Variations by generalized symmetries of local Noether strong currents equivalent to global canonical Noether currents

We will pose the inverse problem question within the Krupka variational sequence framework. In particular, the interplay of inverse problems with symmetry and invariance properties will be exploited considering that the cohomology class of the variational Lie derivative of an equivalence class of forms, closed in the variational sequence, is trivial. We will focalize on the case of symmetries of globally defined field equations which are only locally variational and prove that variations of local Noether strong currents are variationally equivalent to global canonical Noether currents. Variations, taken to be generalized symmetries and also belonging to the kernel of the second variational derivative of the local problem, generate canonical Noether currents - associated with variations of local Lagrangians - which in particular turn out to be conserved along any section. We also characterize the variation of the canonical Noether currents associated with a local variational problem