
=
++
=
++
−
++
−++=−−
( )
[ ]
( )
( )
[ ]
( )
sd
zd
zyxn
zy
zzyxn
z
F
F
sd
yd
zyxn
zy
yzyxn
y
F
F
z
y
,,
1
,,
,,
1
,,
2/122
2/122
=
++
=
∂
∂
=
=
++
=
∂
∂
=
For are tangent to the boundary surfaces A (x0, y0, z0) and B (xf, yf, zf).fird i ,0=
From Transversality Conditions we can see that the rays are normal (transversal) to the
boundary surfaces (see Figure).
Transversality Conditions
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
84. SOLO
Corner Conditions for Geometrical Optics and Fermat’s Principle
( ) ( ) ( ) 22
22
1,,1,,,,,, zyzyxn
xd
zd
xd
yd
zyxnzyzyxF ++=
+
+=
For the Geometrical Optics we obtained:
Let examine the following two cases:
1. The optical path passes between two regions with different refractive indexes n1 to n2
(see Figure)
In region (1) we have:
In region (2) we have:
( ) ( ) 22
11 1,,,,,, zyzyxnzyzyxF ++=
( ) ( ) 22
22 1,,,,,, zyzyxnzyzyxF ++=
( ) ( ) ( )[ ]{
( ) ( ) ( )[ ]}
( ) ( )[ ]
( ) ( )[ ] 0,,,,,,,,
,,,,,,,,
,,,,,,,,,,,,
,,,,,,,,,,,,
222111
222111
22222222222
11111111111
=−+
−+
−−−
−−
dzzyzyxFzyzyxF
dyzyzyxFzyzyxF
dxzyzyxFzzyzyxFyzyzyxF
zyzyxFzzyzyxFyzyzyxF
zz
yy
zy
zy
The Weierstrass-Erdmann necessary condition
at the boundary between the two regions is
where dx, dy, dz are on the boundary between the two regions.
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
85. SOLO
Corner Conditions for Geometrical Optics and Fermat’s Principle (continue – 1)
( ) [ ]
[ ] [ ]
[ ]
( )
sd
xd
zyxn
zy
n
zy
zn
z
zy
yn
yzynFzFyzyzyxF zy
,,
1
11
1,,,,
2/122
2/1222/122
2/122
=
++
=
++
−
++
−++=−−
( )
[ ]
( )
( )
[ ]
( )
sd
zd
zyxn
zy
zzyxn
z
F
F
sd
yd
zyxn
zy
yzyxn
y
F
F
z
y
,,
1
,,
,,
1
,,
2/122
2/122
=
++
=
∂
∂
=
=
++
=
∂
∂
=
( ) ( )
0
21
21 =⋅
− rd
sd
rd
n
sd
rd
n
rayray
where is on the boundary between the two regions andrd
( ) ( )
sd
rd
s
sd
rd
s
rayray 2
:ˆ,
1
:ˆ 21
==
are the unit vectors in the direction of propagation of the rays.
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
86. SOLO
Corner Conditions for Geometrical Optics and Fermat’s Principle (continue – 2)
( ) 0ˆˆ 2211 =⋅− rdsnsn
2211
ˆˆ snsn −Therefore is normal to .rd
Since can be in any direction on the
boundary between the two regions (see Figure )
is parallel to the unit vector
normal to the boundary surface, and we have
rd
2211
ˆˆ snsn − 21
ˆ −n
( ) 0ˆˆˆ 221121 =−×− snsnn
This the Snell’s Law of Geometrical Optics
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
87. SOLO
Corner Conditions for Geometrical Optics and Fermat’s Principle (continue – 3)
2. The optical path is reflected at the boundary.
( ) ( )
( ) 0ˆˆ
21
21 =⋅−=⋅
− rdssrd
sd
rd
sd
rd rayray
n1 = n2 , we obtain
i.e. is normal to , i.e. to the boundary where the
reflection occurs.
