Extensions of adiabatic invariant theory for a charged particle

The standard theory of Hamiltonian dynamics of a charged particle is extended to allow electric and magnetic fields to vary across magnetic field lines or surfaces on the Larmor radius distance scale. After the development of the general theory, the special cases of toroidally nested magnetic surfaces and of axisymmetry are considered. In a further restriction the situation with spatially slowly varying static magnetic fields but spatially rapidly varying static bounded electrostatic potentials is treated. The dynamics of the perpendicular velocity is represented by a nonlinear oscillator. The adiabatic invariant and drift Hamiltonian are constructed near an O point in the perpendicular velocity phase plane. Motion near a separatrix and X point in physical space is also briefly explored.