Noether's theorem for the variational equations

TL;DR

This paper generalizes Noether's theorem to include variational equations, establishing additional constants of motion linked to symmetry groups in Lagrangian systems.

Contribution

It introduces a new action functional framework that extends Noether's theorem to variational equations, revealing extra conserved quantities.

Findings

Derivation of additional constants of motion for variational equations

Extension of Noether's theorem to include variational symmetries

Application to any Lagrangian system with continuous symmetries

Abstract

We introduce an generalized action functional describing the equations of motion and the variational equations for any Lagrangian system. Using this novel scheme we are able to generalize Noether's theorem in such a way that to any $n$-parameter continuous symmetry group of the Lagrangian there exist 1) the usual $n$ constants of motion and 2) $n$ extra constants valid in the variational equations. The new constants are related to the infinitesimal generators of the symmetry transformation by relations similar to the ones that stem from the `nonextended' Noether theorem.