# Calculus of Variations Euler-Lagrange Diff. Eq.

1. Apr 9, 2009

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.

The text asks that I consider a function:

F(y, dy/dx, x) and the integral I = $$\int$$ 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) + $$\xi$$(x), I is unchanged to order $$\xi$$, provided $$\xi$$ 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) + $$\alpha\eta$$(x)
where $$\alpha$$ is small and $$\eta$$(x) arbitrary. If I[y(x)] is to be stationary, then we must have
dI/d$$\alpha$$, evaluated at $$\alpha$$=0, = 0
for all $$\eta$$(x)."

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

2. Apr 9, 2009

I'm not sure that I truly understand what it means to have a function like:

F(y, dy/dx, x). This is a function dependent upon not only x and y but, upon dy/dx as well? If that's correct, what does that mean exactly? In what scenario would one see something like that? Honestly, I'm probably doing pretty well to truly understand the correlation between x and y when y is a function of x: y(x).

Thank you.

3. Apr 9, 2009

### tiny-tim

For example, the energy of a body might be 1/2 mv2 + mgh + Be-kt

that's a function of dh/dt and h and t separately

(the point is that it's the way F is written that matters … once you solve the equation, you could presumably just write F = G(h,t) or even F = H(t) … but so long as it's still written F(dh/dt, h,t) it can be differentiated separately with respect to each of the three variables)

4. Apr 10, 2009