Calculus of Variation - Euler Lagrange Question

Find the minimum value taken by the following two integral expressions where and are both functions of

(a)

(b)

I've had a few stabs at this but my answers are looking dodgy. Could someone please show me the way? ^^

Here is what I've done so far:

(a) Using Euler Lagrange Equation , I get

so

so by integrating the last expression for I get

Applying boundary conditions again, I get

Substituting and rearranging, I end up with as the extremal funcation.

For part (b) I did something similar using as the Euler Langrage equation and ended up with

Here is my working out for part (b):

so using , I get

Taking square roots of both sides and multiplying the whole equation by -1, I get

Using boundary condition , I get

Substituting and rearranging, I get

Using the second boundary condition, I get

Hence,

Re: Calculus of Variation - Euler Lagrange Question

Is y funcion of x?

Otherwise, the two integral are just **numbers**.

Re: Calculus of Variation - Euler Lagrange Question

Sorry, yes. y is a function of x

Re: Calculus of Variation - Euler Lagrange Question

Quote:

Originally Posted by

**yaohui88** Find the minimum value taken by the following two integral expressions where

and

are both functions of

(a)

, y(0)=1, y'(0)=2

(b)

, y(0)=0, y(1)=

I've had a few stabs at this but my answers are looking dodgy. Could someone please show me the way? ^^

Here is what I've done so far:

(a) Using Euler Lagrange Equation

, I get

Good so far, but the next step is, I think, incorrect. Try taking the LHS derivative again. You can't just integrate the way you did, because you don't know what y is.

Quote:

y(0)=1, y'(0)=2 so A=

so by integrating the last expression for

I get

Applying boundary conditions again, I get

Substituting and rearranging, I end up with

as the extremal funcation.

For part (b) I did something similar using

as the Euler Langrage equation and ended up with

Hmm. That's not what I get. What is your differential equation?

Re: Calculus of Variation - Euler Lagrange Question

Hmmm...okay. I've updated the original post to show you what I've done with part (b), and I'm pretty certain I have gone wrong somewhere...Is it the same issue with not being able to integrate like I did because I don't know what is?

As with part (a), I gave it another shot considering your advice. So, I differentiate again and end up with:

Which gives me a second order differential equation:

The solution I get from this is:

which I assume is the extremal function? Then I differentiate to get ?

Re: Calculus of Variation - Euler Lagrange Question

Concerning part a):

I don't know if I agree with your solution of the DE. You've got the right basic functions, but your constants are off. The solution looks like

and hence

Plugging in the initial conditions yields the system of equations

Can you proceed?

As for part b), I can see that you're using the equivalent of Troutman's Equation (4) on page 150 of *Variational Calculus and Optimal Control*, valid when x does not show up explicitly in the integrand of the functional.

Again, your problem is in solving the DE. You cannot simply integrate the way you have done, when y = y(x) is unknown. Your DE is

Try using the integrating factor method.