How to set the variation of an integral to 0?

In summary, the conversation discusses an integral involving a function and a constant, and the need to set the variation of the integral to 0. The correct solution involves applying Euler-Lagrange's equations and taking into account derivatives with respect to the integration variable.
  • #1
phoneketchup
9
0
So I have an integral:

## \delta W=\int_{-\Delta}^\Delta\left[x^2\left(\frac{d\xi}{dx}\right)^2−D_S\xi^2\right]dx ##
Here ##\xi## is a function of ##x## and ##D_S## is a constant. ##\Delta## is just some small ##x##. Now I need to set the variation of ##\delta W## to 0. Do do this I differentiated whatever is inside the bracket and set it to 0. I get:

## x^2\xi″+x\xi′−D_S\xi = 0 ##

However, the answer is:

## \frac{d}{dx}\left(x^2\frac{d\xi}{dx}\right)+D_S\xi = x^2\xi″+2x\xi′+D_S\xi = 0 ##

Where the primes are derivatives with respect to x. As you can see the difference is a factor of 2 in the middle term and that minus sign.

If anyone could point out where I am going wrong, it would be really appreciated.

Thanks!
 
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  • #2
You cannot simply differentiate the integrand with respect to the integration variable. You need to check out a textbook or lecture notes on variational calculus and apply Euler-Lagrange's equations.
 
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Likes phoneketchup
  • #3
Thanks a lot! Got the answer!
 

1. What does it mean to set the variation of an integral to 0?

Setting the variation of an integral to 0 means finding the function that minimizes or maximizes the value of the integral. This is often used in optimization problems where the goal is to find the optimal value of a certain quantity.

2. Why is it important to set the variation of an integral to 0?

Setting the variation of an integral to 0 helps us find the solution to optimization problems and determine the critical points of a function. It also allows us to find the maximum or minimum values of a certain quantity.

3. What is the process for setting the variation of an integral to 0?

The process for setting the variation of an integral to 0 involves taking the derivative of the integral with respect to the variable that we want to optimize, setting it equal to 0, and then solving for the variable.

4. Are there any specific techniques for setting the variation of an integral to 0?

Yes, there are various techniques that can be used to set the variation of an integral to 0. Some common techniques include the method of Lagrange multipliers, the method of substitution, and the method of integration by parts.

5. Can setting the variation of an integral to 0 always guarantee the optimal solution?

No, setting the variation of an integral to 0 does not always guarantee the optimal solution. It can only give us a critical point, which may or may not be the optimal solution. Further analysis and evaluation of the function are needed to determine if the critical point is indeed the maximum or minimum value.

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