Symmetries of the dynamics and Noether theorem in classical mechanics

TL;DR

This paper explores the connection between continuous symmetries in classical mechanics and conservation laws, extending Noether's theorem to cases where the Hamiltonian changes by a total derivative, revealing anomalies in conserved quantities.

Contribution

It generalizes Noether's theorem in Hamiltonian mechanics to include symmetries that alter the Hamiltonian by a total derivative, highlighting the resulting anomalies in conservation laws.

Findings

Symmetries can leave the Hamiltonian invariant up to a total derivative.

Conservation laws may include additional terms due to gauge transformations.

The paper clarifies the structure of conserved quantities in generalized symmetry cases.

Abstract

The aim of this note is to discuss the relation between one-parameter continuous symmetries of the dynamics, defined on physical grounds, and conservation laws. In the Hamiltonian formulation, such symmetries of the dynamics in general leave the Hamiltonian invariant only up to a total derivative $ d G(q)/d t$ . In this more general case, the corresponding formulation of Noether theorem gives that the conservation law displays a sort of ano\-ma\-ly, the constant of motion being the sum of the canonical generator of the symmetry transformations plus the generator $G$ of the gauge transformation $q_i \rightarrow q_i$, $p_i \rightarrow p_i - \partial G/ \partial q_i$.