Particle canonical variables and guiding center Hamiltonian up to second order in the Larmor radius

A generating function, expressed as a power series in the particle Larmor radius, is used to relate an arbitrary set of magnetic field line coordinates to particle canonical variables. A systematic procedure is described for successively choosing the generating function at each order in the Larmor radius so that the transformed particle Hamiltonian is independent of the Larmor phase angle. The particle guiding center Hamiltonian up to second order in the Larmor radius is thereby derived. The analysis includes finite electric fields in which the particle “electric drifts” can be of the order of the particle velocity. The transformations which relate an arbitrary set of toroidal magnetic flux coordinates to particle canonical variables are also discussed.