Toroidal regularization of the guiding center Lagrangian
Abstract
In the Lagrangian theory of guiding center motion, an effective magnetic field B* = B+ (m/e)v _{∥}∇ x b appears prominently in the equations of motion. Because the parallel component of this field can vanish, there is a range of parallel velocities where the Lagrangian guiding center equations of motion are either illdefined or very badly behaved. Moreover, the velocity dependence of B* greatly complicates the identification of canonical variables and therefore the formulation of symplectic integrators for guiding center dynamics. Here, this letter introduces a simple coordinate transformation that alleviates both these problems simultaneously. In the new coordinates, the Liouville volume element is equal to the toroidal contravariant component of the magnetic field. Consequently, the largevelocity singularity is completely eliminated. Moreover, passing from the new coordinate system to canonical coordinates is extremely simple, even if the magnetic field is devoid of flux surfaces. We demonstrate the utility of this approach in regularizing the guiding center Lagrangian by presenting a new and stable onestep variational integrator for guiding centers moving in arbitrary timedependent electromagnetic fields.
 Authors:

 Courant Inst. of Mathematical Sciences, New York, NY (United States)
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Publication Date:
 Research Org.:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Fusion Energy Sciences (FES) (SC24)
 OSTI Identifier:
 1424084
 Alternate Identifier(s):
 OSTI ID: 1409875
 Report Number(s):
 LLNLJRNL737871
Journal ID: ISSN 1070664X; TRN: US1801902
 Grant/Contract Number:
 AC5207NA27344; FG0286ER53223; AC0506OR23100; AC5207NA2734
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Physics of Plasmas
 Additional Journal Information:
 Journal Volume: 24; Journal Issue: 11; Journal ID: ISSN 1070664X
 Publisher:
 American Institute of Physics (AIP)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; Hamiltonian mechanics; Energy; Probability theory; General physics; Plasma gyrokinetics; Stochastic processes; Mathematical physics; Nuclear fusion power; Plasma confinement; Tokamaks; Plasma Physics and Thermonuclear Processes
Citation Formats
Burby, J. W., and Ellison, C. L. Toroidal regularization of the guiding center Lagrangian. United States: N. p., 2017.
Web. doi:10.1063/1.5004429.
Burby, J. W., & Ellison, C. L. Toroidal regularization of the guiding center Lagrangian. United States. doi:10.1063/1.5004429.
Burby, J. W., and Ellison, C. L. Wed .
"Toroidal regularization of the guiding center Lagrangian". United States. doi:10.1063/1.5004429. https://www.osti.gov/servlets/purl/1424084.
@article{osti_1424084,
title = {Toroidal regularization of the guiding center Lagrangian},
author = {Burby, J. W. and Ellison, C. L.},
abstractNote = {In the Lagrangian theory of guiding center motion, an effective magnetic field B* = B+ (m/e)v∥∇ x b appears prominently in the equations of motion. Because the parallel component of this field can vanish, there is a range of parallel velocities where the Lagrangian guiding center equations of motion are either illdefined or very badly behaved. Moreover, the velocity dependence of B* greatly complicates the identification of canonical variables and therefore the formulation of symplectic integrators for guiding center dynamics. Here, this letter introduces a simple coordinate transformation that alleviates both these problems simultaneously. In the new coordinates, the Liouville volume element is equal to the toroidal contravariant component of the magnetic field. Consequently, the largevelocity singularity is completely eliminated. Moreover, passing from the new coordinate system to canonical coordinates is extremely simple, even if the magnetic field is devoid of flux surfaces. We demonstrate the utility of this approach in regularizing the guiding center Lagrangian by presenting a new and stable onestep variational integrator for guiding centers moving in arbitrary timedependent electromagnetic fields.},
doi = {10.1063/1.5004429},
journal = {Physics of Plasmas},
number = 11,
volume = 24,
place = {United States},
year = {2017},
month = {11}
}
Web of Science
Works referenced in this record:
Canonicalization and symplectic simulation of the gyrocenter dynamics in timeindependent magnetic fields
journal, March 2014
 Zhang, Ruili; Liu, Jian; Tang, Yifa
 Physics of Plasmas, Vol. 21, Issue 3
Poisson integrators
journal, December 2004
 Karasözen, B.
 Mathematical and Computer Modelling, Vol. 40, Issue 1112
Variational symplectic algorithm for guiding center dynamics and its application in tokamak geometry
journal, April 2009
 Qin, Hong; Guan, Xiaoyin; Tang, William M.
 Physics of Plasmas, Vol. 16, Issue 4
Variational symplectic algorithm for guiding center dynamics in the inner magnetosphere
journal, May 2011
 Li, Jinxing; Qin, Hong; Pu, Zuyin
 Physics of Plasmas, Vol. 18, Issue 5
Development of variational guiding center algorithms for parallel calculations in experimental magnetic equilibria
journal, April 2015
 Ellison, C. Leland; Finn, J. M.; Qin, H.
 Plasma Physics and Controlled Fusion, Vol. 57, Issue 5
Variational principles of guiding centre motion
journal, February 1983
 Littlejohn, Robert G.
 Journal of Plasma Physics, Vol. 29, Issue 1
Regularization of HamiltonLagrangian Guiding Center Theories
journal, November 1985
 CorreaRestrepo, D.; Wimmel, H. K.
 Physica Scripta, Vol. 32, Issue 5
Hamiltonian theory of guidingcenter motion
journal, May 2009
 Cary, John R.; Brizard, Alain J.
 Reviews of Modern Physics, Vol. 81, Issue 2
Variational approach to lowfrequency kineticMHD in the current coupling scheme
journal, March 2017
 Burby, Joshua W.; Tronci, Cesare
 Plasma Physics and Controlled Fusion, Vol. 59, Issue 4
Hamiltonian guiding center equations in toroidal magnetic configurations
journal, March 2003
 White, Roscoe; Zakharov, Leonid E.
 Physics of Plasmas, Vol. 10, Issue 3
Variational Symplectic Integrator for LongTime Simulations of the GuidingCenter Motion of Charged Particles in General Magnetic Fields
journal, January 2008
 Qin, Hong; Guan, Xiaoyin
 Physical Review Letters, Vol. 100, Issue 3
Nonlinear gyrokinetic theory for finitebeta plasmas
journal, January 1988
 Hahm, T. S.; Lee, W. W.; Brizard, A.
 Physics of Fluids, Vol. 31, Issue 7
Foundations of nonlinear gyrokinetic theory
journal, April 2007
 Brizard, A. J.; Hahm, T. S.
 Reviews of Modern Physics, Vol. 79, Issue 2