Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

Burby, J. W.; Hirvijoki, E.; Leok, M.

doi:10.1007/s00332-023-09891-4

Title: Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

Journal Article · Sat Feb 25 00:00:00 EST 2023 · Journal of Nonlinear Science

DOI:https://doi.org/10.1007/s00332-023-09891-4· OSTI ID:1958642

; Hirvijoki, E.; Leok, M.

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.

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Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: 20180756PRD4; FOA-0002493; 89233218CNA000001

OSTI ID:: 1958642

Alternate ID(s):: OSTI ID: 2318945

Report Number(s):: LA-UR-21-32231; 38; PII: 9891

Journal Information:: Journal of Nonlinear Science, Journal Name: Journal of Nonlinear Science Vol. 33 Journal Issue: 2; ISSN 0938-8974

Publisher:: Springer Science + Business MediaCopyright Statement

Country of Publication:: United States

Language:: English

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