The exact and drift Hamiltonian

A canonical Hamiltonian treatment for the motion of the guiding center of a charged particle in a magnetic field of a toroidal configuration is given. The Hamiltonian of the exact trajectory, first- and second-order corrections to the standard drift Hamiltonian are given in the Boozer coordinates for a vacuum magnetic field with magnetic surfaces. The exact Hamiltonian depends on both the field strength and the shape of the magnetic surfaces (the metric of the magnetic coordinates) while the standard drift Hamiltonian depends only on the field strength. The first-order correction, in gyroradius to system size, to the standard drift Hamiltonian depends in a generic way on the metric of the magnetic coordinates, as does the exact Hamiltonian.