ussc3002 oscillations and waves lecture 10 calculus of variations wayne m. lawton department of...

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]://www.math.nus/~matwml

Tel (65) 6874-2749

1

ROLLE’S THEOREM

2

If f : [a,b] R is continuous,differentiable in (a,b) and.0)(' cf),( bac such that

Proof First, since f and continuous on [a,b] and [a,b] is both closed and bounded, there exists

such that ],[, baxx Mm ]),([max)(]),,([min)( bafxfbafxf Mm

If neither point belongs to (a,b) then f is constant andevery choice of c in (a,b) satisfies .0)(' cfIf ),( baxm then for every }0{\),( mm xbxah

hxfhxf

hmmQ )()(

f(a) = f(b), then

satisfies 00,00 hh QhQh

hence .0lim)(' 0 hhm Qxf Q1. If ? ),( baxM

STATIONARY POINTS

3

Definition A point c in the interior of the domain of a function f is a stationary point (of f) if .0)(' cf

Example 3)(,]1,1[: xxfRf has a stationary

point 0c

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:

but c is not a local extremum.

Q2. Does f have a minimum and maximum ? Are they stationary points ?

LINEAR FUNCTIONALS

4

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.

define Example 1. Let and for

di

i iid yxyxRV

1),(,

Vy RVFy : by .),,()( VxyxxFy

Then yF is a linear functional on V.

Example 2. Let b

adxxgxfgfbaLV )()(),(]),,([2

and for Vg define RVFg : by .),,()( VfgffFg

Example 3. Let b

agfgfgfbaHV ''),(]),,([1


DERIVATIVES

5

Definition A functional RVG :differentiable at

h

uGhvuGhvhuG

dh

d )()(}0lim_{)(

is G teaux

in the direction of

if it is

ifVu

It is called G teaux differentiable at

for every .Vv If V is a Fr chet space and the

in the direction of

G teaux derivative of G is a continuous function of(u,v) and a linear function of v then it is called theFr chet derivative and G is said Fr chet differentiable.

Vv

exists.

a

Vua

is G teaux differentiable at a VuVva

e

e e

EXAMPLES

6

Example 1. Let

.uat

RuG )('be differentiable at

Then the linear function

RRG :

derivative ofand letG

denote the ordinary

defined by RvvuGvF ,)()( '

is the Frechet derivative of

Proof From the definition of an ordinary derivative

Since

RRF :

that

Ru

G at .u

RhvhvohvuGuGvhuG ,),()()()( '

0limlimlim )(

0

)(

0

)(

0

vhvho

hvvhvho

hhvho

hvv

RhvhohvFuGvhuG ,),()()()(

it follows

EXAMPLES

7

define

Example 2. Let and for

di

i iid yxyxRV

1),(,

Vy RVFy : by .),,()( VxyxxFy Recall from example 1.1 on vufoil 4 that yF

linear functional on V. Now defineThen for every

)()(),()( hvFuGyhvuhvuG yhence for everyRemark Note that so that we usually say that iu

Giy

Gy grad However, it is better to think that y merely

is a

RVFG y :RhVvu ,,

yFuGVu )(, '

represents the Frechet derivative of G at u with respect to the inner product ( . , . ) on V.

for some fixed .Vy

EXAMPLES

8

define


di

i iid yxyxRV

1),(,


Now define by

),()()()( 221 vvhhvFuGhvuG u

Then for

Clearly therefore 0lim).(

0

221

h

vvh

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).

RVG :

RhVvu ,,

.),,()( 21 VuuuuG

)(),(221 hovvh

EXAMPLES

9

define


di

i iid yxyxRV

1),(,


Now define by

Where is a d by d matrix – not necessarily

Question 2. Compute the Frechet derivative of G.

RVG :ddRM

.),,()( 21 VuMuuuG

symmetric.

LINEAR FUNCTIONALS

10

Example 5. Let b



Define


RVFG g :

RVG : by b

adxxffG )()( 2

21

Recall that

],[,))(()( 22 baxxfxf


Example 6. Let

for some fixed .Vg

LINEAR FUNCTIONALS

11

Example 7. Let b



Now let

and define

RRH 2: be continuously differentiable

RVG : by b

adxxfxHfG ))(,()(

Question 5. What is the Frechet derivative of G ?

Question 6. What are the conditions for f to be astationary point of G ?

EXAMPLES

12

Example 8. Let b

agfgfgfbaHV ''),(]),,([1


Define RVFG g :


for some fixed .Vg

Example 9. Let RRH 3: be continuously diff.

and define RVG : by b

adxxfxfxHfG ))(),(,()( '


EULER EQUATIONS

13

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

0'

fH

f

Hxdd

Proof. Clearly f is a stationary point hence by Q8,

.0)()(,,0))(( '''

bgagVgdxgggfG

b

a f

HfH

Euler-Lagrange Equation:

Integration by parts yields

.0)()(,,'

bgagVgdxg

b

a f

Hdxd

fH

which implies that f satisfies the EL equation.

GEODESICS

14

Theorem A vector valued function

miii qL

qL

xdd ,...,1,0

with fixed boundary values for )(),( bqaq

is a stationary point for the functional

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

mRbaq ],[:

b

adttqtqtLqG ))(),(,()(

mi

i iqqqtL1

2),,(

Question 9. What is Euler’s equation for this example and why are the solutions straight lines ?

TUTORIAL 10

15

1. Solve each EOM that you derived in tutorial 9.

),,(),,( 1100 yxyx2. Among all curves joining two pointsfind 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.

dydxyxfyxGfJD

yf

xf

),),,(,,()(

4. Derive conditions a function f(x,y) to be a stationary point for a functional of the form

where D is a planar region - use Greens Theorem.

ussc3002 oscillations and waves lecture 10 calculus of variations wayne m. lawton department of...

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