Effective Hamiltonian dynamics via the Maupertuis principle

Hartmut Schwetlick; Daniel C. Sutton; Johannes Zimmer

doi:10.3934/dcdss.2020078

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2020, Volume 13, Issue 4: 1395-1410. Doi: 10.3934/dcdss.2020078

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Effective Hamiltonian dynamics via the Maupertuis principle

1.
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
2.
3 Badgers Hollow, Peparharow Road, Godalming GU7 2PX, United Kingdom

^* Corresponding author: Johannes Zimmer
^* Corresponding author: Johannes Zimmer

Received: December 2017

Revised: September 2018

Published: April 2020

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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.

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Mathematics Subject Classification: Primary: 70H05; Secondary: 37J45, 70H25, 35B27.

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References

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