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 language | English |
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Pages (from-to) | 1395-1410 |
Number of pages | 16 |
Journal | Discrete and Continuous Dynamical Systems Series S |
Volume | 13 |
Issue number | 4 |
Early online date | 1 Apr 2019 |
DOIs | |
Publication status | Published - 1 Apr 2020 |
Keywords
- Effective dynamics
- Hamiltonian dynamics
- Homogenisation
- Maupertuis principle
- Periodic potential
ASJC Scopus subject areas
- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics