Noether's Theorem Under the Legendre Transform
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.
Cite
@article{arxiv.1409.5837,
title = {Noether's Theorem Under the Legendre Transform},
author = {Jonathan Herman},
journal= {arXiv preprint arXiv:1409.5837},
year = {2014}
}
Comments
This paper was submitted to the University of Waterloo in fulfillment of the research paper requirement for the degree of Master of Mathematics in Pure Mathematics