Invariant variational structures on fibered manifolds

The aim of this paper is to present a relatively complete theory of invariance of global, higher-order integral variational functionals in fibered spaces, as developed during a few past decades. We unify and extend recent results of the geometric invariance theory; new results on deformations of extremals are also included. We show that the theory can be developed by means of the general concept of invariance of a differential form in geometry, which does not require different ad hoc modifications. The concept applies to invariance of Lagrangians, source forms and Euler-Lagrange forms, as well as to extremals of the given variational functional. Equations for generators of invariance transformations of the Lagrangians and the Euler-Lagrange forms are characterized in terms of Lie derivatives. As a consequence of invariance, we derive the global Noether's theorem on existence of conserved currents along extremals, and discuss the meaning of conservation equations. We prove a theorem describing extremals, whose deformations by a vector field are again extremals. The general settings and structures we use admit extension of the global invariance theory to variational principles in physics, especially in field theory.