The exact and drift Hamiltonian in a toroidal magnetic field

The exact Hamiltonian and the first‐order corrections to the standard drift Hamiltonian are given in magnetic coordinates for a full static electromagnetic 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 on the field strength only. The first‐order correction to the standard drift Hamiltonian, in gyroradius to system size, depends in a generic way on the metric of the magnetic coordinates, as does the exact Hamiltonian. The finite‐Larmor‐radius effect and the polarization drift are in the first‐order corrections.