Hamiltonian guiding center equations in a toroidal system

A Hamiltonian method to study the guiding center motion of charged particles in a toroidal magnetic system has been developed. It uses a cylindrical coordinate system instead of a magnetic coordinate system on which many conventional standard methods are based. The six-dimensional (6D) Hamiltonian equations for the guiding center motion are derived by a canonical transformation of fast-oscillating variables to slowly varying ones which are guiding center coordinates. It is shown that one of these slowly varying variables, i.e., the action variable conjugated to the fast-oscillating gyrophase is an adiabatic invariant for the tokamak equilibrium magnetic field perturbed by the external time-dependent magnetic field. This allows to reduce the 6D Hamiltonian system to the 4D one. The method is valid for the study of the guiding center motion of particles in time-dependent magnetic and electric fields, especially, ergodic magnetic fields, where spatial and temporal scales of variation are much larger than the gyroradius and the gyroperiod.