Applications of Noether conservation theorem to Hamiltonian systems

TL;DR

This paper demonstrates how Noether's theorem can be directly applied within Hamiltonian systems to derive conservation laws, emphasizing boundary condition invariance and illustrating with classical and quantum examples.

Contribution

It provides a direct Hamiltonian formulation of Noether's theorem, addressing boundary invariance issues and applying it to diverse physical systems.

Findings

Unified Hamiltonian approach to Noether's theorem

Application to classical field theory and quantum dynamics

Clarification of boundary condition invariance

Abstract

The Noether theorem connecting symmetries and conservation laws can be applied directly in a Hamiltonian framework without using any intermediate Lagrangian formulation. This requires a careful discussion about the invariance of the boundary conditions under a canonical transformation and this paper proposes to address this issue. Then, the unified treatment of Hamiltonian systems offered by Noether's approach is illustrated on several examples, including classical field theory and quantum dynamics.