TY - JOUR

T1 - Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

AU - Burby, J. W.

AU - Hirvijoki, E.

AU - Leok, M.

N1 - Funding Information:
The work of JWB was supported by the Los Alamos National Laboratory LDRD program under Project No. 20180756PRD4. It was also supported by a Data-Intensive Scientific Machine Learning grant awarded by the DOE SC program in Advanced Scientific Computing Research (ASCR). The work of EH was supported by the Academy of Finland (Grant No. 315278). Any subjective views or opinions expressed herein do not necessarily represent the views of the Academy of Finland or Aalto University. The work of ML was supported by NSF under grants DMS-1345013, DMS-1813635, by AFOSR under grant FA9550-18-1-0288, and by the DoD under grant HQ00342010023 (Newton Award for Transformative Ideas during the COVID-19 Pandemic).
Funding Information:
The work of JWB was supported by the Los Alamos National Laboratory LDRD program under Project No. 20180756PRD4. It was also supported by a Data-Intensive Scientific Machine Learning grant awarded by the DOE SC program in Advanced Scientific Computing Research (ASCR). The work of EH was supported by the Academy of Finland (Grant No. 315278). Any subjective views or opinions expressed herein do not necessarily represent the views of the Academy of Finland or Aalto University. The work of ML was supported by NSF under grants DMS-1345013, DMS-1813635, by AFOSR under grant FA9550-18-1-0288, and by the DoD under grant HQ00342010023 (Newton Award for Transformative Ideas during the COVID-19 Pandemic).
Publisher Copyright:
© 2023, The Author(s).

PY - 2023/4

Y1 - 2023/4

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85149038312&partnerID=8YFLogxK

U2 - 10.1007/s00332-023-09891-4

DO - 10.1007/s00332-023-09891-4

M3 - Article

AN - SCOPUS:85149038312

SN - 0938-8974

VL - 33

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

IS - 2

M1 - 38

ER -