Hamiltonian formulation of guiding center motion

Abstract

Nonrelativistic guiding center motion in the magnetic field BB(x), with E=0, is studied using Hamiltonian methods. f'J The drift equations are carried to second order in the perpendicular motion. The Hamiltonian methods which are used are described in detail in order to facilitate possible applications. Unusual mathematical techniques are called upon, especially the use of noncanonical coordinates in phase space. Lie transforms are used to carry out the perturbation expansion. Applications in kinetic theory, in the area of adiabatic invariants, and in other areas are anticipated.

Modern gyrokinetic formulation of collisional and turbulent transport in toroidally rotating plasmas

“…In the present paper, the background electromagnetic fields are assumed to slowly vary in the time scale $ ðq=LÞ À2 ðL=v T Þ. [Note that, in this subsection, we follow Littlejohn's work (Littlejohn 1981) where the guiding-center motion equations are derived so as to be valid even for the background fields that rapidly vary with the transit time scale]. Then, d $ q=L is used as an ordering parameter for the perturbation expansion to make the guiding center approximation.…”

“…(1) is represented as a function of ðZ; _ Z; tÞ, where _¼ d=dt denotes the derivative with respect to the time t. Here, following Littlejohn (Littlejohn 1981), the Lagrangian LðZ; _ Z; tÞ is written as…”

“…Here, following Brizard's terminology (Brizard 1989), we refer to the single-particle phase-space variables defined from the equilibrium and perturbed fields as the guiding-center and gyrocenter coordinates, respectively. The guiding-center equations derived by Littlejohn in the high-flow ordering (Littlejohn 1981) are reviewed in Sect. 2.1 and they are modified to treat toroidally rotating plasmas in Sect.…”

Relativistic ponderomotive Hamiltonian

“…The equations of motion (the drift equations) are given implicitly by <5 f Ldt -0 , (13) or explicitly by…”

Toroidal Precession as a Geometric Phase

“…We start from the non-canonical Hamiltonian formulation of guiding center motion found by Littlejohn [20][21][22] . The guiding center phase space, P , is the cartesian product of the physical domain D and the parallel velocity axis, P = D × R. The symplectic structure on P is given by the Lagrange tensor −dϑ, where…”