Calculus of Variations Euler-Lagrange Diff. Eq.

[avocadogirl]

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?

 

[avocadogirl]

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.

 

[tiny-tim]

Hi avocadogirl!

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).

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)

 

[avocadogirl]

Thank you. That does make more sense, especially when thinking of the function in such a way where the components might be differentiable.

Could someone elaborate a little about the paragraph in the text, in its entirety?

Thank you, sincerely.