Equivalent Higher-order Guiding-center Hamiltonian Theories
  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)]. 
A. J. Brizard and N. Tronko
physics.plasm-ph