Why it is interesting?

Variational calculus is the alternative to the usual calculus methods when we want to find functions that minimize something. As an analogy, usually when we search for the extrema of a function, we differentiate the function, set the derivative of the function to zero and find the point that yields the extrema. Similar results can be computed by using variational calculus.In variational calculus we find extrema of functionals which are functions of functions with respect some function (instead of variable). This is extremely important for the Lagrangian formalism.

Concrete

What we know from Calculus

On calculus, when we want to find the extremum of a function, we use the derivative:

We differentiate the function $f(x)$, then demand that the resulting derivative vanishes: $$\frac{d f(x)}{dx} \stackrel{!}{=} 0 , $$ if we solve for $x$, we find an critical point, for this function $f$

Concept of Variational Calculus

Instead of a function $f$ that takes numbers to numbers, the object of interest is a functional, a function of functions of sorts. To find the stationary functions of the functionals, we need to change a bit the differentiation process, and we use the functional derivative or variational derivative, and we equate the result to zero.

In the same way that the to find the extrema of a function one needs to solve a system of algebraic equation, the result of a variational derivative is a system of differential equations, these being ordinary or partial differential equations, depending on the function space being search.

This is the recipe of the variational problem

References

• Calculus of Variations, Gelfand and Fomin
• Calculus of Variations by MacCluer.

Abstract

See Karen Uhlenbeck and the Calculus of Variations by Simon Donaldson

Quotes

Another way of saying a thing is least is to say that if you move the path a little bit at first it does not make any difference. Suppose you were walking around on hills – but smooth hills, since the mathematical things involved correspond to smooth things – and you come to a place where you are lowest, then I say that if you take a small step forward you will not change your height. When you are at the lowest or at the highest point, a step does not make any difference in the altitude in first approximation, whereas if you are on a slope you can walk down the slope with a step and then if you take the step in the opposite direction you walk up. That is the key to the reason why, when you are at the lowest place, taking a step does not make much difference, because if it did make any difference then if you took a step in the opposite direction you would go down. Since this is the lowest point and you cannot go down, your first approximation is that the step does not make any difference. We therefore know that if we move a path a little bit it does not make any difference to the action on a first approximation. "The Character of Physical Law" by R. Feynman