  Techniques for coordinate changes that depend on both dependent and
independent variables are developed and applied to the Maxwell-Vlasov
Hamiltonian theory. Particle coordinate changes with a new velocity variable
dependent on the magnetic field, with spatial coordinates unchanged, are lifted
to transform the noncanonical Poisson bracket and, thus, the field Hamiltonian
structure of the Vlasov-Maxwell equation. Several examples are given including
magnetic coordinates, where the velocity is decomposed into components parallel
and perpendicular to the local magnetic field, and the case of spherical
velocity coordinates. An example of the lifting procedure is performed to
obtain a simplified version of gyrokinetics, where the magnetic moment is used
as a coordinate and the dynamics is reduced by elimination of the electric
field energy in the Hamiltonian.