Also we can write
21
ˆˆ ss − rd
( ) 0ˆˆˆ 2121 =−×− ssn
( ) ( )
( ) 0ˆˆ
21
221121 =⋅−=⋅
− rdsnsnrd
sd
rd
n
sd
rd
n
rayray
In this case, if we substitute in the equation
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
88. 88
SOLO
Hilbert’s Invariant Integral
David Hilbert
(1862 – 1943)
Geometrical Optics and Fermat’s Principle
( ) ( ) ( ) 22
22
1,,1,,,,,, zyzyxn
xd
zd
xd
yd
zyxnzyzyxF ++=
+
+=
For the Geometrical Optics we obtained:
The Hilbert’s Invariant Integral is
( ) ( )( ) ( ) ( )[ ] ( ) ( )( ){
( )
( )
( ) ( )[ ] ( ) ( )( )} xdzyxzzyxyzyxFzyxZzyxz
zyxzzyxyzyxFzyxYzyxyzyxzzyxyzyxF
z
zyxP
zyxP
yC
ffff
,,,,,,,,,,,,
,,,,,,,,,,,,,,,,,,,,
,,
,, 0000
−−
∫ −−
This is known as Hilbert’s Invariant Integral because it is invariant on the path C as long
as this curve remains in the field of the unique extremal solution.
( ) ( ) ( ) ( )zyx
x
z
zyxzzyx
x
y
zyxy ,,,,,,,,,
∂
∂
=
∂
∂
= is the field slope and
( ) ( )
CC
x
z
zyxZ
x
y
zyxY
∂
∂
=
∂
∂
= :,,,:,, is the path C slope at the point (x,y,z) of C
we have on path C ( ) ( ) dx
x
z
dxzyxZzddx
x
y
dxzyxYyd
C
C
C
C
∂
∂
==
∂
∂
== ,,,,,
Calculus of Variations - Problems
89. 89
SOLO
Hilbert’s Invariant Integral (continue – 1)
David Hilbert
(1862 – 1943)
( ) ( ) ( )[ ]{
( )
( )
( ) ( ) }zdzyzyxFydzyzyxFxdzyzyxFzzyzyxFyzyzyxF zy
zyxP
zyxP
zyC
ffff
,,,,,,,,,,,,,,,,,,,,
,,
,, 0000
−−−−∫
The Hilbert’s Invariant Integral is
We can write
( ) [ ]
[ ] [ ] [ ]
( )
sd
xd
zyxn
zy
n
zy
zn
z
zy
yn
yzynFzFyzyzyxF zy ,,
111
1,,,, 2/1222/1222/122
2/122
=
++
=
++
−
++
−++=−−
( )
[ ]
( )
( )
[ ]
( )
sd
zd
zyxn
zy
zzyxn
z
F
F
sd
yd
zyxn
zy
yzyxn
y
F
F
z
y
,,
1
,,
,,
1
,,
2/122
2/122
=
++
=
∂
∂
=
=
++
=
∂
∂
=
Now we can write the Hilbert’s Invariant Integral as
( )
( )
( )
( )
∫∫ ⋅=⋅
ffffffff zyxP
zyxP
zyxP
zyxP
ray
rdsnrd
sd
rd
n
,,
,,
,,
,, 1000010000
ˆ
This is the Lagrange’s Invariant Integral from
Geometrical Optics.
Joseph-Louis Lagrange
(1736-1813)
Integration Path
through a Ray Bundle
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
90. 90
SOLO
Second Order Conditions: Legendre’s Condition for a Weak Minimum
Adrien-Marie Legendre
1752-1833
From
( )
[ ]
( )
sd
yd
zyxn
zy
yzyxn
y
F
Fy ,,
1
,,
2/122
=
++
=
∂
∂
=
we obtain
( )
[ ] [ ] [ ]
( )
[ ] 2/322
2
2/322
2
2/1222/1222
2
1
1
111
,,
zy
zn
zy
yn
zy
n
zy
yzyxn
yy
F
++
+
=
++
−
++
=
++∂
∂
=
∂
∂
From
we obtain
( )
[ ] [ ] 2/3222/122
2
11
,,
zy
zyn
zy
zzyxn
yzy
F
++
−=
++∂
∂
=
∂∂
∂
( )
[ ]
( )
sd
zd
zyxn
zy
zzyxn
z
F
Fz ,,
1
,,
2/122
=
++
=
∂
∂
=
( )
[ ]
( )
[ ] 2/322
2
2/1222
2
1
1
1
,,
zy
yn
zy
zzyxn
zz
F
++
+
=
++∂
∂
=
∂
∂
From those equations we obtain
( )
[ ]
( )
( )
+−
−+
++
=
∂
∂
∂∂
∂
∂∂
∂
∂
∂
= 2
2
2/322
2
22
2
2
2
''
1''
1
1
,,
yyx
zyz
zy
zyxn
z
F
yz
F
zy
F
y
F
F XX
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
91. 91
SOLO
Second Order Conditions: Legendre’s Condition for a Weak Minimum (continue – 1)
( )
[ ]
( )
( )
+−
−+
++
=
∂
∂
∂∂
∂
∂∂
∂
∂
∂
= 2
2
2/322
2
22
2
2
2
''
1''
1
1
,,
yyx
zyz
zy
zyxn
z
F
yz
F
zy
F
y
F
F XX
Let use Sylvester’s Theorem to check if/when is positive
definite (i.e. check that the determinants of all principal minors
are positive)
''XXF
James Joseph
Sylvester
1814-1897
[ ]
( )
( )
+−
−+
++
= 2
2
2/322
''
1''
1
det
1
det
yyx
zyz
zy
n
F XX
[ ]
( )( )[ ]
[ ]
0
1
11
1
2/122
2222
2/322
>
++
=−++
++
=
zy
n
zyyz
zy
n
( ) 01 2
>+ z
According to Sylvester’s Theorem is positive definite''XXF ( )0'' >XXF
According to Legendre’s Condition ,if the Jacobi’s Condition is satisfied
(no conjugate points between P1 and P2), every extremal is a weak minimum.
