Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems
Abstract
M. Kruskal showed that each continuous-time nearly periodic dynamical system admits a formal U(1)-symmetry, generated by the so-called roto-rate. When the nearly periodic system is also Hamiltonian, Noether’s theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analog of Kruskal’s theory. Nearly periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along a U(1)-action. When the limiting rotation is non-resonant, these maps admit formal U(1)-symmetries to all orders in perturbation theory. For Hamiltonian nearly periodic maps on exact presymplectic manifolds, we prove that the formal U(1)-symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether’s theorem. When the unperturbed U(1)-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. As an application of the theory, we use it to develop a novel technique for geometric integration of non-canonical Hamiltonian systems on exact symplectic manifolds.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Feb 25, 2023
- Source ID
- 10.1007/s00332-023-09891-4
Entities
People
- E. Hirvijoki
- J. W. Burby
- Melvin Leok
Organizations
- Air Force Office of Scientific Research
- Los Alamos National Laboratory
- National Science Foundation
- Office of Advanced Scientific Computing Research
- Research Council of Finland
- United States Department of Defense