Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

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

doi:10.1007/s00332-023-09891-4

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.

View Journal Article

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

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

Grant/Contract Number:: 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 Journal Issue: 2 Vol. 33; ISSN 0938-8974

Publisher:: Springer Science + Business MediaCopyright Statement

Country of Publication:: United States

Language:: English

References (24)

Introduction to Mechanics and Symmetry Marsden, Jerrold E.; Ratiu, Tudor S. Texts in Applied Mathematics https://doi.org/10.1007/978-0-387-21792-5	book	January 1999
Mathematical Methods of Classical Mechanics Arnold, V. I. Graduate Texts in Mathematics https://doi.org/10.1007/978-1-4757-2063-1	book	January 1989
On the Γ-convergence of discrete dynamics and variational integrators Müller, S.; Ortiz, M. Journal of Nonlinear Science, Vol. 14, Issue 3 https://doi.org/10.1007/BF02666023	journal	May 2004
Spectral variational integrators Hall, James; Leok, Melvin Numerische Mathematik, Vol. 130, Issue 4 https://doi.org/10.1007/s00211-014-0679-0	journal	December 2014
Long-term analysis of a variational integrator for charged-particle dynamics in a strong magnetic field Hairer, Ernst; Lubich, Christian Numerische Mathematik, Vol. 144, Issue 3 https://doi.org/10.1007/s00211-019-01093-z	journal	January 2020
On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation Iserles, A.; N�rsett, S. P. BIT Numerical Mathematics, Vol. 44, Issue 4 https://doi.org/10.1007/s10543-004-5243-3	journal	December 2004
Noncanonical Hamiltonian mechanics and its application to magnetic field line flow Cary, John R.; Littlejohn, Robert G. Annals of Physics, Vol. 151, Issue 1 https://doi.org/10.1016/0003-4916(83)90313-5	journal	November 1983
INVITED: Slow manifold reduction for plasma science Burby, J. W.; Klotz, T. J. Communications in Nonlinear Science and Numerical Simulation, Vol. 89 https://doi.org/10.1016/j.cnsns.2020.105289	journal	October 2020
Slow manifolds of classical Pauli particle enable structure-preserving geometric algorithms for guiding center dynamics Xiao, Jianyuan; Qin, Hong Computer Physics Communications, Vol. 265 https://doi.org/10.1016/j.cpc.2021.107981	journal	August 2021
An energy-conserving and asymptotic-preserving charged-particle orbit implicit time integrator for arbitrary electromagnetic fields Ricketson, L. F.; Chacón, L. Journal of Computational Physics, Vol. 418 https://doi.org/10.1016/j.jcp.2020.109639	journal	October 2020
Variational principles of guiding centre motion Littlejohn, Robert G. Journal of Plasma Physics, Vol. 29, Issue 1 https://doi.org/10.1017/S002237780000060X	journal	February 1983
General formulas for adiabatic invariants in nearly periodic Hamiltonian systems Burby, J. W.; Squire, J. Journal of Plasma Physics, Vol. 86, Issue 6 https://doi.org/10.1017/S002237782000080X	journal	December 2020
Discrete mechanics and variational integrators Marsden, J. E.; West, M. Acta Numerica, Vol. 10 https://doi.org/10.1017/S096249290100006X	journal	May 2001
Asymptotic Theory of Hamiltonian and other Systems with all Solutions Nearly Periodic Kruskal, Martin Journal of Mathematical Physics, Vol. 3, Issue 4 https://doi.org/10.1063/1.1724285	journal	July 1962
Degenerate variational integrators for magnetic field line flow and guiding center trajectories Ellison, C. L.; Finn, J. M.; Burby, J. W. Physics of Plasmas, Vol. 25, Issue 5 https://doi.org/10.1063/1.5022277	journal	May 2018
A guiding center Hamiltonian: A new approach Littlejohn, Robert G. Journal of Mathematical Physics, Vol. 20, Issue 12 https://doi.org/10.1063/1.524053	journal	December 1979
Hamiltonian formulation of guiding center motion Littlejohn, Robert G. Physics of Fluids, Vol. 24, Issue 9 https://doi.org/10.1063/1.863594	journal	January 1981
Normal stability of slow manifolds in nearly periodic Hamiltonian systems Burby, J. W.; Hirvijoki, E. Journal of Mathematical Physics, Vol. 62, Issue 9 https://doi.org/10.1063/5.0054323	journal	September 2021
Hamiltonian perturbation theory: periodic orbits, resonances and intermittency Lochak, P. Nonlinearity, Vol. 6, Issue 6 https://doi.org/10.1088/0951-7715/6/6/003	journal	November 1993
Discrete Hamiltonian variational integrators Leok, M.; Zhang, J. IMA Journal of Numerical Analysis, Vol. 31, Issue 4 https://doi.org/10.1093/imanum/drq027	journal	March 2011
Explicit symplectic approximation of nonseparable Hamiltonians: Algorithm and long time performance Tao, Molei Physical Review E, Vol. 94, Issue 4 https://doi.org/10.1103/PhysRevE.94.043303	journal	October 2016
Variational Symplectic Integrator for Long-Time Simulations of the Guiding-Center Motion of Charged Particles in General Magnetic Fields Qin, Hong; Guan, Xiaoyin Physical Review Letters, Vol. 100, Issue 3 https://doi.org/10.1103/PhysRevLett.100.035006	journal	January 2008
Hamiltonian theory of guiding-center motion Cary, John R.; Brizard, Alain J. Reviews of Modern Physics, Vol. 81, Issue 2 https://doi.org/10.1103/RevModPhys.81.693	journal	May 2009
Implicit-Explicit Variational Integration of Highly Oscillatory Problems Stern, Ari; Grinspun, Eitan Multiscale Modeling & Simulation, Vol. 7, Issue 4 https://doi.org/10.1137/080732936	journal	January 2009

Similar Records

Approximation of nearly-periodic symplectic maps via structure-preserving neural networks

Journal Article · Mon May 22 20:00:00 EDT 2023 · Scientific Reports · OSTI ID:2426510

Normal stability of slow manifolds in nearly periodic Hamiltonian systems

Journal Article · Sun Sep 12 20:00:00 EDT 2021 · Journal of Mathematical Physics · OSTI ID:1828720

Related Subjects

97 MATHEMATICS AND COMPUTING
Mathematics

Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

Citation Formats

References (24)

Similar Records

Related Subjects