  We investigate the dynamical behavior of rings around bodies whose shapes
depart considerably from that of a sphere. To this end, we have developed a new
self-gravitating discrete element N-body code, and employed a local simulation
method to simulate a patch of the ring. The central body is modeled as a
symmetric (oblate or prolate) ellipsoid, or defined through the characteristic
frequencies (circular, vertical, epicyclic) that represent its gravitational
field. Through our simulations we explore how a ring's behavior --
characterized by dynamical properties like impact frequency, granular
temperature, number density, vertical thickness and radial width -- varies with
the changing gravitational potential of the central body. We also contrast
properties of rings about large central bodies (e.g. Saturn) with those of
smaller ones (e.g. Chariklo). Finally, we investigate how the characteristic
frequencies of a central body, restricted to being a solid of revolution with
an equatorial plane of symmetry, affect the ring dynamics. The latter process
may be employed to qualitatively understand the dynamics of rings about any
symmetric solids of revolution.
