Invariant Variational Structures

Abstract

Let X be any manifold, W an open set in X, and let α: W → X be a smooth mapping. A differential form η, defined on the set α(W) in X, is said to be invariant with respect to α, if the transformed form $\alpha \ast \eta$ coincides with η, that is, if $\alpha \ast \eta = \eta$ on the set $W \cap \alpha (W)$; in this case, we also say that α is an invariance transformation of η. A vector field, whose local one-parameter group consists of invariance transformations of η, is called the generator of invariance transformations. In this chapter, these definitions are extended to variational structures (Y, ρ) and to the integral variational funtionals associated with them. Our objective is to study invariance properties of the form ρ and other differential forms, associated with ρ, the Lagrangian $\lambda$, and the Euler-Lagrange from $E_{\lambda }$. The class of transformations we consider are auto-morphisms of fibred manifolds and their jet prolongations. This part of the variational theory represents a notable extension of the classical coordinate concepts and methods to topologically nontrivial fibred manifolds that cannot be covered by a single chart. The geometric coordinate-free structure of the infinitesimal first variation formula leads in several consequences, such as the geometric invariance criteria of the Lagrangians and the Euler-Lagrange forms, a global theorem on the conservation law equations, and the relationship between extremals and conservation laws. Resuming, we can say that these results as a whole represent an extension of the classical Noether’s theory to higher-order variational functional on fibred manifolds