Post-Newtonian Hamiltonian dynamics: applications to stationary
  spacetimes and statistical mechanics
  Although the post-Newtonian Lagrangian formalism is widely used in
relativistic dynamical and statistical studies of test bodies moving around
arbitrary mass distributions, the corresponding general Hamiltonian formalism
is still relatively uncommon, being restricted basically to the case of N-body
problems. Here, we present a consistent Hamiltonian formalism for the dynamics
of test particles in spacetimes with arbitrary energy-momentum distributions in
the first post-Newtonian (1PN) approximation. We apply our formalism to orbital
motion in stationary axisymmetric spacetimes and obtain the 1PN relativistic
corrections to the radial and vertical epicyclic frequencies for quasi-circular
equatorial motion, a result potentially interesting for galactic dynamics. For
the case of razor-thin disk configurations, we obtain an approximated third
integral which could be used to determine analytically the envelope of nearly
equatorial orbits. One of the main advantages of this 1PN analysis is the
explicit presence of frame-dragging effects in all pertinent expressions,
allowing some qualitative predictions in rotating spacetimes. We finish
discussing the 1PN collisionless Boltzmann equation in terms of the Hamiltonian
canonical variables and its relation with previous results in the literature.
