Counter-rotating stellar discs around a massive black hole:
  self-consistent, time-dependent dynamics
  We formulate the collisionless Boltzmann equation (CBE) for dense star
clusters that lie within the radius of influence of a massive black hole in
galactic nuclei. Our approach to these nearly Keplerian systems follows that of
Sridhar and Touma (1999): Delaunay canonical variables are used to describe
stellar orbits and we average over the fast Keplerian orbital phases. The
stellar distribution function (DF) evolves on the longer time scale of
precessional motions, whose dynamics is governed by a Hamiltonian, given by the
orbit-averaged self-gravitational potential of the cluster. We specialize to
razor-thin, planar discs and consider two counter-rotating ("$\pm$")
populations of stars. To describe discs of small eccentricities, we expand the
$\pm$ Hamiltonian to fourth order in the eccentricities, with coefficients that
depend self-consistently on the $\pm$ DFs. We construct approximate $\pm$
dynamical invariants and use Jeans' theorem to construct time-dependent $\pm$
DFs, which are completely described by their centroid coordinates and shape
matrices. When the centroid eccentricities are larger than the dispersion in
eccentricities, the $\pm$ centroids obey a set of 4 autonomous ordinary
differential equations. We show that these can be cast as a two-degree of
freedom Hamiltonian system which is nonlinear, yet integrable. We study the
linear instability of initially circular discs and derive a criterion for the
counter-rotating instability. We then explore the rich nonlinear dynamics of
counter-rotating discs, with focus on the variety of steadily precessing
eccentric configurations that are allowed. The stability and properties of
these configurations are studied as functions of parameters such as the disc
mass ratios and angular momentum.
