  Equivalent guiding-center Hamiltonian theories are constructed based on
higher-order Lie-transform perturbation methods. Higher-order guiding-center
theories are distinguished on the basis of whether correction terms associated
with magnetic-field nonuniformity appear either in the guiding-center
symplectic (Poisson-bracket) structure, in the guiding-center Hamiltonian, or
both. These theories are called equivalent because they describe the same
guiding-center magnetic-moment invariant. The present work presents the
detailed derivations of results that were summarized in a recent paper by
Tronko and Brizard [Phys. Plasmas 22, 112507 (2015)].
