USSC3002 Oscillations and Waves Lecture 10 Calculus of Variations

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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 matwml_at_nus.edu.sg http//www.math.nus/matw
ml Tel [phone]
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ROLLES THEOREM
If f a,b ?R is continuous,differentiable in
(a,b) and

such that
f(a) f(b), then
Proof First, since f and continuous on a,b and

a,b is both closed and bounded, there exists
such that
If neither point belongs to (a,b) then f is
constant and
every choice of c in (a,b) satisfies
If
then for every
satisfies
hence
Q1. If
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STATIONARY POINTS
Definition A point c in the interior of the
domain of a function f is a stationary point (of
f) if

The proof of Rolles 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 foll
owing example shows that the converse is not tru
e
Example
has a stationary
point
but c is not a local extremum.
Q2. Does f have a minimum and maximum ?
Are they stationary points ?
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LINEAR FUNCTIONALS

Definition Let V be a vector space over the field
R. A function F V ? R is called a functional a
nd F is called a linear functional if it is line
ar.
Example 1. Let
and for
define
by
Then
is a linear functional on V.
Example 2. Let
and for
define
by
Example 3. Let
and for
define
by
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FRECHET DERIVATIVE
Definition A functional
is differentiable
at a point
if there exists a linear functional
with
Here
pronounced little oh of h means that
If G is differentiable at
then F is called the
derivative of G at
and we write
to show its dependence on
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EXAMPLES
Example 1. Let
be differentiable at
and let
denote the ordinary
at
Then the linear function
derivative of
defined by
is the Frechet derivative of
at
Proof From the definition of an ordinary
derivative
Since
it follows
that
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EXAMPLES

Example 2. Let
and for
define
by
Recall from example 1.1 on vufoil 4 that
is a
linear functional on V. Now define
Then for every
for some fixed
hence for every
Remark Note that
so that we usually say that
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.
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EXAMPLES

Example 3. Let
and for
define
by
Now define
by
Then for
Clearly
therefore
Question 1. Compare this example to the previous
example. For which example is the Frechet deri
vative constant when considered as a function
of u (that maps V into linear functionals on V).
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EXAMPLES

Example 4. Let
and for
define
by
Now define
by
Where
is a d by d matrix not necessarily
symmetric.
Question 2. Compute the Frechet derivative of G.
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LINEAR FUNCTIONALS

Example 5. Let
and for
define
by
Define
for some fixed
Question 3. Compute the Frechet derivative of G.
by
Example 6. Let
Recall that
Question 4. Compute the Frechet derivative of G.
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LINEAR FUNCTIONALS

Example 7. Let
and for
define
by
Now let
be continuously differentiable
and define
by
Question 5. What is the Frechet derivative of G ?
Question 6. What are the conditions for f to be
a
stationary point of G ?
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EXAMPLES
Example 8. Let
and for
define
by
Define
for some fixed
Question 7. What is the Frechet derivative of G ?
Example 9. Let
be continuously diff.
and define
by
Question 8. What is the Frechet derivative of G ?
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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 con
sists 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
Euler-Lagrange Equation
Proof. Clearly f is a stationary point hence by
Q8,
Integration by parts yields
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which implies that f satisfies the EL equation.
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GEODESICS
Theorem A vector valued function
is a stationary point for the functional
with fixed boundary values for
if and only if
Example 10 The distance between two q(a) and
q(b) along a path q that connects is a functional
G where
Question 9. What is Eulers equation for this
example and why are the solutions straight lines
?
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TUTORIAL 10

• Solve each EOM that you derived in tutorial 9.

2. Among all curves joining two points
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 shortest time.
4. Derive conditions a function f(x,y) to be a
stationary point for a functional of the form
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where D is a planar region - use Greens Theorem.