\def\sles{\lower2pt\hbox{$\buildrel {\scriptstyle <} \over {\scriptstyle\sim}$}} \def\sgreat{\lower2pt\hbox{$\buildrel {\scriptstyle >} \over {\scriptstyle\sim}$}}
Studies of inclined rings inside galaxy potentials have mostly considered the influence of self--gravity and viscous dissipation separately. In this study, we construct models of highly-inclined (``polar'') rings in an external potential including both self--gravity and dissipation due to a drag force. We do not include pressure forces and thus ignore shock heating that dominates the evolution of gaseous rings inside strongly nonspherical potentials. We adopt an oblate spheroidal scale--free logarithmic potential with axis ratio $q=0.85$ and an initial inclination of 80$^\circ$ for the self--gravitating rings. We find that stellar (dissipationless) rings suffer from mass loss during their evolution. Mass loss also drives a secular change of the mean inclination toward the poles of the potential. As much as half of the ring mass escapes in the process and forms an inner and an outer shell of precessing orbits. If the remaining mass is more than $\sim$ 0.02 of the enclosed galaxy mass, rings remain bound and do not fall apart from differential precession. The rings precess at a constant rate for more than a precession period $\tau_p$ finding the configuration predicted by Sparke (1986) which warps at larger radii toward the poles of the potential. We model shear viscosity with a velocity-dependent drag force and find that nuclear inflow dominates over self--gravity if the characteristic viscous inflow time scale $\tau_{vi}$ is shorter than $\sim 25 \tau_p$. Rings with $\tau_{vi}/\tau_p \ \sles \ 25$ collapse toward the nucleus of the potential within one precession period independent of the amount of self--gravity. Our results imply that stars and gas in real polar rings exhibit markedly different dynamical evolutions.
\noindent This work is supported in part by NASA grant NAGW--1510, NSF grant AST 91--48279, and by a CfA Postdoctoral fellowship.
\noindent Sparke, L. S. 1986, MNRAS, 219, 657