  Two complementary representations of higher-order guiding-center theory are
presented, which are distinguished by whether higher-order corrections due to
magnetic-field nonuniformity appear in the guiding-center Poisson bracket or
the guiding-center Hamiltonian. The equivalence of these two representations
implies that the guiding-center gyroaction adiabatic invariant and the
guiding-center Hamilton equations of motion are representation-invariants. By
using a new perturbative hierarchy in guiding-center theory, second-order
corrections are introduced in Hamiltonian guiding-center theory without the
need to calculate the guiding-center transformation at second order.

Plasma Physics