This tag is for problems relating to the calculus of variations that deal with maximizing or minimizing functionals. This problem is a generalization of the problem of finding extrema of functions of several variables. In fact, these variables will themselves be functions and we will be finding extrema of “functions of functions” or functionals.
The calculus of variations seeks to minimize or maximize an entire functional's worth of parameters instead of changing just one parameter. It achieves this by applying standard calculus techniques to the integral of a functional, thereby reducing $\mathbb R \to \mathbb R$ to just one parameter $\in\mathbb R$.
In symbols one considers $\displaystyle\max \int f(x) \ker(x)$ $dx$ rather than $\max f(s)$.
Two famous applications of the calculus of variations are the brachistochrone problem and deriving the catenary shape of a rope hanging between two poles.
Some basic problems in the calculus of variations are:
$(i)$ find minimizers
$(ii)$ find necessary conditions which minimizers must satisfy
$(iii)$ find solutions (extremals) which satisfy the necessary conditions
$(iv)$ find sufficient conditions which guarantee that such solutions are minimizers
$(v)$ qualitative properties of minimizers, like regularity properties
$(vi)$ how do the minimizers depend on parameters?
$(vii)$ stability of extremals depending on parameters.
Application: A huge number of problems in the calculus of variations have their origin in physics where one has to minimize the energy associated to the problem under consideration. Nowadays many problems come from economics.
References:
https://en.wikipedia.org/wiki/Calculus_of_variations
