VARIATIONALLY EQUIVALENT PROBLEMS AND VARIATIONS OF NOETHER CURRENTS

Abstract

We consider systems of local variational problems defining nonvanishing cohomology classes. Symmetry properties of the Euler–Lagrange expressions play a fundamental role since they introduce a cohomology class which adds up to Noether currents; they are related with invariance properties of the first variation, thus with the vanishing of a second variational derivative. In particular, we prove that the conserved current associated with a generalized symmetry, assumed to be also a symmetry of the variation of the corresponding local inverse problem, is variationally equivalent to the variation of the strong Noether current for the corresponding local system of Lagrangians. This current is conserved and a sufficient condition will be identified in order that such a current be global.