Variational problem

(See @olver86 page 243)
Let $\mathrm{\Omega }\subset {\mathbb{R}}^p$ be an open, connected subset with smooth boundary $\partial \mathrm{\Omega }$. A variational problem consists of finding the extrema of a functional

in some class of functions $u:{\mathbb{R}}^p\to {\mathbb{R}}^q$, where $L$ is called the Lagrangian of the variational problem $J$ and it is a smooth function of $x$, $u(x)$ and their derivatives. Indeed, the Lagrangian should be thought as a horizontal 1-form, instead of a function. See Lagrangian Mechanics#Jet space approach.

The search of the extrema of this functional can be shown to be equivalent (by means of the variational derivative) to the Euler-Lagrange equations.

In this context we have variational symmetrys, which have some relation to the generalized symmetries of the associated Euler-Lagrange equations.

I think this can be formalized in terms of the jet bundle $J^n({\mathbb{R}}^p,{\mathbb{R}}^q)$ by means of the variational bicomplex, ...

Related: Classical Field Theory.