Noether's Theorem Under the Legendre Transform

TL;DR

This paper explores how the Legendre transform links Noether's theorem in Hamiltonian and Lagrangian mechanics, clarifying symmetry and conservation concepts across both frameworks and conditions for their correspondence.

Contribution

It provides a detailed analysis of the relationship between symmetries and conserved quantities in Hamiltonian and Lagrangian mechanics via the Legendre transform, including less restrictive conditions.

Findings

Legendre transform establishes a correspondence between symmetries and conserved quantities in both frameworks.

Conditions identified for preserving the symmetry-conservation correspondence under less restrictive definitions.

Clarification of the differences and connections between Hamiltonian and Lagrangian symmetry concepts.

Abstract

In this paper we demonstrate how the Legendre transform connects the statements of Noether's theorem in Hamiltonian and Lagrangian mechanics. We give precise definitions of symmetries and conserved quantities in both the Hamiltonian and Lagrangian frameworks and discuss why these notions in the Hamiltonian framework are somewhat less rigid. We explore conditions which, when put on these definitions, allow the Legendre transform to set up a one-to-one correspondence between them. We also discuss how to preserve this correspondence when the definitions of symmetries and conserved quantities are less restrictive.