1. 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@nus.edu.sg http://www.math.nus/~matwml Tel (65) 6874-2749 1

2. ROLLE’S THEOREM 2 If f : [a,b]  R is continuous,differentiable in (a,b) and such that 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 Ifthen for every f(a) = f(b), then satisfies henceQ1. If

3. STATIONARY POINTS 3 Definition A point c in the interior of the domain of a function f is a stationary point (of f) if Examplehas a stationary point 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 ?

4. 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. Letand for by Thenis a linear functional on V. Example 2. Let and fordefineby Example 3. Let and fordefineby

5. DERIVATIVES 5 Definition A functional differentiable at is G teaux in the direction of if it is if It is called G teaux differentiable at for everyIf 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 the Fr chet derivative and G is said Fr chet differentiable. exists. is G teaux differentiable at

6. EXAMPLES 6 Example 1. Let at be differentiable at Then the linear functionderivative of and letdenote the ordinary defined by is the Frechet derivative of Proof From the definition of an ordinary derivative Since that at it follows

7. EXAMPLES 7 define Example 2. Let and for by Recall from example 1.1 on vufoil 4 that linear functional on V. Now define Then for every hence for every Remark Note that so that we usually say that However, it is better to think that y merely is a represents the Frechet derivative of G at u with respect to the inner product (.,. ) on V. for some fixed

8. EXAMPLES 8 define Example 3. Letand for by Now defineby Then for Clearlytherefore 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).

9. EXAMPLES 9 define Example 4. Letand for by Now defineby Whereis a d by d matrix – not necessarily Question 2. Compute the Frechet derivative of G. symmetric.

10. LINEAR FUNCTIONALS 10 Example 5. Let and fordefineby Define Question 3. Compute the Frechet derivative of G. by Recall that Question 4. Compute the Frechet derivative of G. Example 6. Let for some fixed

11. LINEAR FUNCTIONALS 11 Example 7. Let and fordefineby Now let and define be continuously differentiable 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 ?

12. EXAMPLES 12 Example 8. Let and fordefineby Define Question 7. What is the Frechet derivative of G ? for some fixed Example 9. Letbe continuously diff. and defineby Question 8. What is the Frechet derivative of G ?

13. 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 Proof. Clearly f is a stationary point hence by Q8, Euler-Lagrange Equation: Integration by parts yields which implies that f satisfies the EL equation.

14. GEODESICS 14 Theorem A vector valued function with fixed boundary values for 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 Question 9. What is Euler’s equation for this example and why are the solutions straight lines ?

15. TUTORIAL 10 15 1.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 < a) in the shortest time. 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.