Hamiltonian theory of guiding-center motion

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John R. Cary
Center for Integrated Plasma Studies and Department of Physics, University of Colorado, Boulder, Colorado 80309-0390, USA and Tech-X Corporation, Boulder, Colorado 80303, USA

Alain J. Brizard
Department of Chemistry and Physics, Saint Michael's College, Colchester, Vermont 05439, USA

Published 22 May 2009

Guiding-centertheory provides the reduced dynamical equations for the motion ofcharged particles in slowly varying electromagnetic fields, when the fieldshave weak variations over a gyration radius (or gyroradius) inspace and a gyration period (or gyroperiod) in time. Canonicaland noncanonical Hamiltonian formulations of guiding-center motion offer improvements overnon-Hamiltonian formulations: Hamiltonian formulations possess Noether's theorem (hence invariants followfrom symmetries), and they preserve the Poincaré invariants (so thatspurious attractors are prevented from appearing in simulations of guiding-centerdynamics). Hamiltonian guiding-center theory is guaranteed to have an energyconservation law for time-independent fields—something that is not true ofnon-Hamiltonian guiding-center theories. The use of the phase-space Lagrangian approachfacilitates this development, as there is no need to transforma priori to canonical coordinates, such as flux coordinates, whichhave less physical meaning. The theory of Hamiltonian dynamics isreviewed, and is used to derive the noncanonical Hamiltonian theoryof guiding-center motion. This theory is further explored within thecontext of magnetic flux coordinates, including the generic form alongwith those applicable to systems in which the magnetic fieldslie on nested tori. It is shown how to returnto canonical coordinates to arbitrary accuracy by the Hazeltine-Meiss methodand by a perturbation theory applied to the phase-space Lagrangian.This noncanonical Hamiltonian theory is used to derive the higher-ordercorrections to the magnetic moment adiabatic invariant and to computethe longitudinal adiabatic invariant. Noncanonical guiding-center theory is also developedfor relativistic dynamics, where covariant and noncovariant results are presented.The latter is important for computations in which it isconvenient to use the ordinary time as the independent variablerather than the proper time. The final section uses noncanonicalguiding-center theory to discuss the dynamics of particles in systemsin which the magnetic-field lines lie on nested toroidal fluxsurfaces. A hierarchy in the extent to which particles moveoff of flux surfaces is established. This hierarchy extends fromno motion off flux surfaces for any particle to noaverage motion off flux surfaces for particular types of particles.Future work in magnetically confined plasmas may make use ofthis hierarchy in designing systems that minimize transport losses.

URL:	http://link.aps.org/doi/10.1103/RevModPhys.81.693
DOI:	10.1103/RevModPhys.81.693
PACS:	52.35.Mw; 52.35.Qz; 52.35.Ra 52.35.Mw Nonlinear phenomena: plasma waves, wave propagation and other interactions 52.35.Qz Plasma microinstabilities 52.35.Ra Plasma turbulence YEAR: 2009
KEYWORDS:	particle beams, plasma instability, plasma toroidal confinement, plasma waves

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