I'm in dire need of help in understanding calculus of variations. My professor uses the Mathews and Walker text, second edition, entitled Mathematical Methods of Physics and, he has a tendency to skip around from chapters found towards the beginning of the text to those nearer the end. I couldn't say if that was an intended property of this text but, anything beyond freshman level kinematics or electricity and magnetism is only offered on a two-year rotation at the college which I attend so, my exposure to potential prerequisites for the subject matter covered in the course is spotty, at best.(adsbygoogle = window.adsbygoogle || []).push({});

The text asks that I consider a function:

F(y, dy/dx, x) and the integral I = [tex]\int[/tex] F(y, dy/dx, x) dx, evaluated from a to b = I[y(x)]

Then, the text indicates that the objective would be to choose the function y(x) such that I[y(x)] is either a maximum or a minimum...("or (more generally) staionary.")

It continues:

"That is, we want to find a y(x) such that if we replace y(x) by y(x) + [tex]\xi[/tex](x), I is unchanged to order [tex]\xi[/tex], provided [tex]\xi[/tex] is sufficiently small.

In order to reduce this problem to the familiar one of making an ordinary function stationary, consider the replacement

y(x) y(x) + [tex]\alpha\eta[/tex](x)

where [tex]\alpha[/tex] is small and [tex]\eta[/tex](x) arbitrary. If I[y(x)] is to be stationary, then we must have

dI/d[tex]\alpha[/tex], evaluated at [tex]\alpha[/tex]=0, = 0

for all [tex]\eta[/tex](x)."

Could someone offer a "dumbed-down" explanation of what the text attempts to communicate?

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# Calculus of Variations Euler-Lagrange Diff. Eq.

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