Noether's Theorem Under the Legendre Transform

2014/09/20 by Jonathan Herman, Herman, Jonathan · 1 citation
Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Protein Structure and Dynamics #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1409.5837

openalex publication_date 2014/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Weak Moment Maps in Multisymplectic Geometry

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