USSC3002 Oscillations and Waves
Lecture 10 Calculus of Variations
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected]
http://www.math.nus/~matwml
Tel (65) 6874-2749
1
ROLLE’S THEOREM
If f : [a,b] R is continuous,differentiable in (a,b) and
f(a) = f(b), then  c  ( a, b) such that f ' (c)  0.
Proof First, since f and continuous on [a,b] and
[a,b] is both closed and bounded, there exists
xm , xM [a, b] such that
f ( xm )  min f ([a, b]), f ( xM )  max f ([a, b])
If neither point belongs to (a,b) then f is constant and
every choice of c in (a,b) satisfies f ' (c)  0.
If xm  (a, b) then for every h  (a  xm , b  xm ) \ {0}
f ( xm  h )  f ( xm )
Qh 
satisfies h  0  Qh  0, h  0  Qh  0
h
hence f ' ( xm )  lim h0 Qh  0. Q1. If
xM  (a, b) ?
2
STATIONARY POINTS
Definition A point c in the interior of the domain of a
function f is a stationary point (of f) if f ' (c)  0.
The proof of Rolle’s Theorem makes use the fact
that if f achieves a local extremum (min or max) at c
then c is a stationary point of f. The following example
shows that the converse is not true:
Example f : [1,1]  R, f ( x)  x has a stationary
point c  0 but c is not a local extremum.
3
Q2. Does f have a minimum and maximum ?
Are they stationary points ?
3
LINEAR FUNCTIONALS
Definition Let V be a vector space over the field R.
A function F : V  R is called a functional and F is
called a linear functional if it is linear.
Example 1. Let V  R , ( x, y)  i 1 xi yi and for
y  V define Fy : V  R by Fy ( x)  ( x, y ), x V .
Then Fy is a linear functional on V.
d
i d
b
Example 2. Let V  L ([ a, b]), ( f , g )   f ( x) g ( x) dx
a
and for g V define Fg : V  R by Fg ( f )  ( f , g ), f V .
b
1
Example 3. Let V  H ([ a, b]), ( f , g )   f g  f ' g '
a
and for g V define Fg : V  R by Fg ( f )  ( f , g ), f V .
4
2
DERIVATIVES
Definition A functional G : V  R is Gâ teaux
differentiable at u V in the direction of v V if
d
G (u  hv)  G (u )
exists.
G (u  h v)  lim_{ h  0}
dh
h
It is called G â teaux differentiable at
u V if it is
is Gâ teaux differentiable at u V in the direction of
v V for every vV . If V is a Fr echet space and the
G âteaux derivative of G is a continuous function of
(u,v) and a linear function of v then it is called the

Frechet
derivative and G is said Frechet differentiable.
5
EXAMPLES
Example 1. Let G : R  R be differentiable at
'
u  R and let G (u)  R denote the ordinary
derivative of G at u. Then the linear function
F : R  R defined by F (v)  G ' (u) v, v  R
is the Frechet derivative of
G
at u.
Proof From the definition of an ordinary derivative
G(u  h v)  G(u)  G ' (u)hv  o(hv), v, h  R
Since lim
h0
o(h v)
h
 v lim
h0
o( h v)
hv
 v lim
hv0
o( h v)
hv
 0 it follows
that G (u  h v)  G (u )  F (hv)  o(h), v, h  R
6
EXAMPLES
Example 2. Let V  R , ( x, y)   xi yi and for
i 1
y  V define Fy : V  R by Fy ( x)  ( x, y ), x V .
d
i d
Recall from example 1.1 on vufoil 4 that Fy is a
linear functional on V. Now define G  Fy : V  R
for some fixed y  V . Then for every u, v  V , h  R
G (u  hv)  (u  hv, y )  G (u )  Fy (hv)
hence for every u V , G ' (u)  Fy
G
Remark Note that yi   ui so that we usually say that
y  grad G However, it is better to think that y merely
represents the Frechet derivative of G at u
with respect to the inner product ( . , . ) on V.
7
EXAMPLES
Example 3. Let V  R , ( x, y)  i 1 xi yi and for
y  V define Fy : V  R by Fy ( x)  ( x, y ), x V .
i d
d
Now define G : V  R by G(u )  12 (u, u ), u V .
Then for u, v  V , h  R
1 2
G (u  hv)  G (u )  Fu (hv)  2 h (v, v)
Clearly lim
h0
1 h 2 ( v.v )
2
h
 0 therefore
1
2
h 2 ( v, v )  o ( h )
Question 1. Compare this example to the previous
example. For which example is the Frechet
derivative constant when considered as a function
of u (that maps V into linear functionals on V).
8
EXAMPLES
Example 4. Let V  R , ( x, y)  i 1 xi yi and for
y  V define Fy : V  R by Fy ( x)  ( x, y ), x V .
d
i d
Now define G : V  R by G(u)  12 (u, Mu ), u V .
d d
Where M  R
is a d by d matrix – not necessarily
symmetric.
Question 2. Compute the Frechet derivative of G.
9
LINEAR FUNCTIONALS
b
Example 5. Let V  L ([ a, b]), ( f , g )   f ( x) g ( x) dx
a
and for g V define Fg : V  R by Fg ( f )  ( f , g ), f V .
2
Define G  Fg : V  R for some fixed g  V .
Question 3. Compute the Frechet derivative of G.
Example 6. Let G : V  R by G ( f ) 
1
2

