Effective Hamiltonian dynamics via the Maupertuis principle
| Autor: | Johannes Zimmer, Hartmut Schwetlick, Daniel C. Sutton |
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| Rok vydání: | 2020 |
| Předmět: |
Hamiltonian mechanics
Applied Mathematics 010102 general mathematics Mathematical analysis Dynamical Systems (math.DS) 01 natural sciences Microscopic scale 010101 applied mathematics symbols.namesake Maupertuis' principle Saddle point Subsequence FOS: Mathematics symbols Discrete Mathematics and Combinatorics Initial value problem Boundary value problem Mathematics - Dynamical Systems 0101 mathematics Hamiltonian (quantum mechanics) Analysis Mathematics |
| Zdroj: | Discrete & Continuous Dynamical Systems - S. 13:1395-1410 |
| ISSN: | 1937-1179 |
| DOI: | 10.3934/dcdss.2020078 |
| Popis: | We consider the dynamics of a Hamiltonian particle forced by a rapidly oscillating potential in \begin{document}$ m $\end{document} -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. |
| Databáze: | OpenAIRE |
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