Limitations of the Hamiltonian Treatment for Collisionless Astrophysical Accretion Flows

In the Hamiltonian treatment of purely mechanical systems, the canonical and actual momentum of a particle are the same. In contrast, for a plasma of charged particles and electromagnetic fields, those two momenta are different. We show how this distinction is fundamental in identifying the limitations of a recent attempt by Binney (2003) to rule out two-temperature collisionless astrophysical accretion flows from Hamiltonian theory. This illustrates the Hamiltonian method for astrophysical plasmas, its relation to the equations of motion, and its role in practical calculations. We also discuss how the complete Hamiltonian treatment of a plasma should couple the particle motion to a fully dynamical treatment of the electromagnetic fields. Our results stand independent from the discussion of Quataert (2003) who argued that time scale calculated in Binney (2003) is not the equipartition time as claimed.