Effective Hamiltonian Dynamics via the Maupertuis Principle

TL;DR

This paper introduces a new averaging method for Hamiltonian systems with rapidly oscillating potentials using the Maupertuis principle, providing insights into the convergence of solutions in various boundary conditions and dimensions.

Contribution

It proposes an alternative averaging technique via the Maupertuis principle and explores its relation to the Hamilton-Jacobi approach, including convergence results and a novel connection in higher dimensions.

Findings

Solutions converge uniformly in 1D with fixed total energy.

Limit solutions depend on subsequences when initial velocity is fixed.

In higher dimensions, minimisers and saddle points coincide for the two approaches.

Abstract

We consider the dynamics of a Hamiltonian particle forced by a rapidly oscillating potential in $\dim$-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.