Guiding center dynamics as motion on a formal slow manifold in loop space

Burby, J. W.

doi:10.1063/1.5119801

Guiding center dynamics as motion on a formal slow manifold in loop space

Journal Article · Thu Jan 16 00:00:00 EST 2020 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.5119801· OSTI ID:1660591

^[1]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Since the late 1950s, the dynamics of a charged particle’s “guiding center” in a strong, inhomogeneous magnetic field have been understood in terms of near-identity coordinate transformations. The basic idea has been to approximately transform away the coupling between the fast gyration around magnetic fields lines and the remaining slow dynamics. This basic understanding now serves as a foundation for describing the kinetic theory of strongly magnetized plasmas. I illustrate a new way to understand guiding center dynamics that does not involve complicated coordinate transformations. Starting from a dynamical system formulation of the motion of parameterized loops in a charged particle’s phase space, I identify a formal slow manifold in loop space. Dynamics on this formal slow manifold are equivalent to guiding center dynamics to all orders in perturbation theory. After demonstrating that loop space dynamics comprises an infinite-dimensional noncanonical Hamiltonian system, I recover the well-known Hamiltonian formulation of guiding center motion by restricting the (pre)symplectic structure on loop space to the finite-dimensional guiding center formal slow manifold.

View Accepted Manuscript (DOE)

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

Sponsoring Organization:: USDOE; USDOE Laboratory Directed Research and Development (LDRD) Program

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 1660591

Alternate ID(s):: OSTI ID: 1591967

Report Number(s):: LA-UR--19-24299

Journal Information:: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 1 Vol. 61; ISSN 0022-2488

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

References (24)

Geometric singular perturbation theory for ordinary differential equations Fenichel, Neil Journal of Differential Equations, Vol. 31, Issue 1 https://doi.org/10.1016/0022-0396(79)90152-9	journal	January 1979
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
Adiabatic charged-particle motion Northrop, Theodore G. Reviews of Geophysics, Vol. 1, Issue 3 https://doi.org/10.1029/RG001i003p00283	journal	January 1963
Analysis of the accuracy and convergence of equation-free projection to a slow manifold Zagaris, Antonios; Gear, C. William; Kaper, Tasso J. ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 43, Issue 4 https://doi.org/10.1051/m2an/2009026	journal	July 2009
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
Geometric gyrokinetic theory for edge plasmas Qin, H.; Cohen, R. H.; Nevins, W. M. Physics of Plasmas, Vol. 14, Issue 5 https://doi.org/10.1063/1.2472596	journal	May 2007
Gyrosymmetry: Global considerations Burby, J. W.; Qin, H. Physics of Plasmas, Vol. 19, Issue 5 https://doi.org/10.1063/1.4719700	journal	May 2012
Automation of the guiding center expansion Burby, J. W.; Squire, J.; Qin, H. Physics of Plasmas, Vol. 20, Issue 7 https://doi.org/10.1063/1.4813247	journal	July 2013
Hamiltonian perturbation theory in noncanonical coordinates Littlejohn, Robert G. Journal of Mathematical Physics, Vol. 23, Issue 5 https://doi.org/10.1063/1.525429	journal	May 1982
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
Hamiltonian Theory of Guiding Center Bounce Motion Littlejohn, Robert G. Physica Scripta, Vol. T2A https://doi.org/10.1088/0031-8949/1982/t2a/015	journal	January 1982
Conservative magnetic moment of runaway electrons and collisionless pitch-angle scattering Liu, Chang; Qin, Hong; Hirvijoki, Eero Nuclear Fusion, Vol. 58, Issue 10 https://doi.org/10.1088/1741-4326/aad2a5	journal	August 2018
Fluids and Plasmas: Geometry and Dynamics Marsden, Jerrold E. https://doi.org/10.1090/conm/028/751980	book	January 1984
Adiabatic Invariants of Periodic Classical Systems Gardner, C. S. Physical Review, Vol. 115, Issue 4 https://doi.org/10.1103/physrev.115.791	journal	August 1959
Phase anholonomy in the classical adiabatic motion of charged particles Littlejohn, Robert G. Physical Review A, Vol. 38, Issue 12 https://doi.org/10.1103/physreva.38.6034	journal	December 1988
Application of a Nonlinear WKB Method to the Korteweg–DeVries Equation Miura, Robert M.; Kruskal, Martin D. SIAM Journal on Applied Mathematics, Vol. 26, Issue 2 https://doi.org/10.1137/0126036	journal	March 1974
Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes Gear, C. W.; Kaper, T. J.; Kevrekidis, I. G. SIAM Journal on Applied Dynamical Systems, Vol. 4, Issue 3 https://doi.org/10.1137/040608295	journal	January 2005
Asymptotics of a Slow Manifold Vanneste, J. SIAM Journal on Applied Dynamical Systems, Vol. 7, Issue 4 https://doi.org/10.1137/070710081	journal	January 2008
Slow Manifolds MacKay, R. S. Energy Localisation and Transfer https://doi.org/10.1142/9789812794864_0003	book	February 2004
On the Existence of a Slow Manifold Lorenz, Edward N. Journal of the Atmospheric Sciences, Vol. 43, Issue 15 https://doi.org/10.1175/1520-0469(1986)043<1547:oteoas>2.0.co;2	journal	August 1986
On the Nonexistence of a Slow Manifold Lorenz, E. N.; Krishnamurthy, V. Journal of the Atmospheric Sciences, Vol. 44, Issue 20 https://doi.org/10.1175/1520-0469(1987)044<2940:otnoas>2.0.co;2	journal	October 1987
The Slow Manifold—What Is It? Lorenz, Edward N. Journal of the Atmospheric Sciences, Vol. 49, Issue 24 https://doi.org/10.1175/1520-0469(1992)049<2449:tsmii>2.0.co;2	journal	December 1992
Automation of The Guiding Center Expansion Burby, J. W.; Squire, J.; Qin, H. arXiv https://doi.org/10.48550/arxiv.1312.3973	text	January 2013
Gyrosymmetry: Global Considerations Burby, J. W.; Qin, H. arXiv https://doi.org/10.48550/arxiv.1312.3975	text	January 2013

Similar Records

Hamiltonian theory of guiding-center motion

Journal Article · Wed Apr 15 00:00:00 EDT 2009 · Reviews of Modern Physics · OSTI ID:22038451

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

Stochasticity, superadiabaticity, and the theory of adiabatic invariants and guiding center motion

Technical Report · Wed Jul 01 00:00:00 EDT 1981 · OSTI ID:6181231

Related Subjects

97 MATHEMATICS AND COMPUTING
Coordinate transformations
Differentiable manifold
Dynamical systems
Hamiltonian mechanics
Kinetic theory
Magnetic fields
Magnetic fusion energy
Mathematics
Perturbation theory
Plasma dynamics

Guiding center dynamics as motion on a formal slow manifold in loop space

Citation Formats

References (24)

Similar Records

Related Subjects