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Hamiltonian theory of guiding-center motion

John R. Cary and Alain J. Brizard

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

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Rev. Mod. Phys. 81, 693 – Published 22 May, 2009

DOI: https://doi.org/10.1103/RevModPhys.81.693

Abstract

Guiding-center theory provides the reduced dynamical equations for the motion of charged particles in slowly varying electromagnetic fields, when the fields have weak variations over a gyration radius (or gyroradius) in space and a gyration period (or gyroperiod) in time. Canonical and noncanonical Hamiltonian formulations of guiding-center motion offer improvements over non-Hamiltonian formulations: Hamiltonian formulations possess Noether’s theorem (hence invariants follow from symmetries), and they preserve the Poincaré invariants (so that spurious attractors are prevented from appearing in simulations of guiding-center dynamics). Hamiltonian guiding-center theory is guaranteed to have an energy conservation law for time-independent fields—something that is not true of non-Hamiltonian guiding-center theories. The use of the phase-space Lagrangian approach facilitates this development, as there is no need to transform a priori to canonical coordinates, such as flux coordinates, which have less physical meaning. The theory of Hamiltonian dynamics is reviewed, and is used to derive the noncanonical Hamiltonian theory of guiding-center motion. This theory is further explored within the context of magnetic flux coordinates, including the generic form along with those applicable to systems in which the magnetic fields lie on nested tori. It is shown how to return to canonical coordinates to arbitrary accuracy by the Hazeltine-Meiss method and by a perturbation theory applied to the phase-space Lagrangian. This noncanonical Hamiltonian theory is used to derive the higher-order corrections to the magnetic moment adiabatic invariant and to compute the longitudinal adiabatic invariant. Noncanonical guiding-center theory is also developed for relativistic dynamics, where covariant and noncovariant results are presented. The latter is important for computations in which it is convenient to use the ordinary time as the independent variable rather than the proper time. The final section uses noncanonical guiding-center theory to discuss the dynamics of particles in systems in which the magnetic-field lines lie on nested toroidal flux surfaces. A hierarchy in the extent to which particles move off of flux surfaces is established. This hierarchy extends from no motion off flux surfaces for any particle to no average motion off flux surfaces for particular types of particles. Future work in magnetically confined plasmas may make use of this hierarchy in designing systems that minimize transport losses.

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Rev. Mod. Phys. 81, 693– Published 22 May, 2009

    © 2009 American Physical Society

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