Hamiltonian theory of guiding-center motion

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 gyroradius͒ in space and a gyration period ͑or gyroperiod͒ 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 symmetries͒, and they preserve the Poincaré invariants ͑so that spurious attractors are prevented from appearing in simulations of guiding-center dynamics͒. 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.

Probabilistic approach to nonlinear wave-particle resonant interaction

“…The system also contains an electrostatic wave propagating along this magnetic field. Thus, transverse particle motion (gyrorotation around magnetic field) is not perturbed by the wave field-aligned electric field, and we can use a guidingcenter approximation for the particle Hamiltonian [52]:…”

A novel technique for rapid L* calculation using UBK coordinates

“…For nonrelativistic particles, g % 1. In the relativistic case, g remains nearly constant (i.e., B m changes little during the drift) because the potential energy becomes orders of magnitude smaller than the kinetic energy term (In guiding-center formulation, the electric potential term is an order of magnitude smaller than the terms with B, otherwise the second adiabatic invariant would not be conserved [Brizard and Chan, 1999;Tao et al, 2007;Cary and Brizard, 2009].). Therefore, particles move along straight lines in (U, B m ) space while preserving the second adiabatic invariant, J.…”

Monte Carlo method and High Performance Computing for solving Fokker–Planck equation of minority plasma particles

“…His work was subsequently extended on many occasions. We encourage the reader especially to the paper by Brizard (1995), and to the paper by Cary & Brizard (2009) (and the references therein) where also the history of the guiding-center theory is outlined in detail. In this contribution, we are mostly following these aforementioned works.…”