A study of the guiding center approximation

A Hamiltonian treatment for the motion of a charged particle in a toroidal magnetic field is given. Assuming the plasma equilibrium, Boozer coordinates are used. The Hamiltonian of the exact trajectory is given in the guiding center coordinates. The higher order corrections to the standard drift Hamiltonian are derived. It is shown that the exact Hamiltonian depends on both the field strength and the shape of the magnetic surfaces (the metric of Boozer coordinates) while the standard drift Hamiltonian depends only on the field strength. The first order correction to the standard drift Hamiltonian, in gyroradius to system size, depends in a generic way on the shape of the magnetic surface, as does the exact Hamiltonian. Numerical calculations are done to compare the trajectories of the exact Hamiltonian, the standard drift Hamiltonian and the higher order drift Hamiltonian for a quasihelical symmetric vacuum field. The numerical results show that the difference in phase space structure between the exact and standard drift Hamiltonian can be predicted by the drift Hamiltonian with the first order corrections