Hamiltonian equations of motion for a nonrelativistic charged particle
in magnetospheric hydromagnetic perturbations are derived. The equati
ons are gyroaveraged, allowing much larger time steps in numerical sol
utions of the equations of motion compared to integrating the full Lor
entz equations of motion, but they contain finite-gyroradius effects t
o all orders in k(perpendicular to)rho, where k(perpendicular to) is t
he perpendicular wave number and rho is the particle gyroradius. The f
inite-gyroradius effects are essential for the important class of part
icles which undergo magnetic drift-bounce resonances with the waves. T
he equations are derived by finding a Lie transform of the perturbed g
uiding center phase-space Lagrangian to a new Lagrangian which is inde
pendent of the gyrophase angle. The resulting Euler-Lagrange equations
contain nonlinear terms which automatically preserve the Hamiltonian
properties of the original Lorentz system, such as conservation of ene
rgy (for static systems) and conservation of phase-space volume. The H
amiltonian conservation properties are useful for checking the accurac
y of numerical integration schemes and they are essential for the use
of Poincare surface-of-section plots. Compared to more traditional can
onical Hamiltonian methods, the phase-space Lagrangian Lie transform m
ethods allow general, noncanonical phase-space coordinates and transfo
rmations. This results in more power and flexibility in finding conven
ient forms for the final equations of motion. The results are given in
coordinate-free form and in terms of magnetic field coordinates. Appl
ications of these results to calculations of hydromagnetic wave-induce
d particle motion in the inner magnetosphere are discussed.