Question

In physics there's an incredibly useful and important idea called the least action principle. Essentially this says that an object takes the path that takes the least amount of action. Action is a specific quantity, it is the kinetic energy minus the potential energy.

Now this is really part of a more general concept in which this is only a single application; I will now outline the mathematics of how to minimize a functional and derive the Euler-Lagrange equation of the calculus of variations. Hopefully by the end you understand and are able to apply this! =P

@praxer

Answer 1

awesome :) lemme have it

Answer 2

Alright, so what is a functional? It's an integral of a kind of "function of functions" so it looks something like this: \[\Large J[y(x)]=\int\limits_A^BF(x,y, \dot y)dx\] The notation with the J is just how it's done, and the dot on the y simply means the first derivative. This is a functional.

Alright, so now we want to find out what y minimizes the value of this integral out of all possible values of y. How do we do that? We do this by comparing this to an infinite number of other possible functions surprisingly. =P

So let's say that in a sense we already have the function, we will call it y, and then compare it to another arbitrary function that meets only one condition

Here, the arbitrary function is n(x) and n(a)=n(b)=0.

Sorry if this is a little slow. =P