In chapter 6 of Marion and Thornton’s Classical dynamics they introduce “some methods in the calculus of variations”.
They start out setting J = the integral from x1 to x2 of f{y(x),y’(x);x}dx and then proceed to do things to this poor functional until they wind up with Euler?s equation. I can follow the individual steps they take to get to the final destination but I’m not getting the big picture.
Why is y’ in this equation? If you minimize f(y(x)) why do you have to minimize the slope?
F is called a functional or a function of functions but why is do you have to use this versus just y and y’?
Where did the origanal equation for J come from?
What do they mean by a “stationary value”? Is this like a maximum or minimum in regular calc?
I have a lot more questions but I’ll hold off until I see if I get any takers on these 3.
Thanks in advance if you can help me understand this.
Can write integrals and other math symbols on this board?
y’(x) is used because your ultimate goal is to find paths (or places) of least action. That means you basically want to find the possible paths of a function and given a point of y(x),x there might be more than one derivative associated.
I don’t quite understand when you ask, “Where did the original equation for J come from?” There is a pure mathematical artiface about variational calculus.
Without looking at the chapter, I would imagine the stationary value they are referring to would be where the gradient of the function is equal to zero.
The equation for J is chosen so that Euler’s equations come out right. You do it because it works.
The action integral is stationary in the sense that paths (or functions y(x)) that are close to the stationary path have nearly the same integral. It is like finding a minimum in calculus, but it is a minimum wrt an infinite number of possible variations of the function.
but it is a minimum wrt an infinite number of possible variations of the function.
I think the answer to my problem is somewhere in that statement. I’m gonna have to think on that for awhile. You wouldn’t have another sentence that would help a little would you?
Ordinary calculus answers questions like “What value of x maximises this function?”
Let us consider a famous problem in the calculus of variations: given 2 points, one higher than the other but not vertically above it, what curve minimises the travel time. That is, for each curve joining the points, we can calculate the time it would take for a particle to slide down this curve. Calculus of variations solves the problem “What curve minimises this time?”
When you’re doing a calculus of variations problem, you’re trying to minimize something. What it is you’re trying to minimize depends on the problem you’re doing: In Lagrangian mechanics, you’re specifically trying to minimize the action, but you might also want to minimize the total length of a path, for instance, or the time it takes you to get somewhere, or the energy of an object in a certain configuration, or any of a number of other things. The f is the thing that you’re trying to minimize, and y(x) is the function you’re trying to find that minimizes f. Saying f(y(x), y’(x), x) just means that f is allowed to depend on the function, its derivative, and the independent variable. It might not necessarily depend on all of those things, but it can. If it depends on more things (like, for instance, the second derivative), then you’ve got to use slightly different (and more complicated) methods.
It is like finding the minimum value of a function: Near a minimum, a function won’t change very much (it’s “stationary”), and the same is true of a functional near a minimal path.