# Second variational derivative of local variational problems and conservation laws

## Marcella Palese, Ekkehart Winterroth, and E. Garrone

Address:

Corresponding author: Department of Mathematics, University of Torino via C. Alberto 10, I-10123 Torino, Italy

Doctoral School of Science and Innovative Technologies fellow University of Torino, Italy

E-mail: marcella.palese@unito.it

Abstract: We consider cohomology defined by a system of local Lagrangian and investigate under which conditions the variational Lie derivative of associated local currents is a system of conserved currents. The answer to such a question involves Jacobi equations for the local system. Furthermore, we recall that it was shown by Krupka et al. that the invariance of a closed Helmholtz form of a dynamical form is equivalent with local variationality of the Lie derivative of the dynamical form; we remark that the corresponding local system of Eulerâ€“Lagrange forms is variationally equivalent to a global one.

AMSclassification: primary 55N30; secondary 55R10, 58A12, 58A20, 58E30, 70S10.

Keywords: fibered manifold, jet space, Lagrangian formalism, variational sequence, second variational derivative, cohomology, symmetry, conservation law.