(See @olver86 page 243)
Let $\Omega \subset \mathbb{R}^p$ be an open, connected subset with smooth boundary $\partial \Omega$. A variational problem consists of finding the extrema of a functional
$$ J[u]=\int_{\Omega}L(x,u^{(n)})dx $$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, ...
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Author of the notes: Antonio J. Pan-Collantes
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