Two complementary representations of higher-order guiding-center theory arepresented, which are distinguished by whether higher-order corrections due tomagnetic-field nonuniformity appear in the guiding-center Poisson bracket orthe guiding-center Hamiltonian. The equivalence of these two representationsimplies that the guiding-center gyroaction adiabatic invariant and theguiding-center Hamilton equations of motion are representation-invariants. Byusing a new perturbative hierarchy in guiding-center theory, second-ordercorrections are introduced in Hamiltonian guiding-center theory without theneed to calculate the guiding-center transformation at second order.
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