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Title: Applications of Noether conservation theorem to Hamiltonian systems

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.

Authors:
Publication Date:
OSTI Identifier:
22617379
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics; Journal Volume: 372; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUNDARY CONDITIONS; CANONICAL TRANSFORMATIONS; CONSERVATION LAWS; FIELD THEORIES; HAMILTONIANS; LAGRANGIAN FUNCTION; SYMMETRY

Citation Formats

Mouchet, Amaury, E-mail: mouchet@phys.univ-tours.fr. Applications of Noether conservation theorem to Hamiltonian systems. United States: N. p., 2016. Web. doi:10.1016/J.AOP.2016.05.016.
Mouchet, Amaury, E-mail: mouchet@phys.univ-tours.fr. Applications of Noether conservation theorem to Hamiltonian systems. United States. doi:10.1016/J.AOP.2016.05.016.
Mouchet, Amaury, E-mail: mouchet@phys.univ-tours.fr. 2016. "Applications of Noether conservation theorem to Hamiltonian systems". United States. doi:10.1016/J.AOP.2016.05.016.
@article{osti_22617379,
title = {Applications of Noether conservation theorem to Hamiltonian systems},
author = {Mouchet, Amaury, E-mail: mouchet@phys.univ-tours.fr},
abstractNote = {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.},
doi = {10.1016/J.AOP.2016.05.016},
journal = {Annals of Physics},
number = ,
volume = 372,
place = {United States},
year = 2016,
month = 9
}
  • Motivated by two time physics theory we revisited the Noether theorem for Hamiltonian constrained systems. Our review presents a novel method to show that the gauge transformations are generated by the conserved quantities associated with the first class constraints.
  • Onset in thermoacoustic engines, the transition to spontaneous self-generation of oscillations, is studied here as both a dynamical critical transition and a limiting heat engine behavior. The critical transition is interesting because it occurs for both dissipative and conservative systems, with common scaling properties. When conservative, the stable oscillations above the critical point also implement a reversible engine cycle satisfying Carnot{close_quote}s theorem, a universal conservation law for entropy flux. While criticality in equilibrium systems is naturally associated with symmetries and universal conservation laws, these are usually exploited with global minimization principles, which dynamical critical systems may not have if dissipationmore » is essential to their criticality. Acoustic heat engines furnish an example connecting equilibrium methods with dynamical and possibly even dissipative critical transitions: A reversible engine is shown to map, by a change of variables, to an equivalent system in apparent thermal equilibrium; a Noether symmetry in the equilibrium field theory implies Carnot{close_quote}s theorem for the engine. Under the same association, onset is shown to be a process of spontaneous symmetry breaking and the scaling of the quality factor predicted for both the reversible {ital and irreversible} engines is shown to arise from the Ginzburg-Landau description of the broken phase. {copyright} {ital 1998} {ital The American Physical Society}« less