b
a
f 2 ( x) dx
2
2
f
(
x
)

(
f
(
x
))
, x [a, b]
Recall that
Question 4. Compute the Frechet derivative of G.
10
LINEAR FUNCTIONALS
b
Example 7. Let V  L ([ a, b]), ( f , g )   f ( x) g ( x) dx
2
a
and for g V define Fg : V  R by Fg ( f )  ( f , g ), f V .
2
H
:
R
 R be continuously differentiable
Now let
b
and define G : V  R by G ( f )   H ( x, f ( x)) dx
a
Question 5. What is the Frechet derivative of G ?
Question 6. What are the conditions for f to be a
stationary point of G ?
11
EXAMPLES
b
Example 8. Let V  H ([ a, b]), ( f , g )   f g  f ' g '
1
a
and for g V define Fg : V  R by Fg ( f )  ( f , g ), f V .
Define G  Fg : V  R for some fixed g  V .
Question 7. What is the Frechet derivative of G ?
3
H
:
R
 R be continuously diff.
Example 9. Let
b
'
G
(
f
)

H
(
x
,
f
(
x
),
f
( x)) dx
and define G : V  R by

a
Question 8. What is the Frechet derivative of G ?
12
EULER EQUATIONS
Theorem Fix real numbers A and B and let V(A,B) be
the subset of the set V in Example 9 that consists of
functions in V that satisfy f(a) = A, f(b) = B.
Let H and G be as in Example 9. If f is an extreme
point (minimum or maximum) for G then H satisfies
d H
dx f'
H
f
Euler-Lagrange Equation:
 0
Proof. Clearly f is a stationary point hence by Q8,


b H
a f
G ( f )( g )  
'
g   Hf ' g ' dx  0, g  V , g (a)  g (b)  0.
Integration by parts yields

b H
a f

 dxd
H
f
'
g dx,
g  V , g (a)  g (b)  0.
which implies that f satisfies the EL equation.
13
GEODESICS
m
q
:
[
a
,
b
]

R
Theorem A vector valued function
is a stationary point for the functional
b
G (q)   L(t , q(t ), q (t )) dt
a
with fixed boundary values for q (a ), q (b)
L
d L
if and only if d x  q   q  0, i  1,..., m
i
i
Example 10 The distance between two q(a) and q(b)
along a path q that connects is a functional G where
L(t , q, q ) 

i m
i 1
qi2
Question 9. What is Euler’s equation for this
example and why are the solutions straight lines ?
14
TUTORIAL 10
1. Solve each EOM that you derived in tutorial 9.
2. Among all curves joining two points ( x0 , y0 ), ( x1 , y1 ),
find the one that generates the surface of
minimum area when rotated around the x-axis.
3. Starting from a point P = (a,A), a heavy particle
slides down a curve in the vertical plane. Find the
curve such that the particle reaches the vertical
line x = b (b < a) in the shortest time.
4. Derive conditions a function f(x,y) to be a
stationary point for a functional of the form
J ( f )   G( x, y, f ( x, y),
D
f
x
,
f
y
) dx dy
where D is a planar region - use Greens Theorem. 15