( )0'' >XXF
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
92. SOLO
The Weierstrass Necessary Condition for a Strong Minimum (Maximum)
( ) ( ) ( ) 22
22
1,,1,,,,,, zyzyxn
xd
zd
xd
yd
zyxnzyzyxF ++=
+
+=
For the Geometrical Optics we obtained:
Weierstrass E Function is defined as
( ) ( ) ( ) ( ) ( ) ( ) ( )zyzyxFzZzyzyxFyYzyzyxFZYzyzyxFZYzyzyxE zy ,,,,,,,,,,,,,,,,,,:,,,,,, −−−−−=
[ ] [ ] ( )
[ ]
( )
[ ]
[ ]
[ ]
( ) ( )[ ]
[ ]
[ ]
( )
[ ]
[ ] [ ]
( )InequalitySchwarz
zyZY
zZyY
ZYn
zZyY
zy
n
ZYn
zzZyyYzy
zy
n
ZYn
zy
zn
zZ
zy
yn
yYzynZYn
0
11
1
11
1
1
1
1
1
1
1
''
1
11
2/1222/122
2/122
2/122
2/122
22
2/122
2/122
2/1222/122
2/1222/122
≥
++++
++
−++=
++
++
−++=
−+−+++
++
−++=
++
−−
++
−−++−++=
According to Weierstrass Condition if the Jacobi Condition
(no conjugate points between and ) is satisfied every extremal is a strong minimum.
( )( )0',',',',',',,, ≥ZYXzyxzyxE
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
93. 93
SOLO
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
+−
=
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
94. 94
SOLO
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
−=+−−=
+−
+
+−
+
+−
−=
++−=
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
95. 95
SOLO
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
+−
∂
∂
−=
∂
∂
−=
+−
∂
∂
−=
∂
∂
−=
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
96. 96
SOLO
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
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
97. 97
SOLO
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
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
98. 98
William Rowan Hamilton (1805-1855)
Canonical Equations of Motion 1835
where H is the Hamiltonian defined as:
Hamilton-Jacobi Theory
SOLO
CALCULUS OF VARIATIONS
William Rowan
Hamilton
1805-1855
Carl Gustav
Jacob Jacobi
1804-1851
Hamilton-Jacobi Equation
99. 99
SOLO
Hamilton-Jacobi Equation (continue – 1)
( ) ( ) ( ) [ ]
( )
[ ]
( )
[ ]
( )
[ ]
( )
sd
xd
zyxn
zy
zyxn
zy
zzyxn
z
zy
yzyxn
y
zyzyxnFzFyzyzyxFzyxS zyx
,,
1
,,
1
,,
1
,,
1,,,,,,,,
2/1222/1222/122
2/122
=
++
=
++
−
++
−
++=−−=
( ) ( )
[ ]
( )
sd
yd
zyxnp
zy
yzyxn
y
F
zyxS yy ,,:
1
,,
,, 2/122
==
++
=
∂
∂
=
( ) ( )
[ ]
( )
sd
zd
zyxnp
zy
zzyxn
z
F
zyxS zz ,,:
1
,,
,, 2/122
==
++
=
∂
∂
=
We obtain
sn
sd
rd
nS ray
ˆ==∇
From this
22
ˆˆ nssnSS =⋅=∇⋅∇
We recovered again the Eikonal Equation
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
101. 101
SOLO
If the regularity condition (optical ray not intersecting) doesn’t hold, the optical ray
may not be a minimum, as we can see from the Figure, where the optical ray reflected
from the planar mirror and reaches the point P2 (P1MP2) is longer than the direct ray
from P1 to P2.
