Effective Hamiltonian Dynamics via the Maupertuis Principle

Hartmut Schwetlick, Daniel Sutton, Johannes Zimmer

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the dynamics of a Hamiltonian particle forced by a rapidly oscillating potential in m-dimensional space. As alternative to the established approach of averaging Hamiltonian dynamics by reformulating the system as Hamilton-Jacobi equation, we propose an averaging technique via reformulation using the Maupertuis principle. We analyse the result of these two approaches for one space dimension. For the initial value problem the solutions converge uniformly when the total energy is fixed. If the initial velocity is fixed independently of the microscopic scale, then the limit solution depends on the choice of subsequence. We show similar results hold for the one-dimensional boundary value problem. In the higher dimensional case we show a novel connection between the Hamilton-Jacobi and Maupertuis approaches, namely that the sets of minimisers and saddle points coincide for these functionals.

Original languageEnglish
Pages (from-to)1395-1410
Number of pages16
JournalDiscrete and Continuous Dynamical Systems Series S
Volume13
Issue number4
Early online date1 Apr 2019
DOIs
Publication statusPublished - 1 Apr 2020

Bibliographical note

Funding Information:
Acknowledgments. The authors are grateful for funding of an EPSRC network grant (EP/F03685X/1), which stimulated the research described here. DCS and JZ thank the EPSRC for funding the project EP/K027743/1. JZ also received funding through the Leverhulme Trust (RPG-2013-261) and a Royal Society Wolfson Research Merit Award. All authors thank the reviewers for helpful suggestions.

Publisher Copyright:
© 2019 American Institute of Mathematical Sciences. All rights reserved.

Keywords

  • Effective dynamics
  • Hamiltonian dynamics
  • Homogenisation
  • Maupertuis principle
  • Periodic potential

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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