Second variational derivative of gauge-natural invariant Lagrangians and conservation laws

TL;DR

This paper explores the second variational derivative of gauge-natural invariant Lagrangians, linking its vanishing to conserved currents via the Second Noether Theorem and the gauge-natural Jacobi morphism.

Contribution

It introduces a novel connection between the second variational derivative and conserved currents in gauge-natural theories, expanding the theoretical framework of gauge invariance and conservation laws.

Findings

Vanishing second variational derivative implies a conserved current.

A covariant strongly conserved current is associated with the deformed Lagrangian.

The approach generalizes the understanding of conservation laws in gauge-natural field theories.

Abstract

We consider the second variational derivative of a given gauge-natural invariant Lagrangian taken with respect to (prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal automorphisms. By requiring such a second variational derivative to vanish, {\em via} the Second Noether Theorem we find that a covariant strongly conserved current is canonically associated with the deformed Lagrangian obtained by contracting Euler--Lagrange equations of the original Lagrangian with (prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal automorphisms lying in the kernel of the generalized gauge-natural Jacobi morphism.