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
102. 102
SOLO
On other example is given in Figure bellow on the rays from a point source refracted
by a lens. The refracted rays form an envelope called caustic. The point P’2 where the
refracted ray touches the caustic is called a conjugate point. From the Figure we can
see that this point is reached by, at least, two rays with different optical paths.
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
103. 103
SOLO
Example of the stationarity of the Fermat’s Principle
Suppose that we have a elliptical mirror and a point source locate at one of it’s foci P1.
The elliptical mirror has the following properties:
1. The sum of the distances from the two foci to any point R on the ellipse is
constant.
2121 PRRPPRRP EE +=+
2. The normal at any point R on the ellipse bisects the angle P1RP2.
2P
1P
Point
Source
Elliptic
Mirror
RER
Rnˆ
ERsˆ
Rsˆ
According to Snell’s Law, all the rays
originated at the focus P1 will be reflected
by the elliptical mirror and intersect at the
second foci P2.
Since the rays travel in the same media
and the geometrical paths are equal, the
optical paths will be equal also.
( ) ( )2121 PRRPnPRRPn EE +=+
Since all the optical paths reflected by the mirror reach the point P2, we call P2
the conjugate point to P1.
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
104. 104
SOLO
Example of the stationarity of the Fermat’s Principle (continue – 1)
Now replace the elliptical mirror with a planar one normal to at the point R.Rnˆ
2P
1P
Point
Source
Planar
Mirror
Elliptic
Mirror
RER
PR
Rnˆ
ERsˆ
PRsˆ
Rsˆ
For this reason the ray will be reflected at R and reach the point P2, in the same
way as for the elliptical mirror.
RP1
From the Figure we can see that:
( ) ( ) 222121 PRRRPRPRRRPnPRRPn PPEEPPE +<←+<+
In this case the Fermat’s Principle will
give a minimum for the optical path
( )21 PRRPn +
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
105. 105
SOLO
Example of the stationarity of the Fermat’s Principle (continue – 3)
Now replace the elliptical mirror with a circular one normal to at the point R
(The mirror diameter is smaller than the maximum axis of the ellipse).
Rnˆ
For this reason the ray will be reflected at R and reach the point P2, in the same
way as for the elliptical mirror.
RP1
From the Figure we can see that:
In this case the Fermat’s Principle will
give a maximum for the optical path
( )21 PRRPn +
2P
1P
Point
Source
Elliptic
Mirror
Circular
Mirror
R
CR
ER
RnˆCRsˆ
ERsˆ
PRsˆ
Rsˆ
( ) ( ) 222121 PRRRPRPRRPnPRRPn EECCCC +<←+>+
Geometrical Optics and Fermat Principle
Calculus of Variations - Problems
Return to Table of Content
106. 106
SOLO Calculus of Variations - Problems
References
Bryson & Ho “Applied Optimal Control”, Ginn and Company, 1969,
Sec 3.1, pg. 91-95
George Leitmann “The Calculus of Variations and Optimal Control –
An Introduction”, Plenum Press, 1981
O. Bolza, “Lectures on the Calculus of Variations”, Dover Publications, New York,
1961, Republication of a work published by Univ. of Chicago 1904,
pp.1,27-28,41,48-49,64, 77-78, 153
W.S. Kimball, “Calculus of Variations, by Parallel Displacement”, Butterworths
Scientific Publications, 1952, § 13, pp.432-475
L.E. Elsgolc, , “Calculus of Variations”, Pergamon Press, Addison-Wesley, 1962,
pp.37-38
I.M. Gelfand, S.V. Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp.20-21
H. Sagan, “Introduction to Calculus of Variations”, Dover Publication, New York, 1969,
pp.62-65
H. Tolle, “Optimization Methods”, Springer Verlag, Berlin Heidelberg New York, 1975,
pp.15-18
107. 107
SOLO Calculus of Variations - Problems
References
S. Hermelin, “Calculus of Variations”, http://www.solohermelin.com
Math Folder
S. Hermelin, “Foundation of Geometrical Optics”, http://www.solohermelin.com
Optics Folder
Return to Table of Content
108. February 23, 2015 108
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA