Hamiltonian equations of motion for a nonrelativistic charged particle in magnetospheric hydromagnetic perturbations are derived. The equations are gyroaveraged, allowing much larger time steps in numerical solutions of the equations of motion compared to integrating the full Lorentz equations of motion, but they contain finite-gyroradius effects to all orders in k⊥&rgr;, where k⊥ is the perpendicular wave number and &rgr; is the particle gyroradius. The finite-gyroradius effects are essential for the important class of particles which undergo magnetic drift-bounce resonances with the waves. The equations are derived by finding a Lie transform of the perturbed guiding center phase-space Lagrangian to a new Lagrangian which is independent 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 energy (for static systems) and conservation of phase-space volume. The Hamiltonian conservation properties are useful for checking the accuracy of numerical integration schemes and they are essential for the use of Poincar¿ surface-of-section plots. Compared to more traditional canonical Hamiltonian methods, the phase-space Lagrangian Lie transform methods allow general, noncanonical phase-space coordinates and transformations. This results in more power and flexibility in finding convenient forms for the final equations of motion. The results are given in coordinate-free form and in terms of magnetic field coordinates. Applications of these results to calculations of hydromagnetic wave-induced particle motion in the inner magnetosphere are discussed. ¿ 1998 American Geophysical Union