dq N dp N Ο const, (3a) A numerical method for the time evolution of systems described by Liouville-type equations is derived. The algorithm uses a lattice
of numerical markers, which follow exactly Hamiltonian trajectories, to represent the operator d/dt in moving (i.e., Lagrangian) coordinates. However, nonconservative effects such as particle drag, cre-In the above, β¬β« [t] refers to any connected, moving collecation, and annihilation are allowed in the evolution of the physical tion of phase points. The N Οͺ 2 intermediate integralsdistribution function, which is itself represented according to a Ν³f decomposition. Further, the method is suited to the study of a gen-which we have not included-are written more naturally eral class of systems involving the resonant interaction of energetic using differential forms, and we refer the reader to the particles with plasma waves. Detailed results are presented for both text by Arnold [1] for a general discussion. These quantities the classic bump-on-tail problem and the beam-driven TAE instabilare time (and canonical transformation) invariant.
ity. In both cases, the algorithm yields exceptionally smooth, low-However, if one wants to model the motion of particles noise evolution of wave energy, especially in the linear regime.
Phenomena associated with the nonlinear regime are also which can be effectively created or destroyed, then the described.