NONCANONICAL HAMILTONIAN METHODS FOR PARTICLE MOTION IN MAGNETOSPHERIC HYDROMAGNETIC-WAVES

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.