A long time ago I spent some time analyzing the torus (doughnut or bagel).

I managed to develop (and then forget) differential equations for geodesics (shortest distance curves), but never managed to develop an analytical expression for the gravitational field.

Can anybody provide differential equations for geodesics? I would like to avoid redoing the work. I would also be interested in equations for geodesics on a spheroid with three unequal axes, which I think are more difficult to develop than equations for the torus.

I do not think it is possible to develop an analytical expression for the gravitational field, but would like to know if anybody thinks otherwise. Actually, I think that an expression for the field (or the potential) due to mass uniformly distributed on a circle would be enough to solve this problem.

Has anybody here have any good ideas on the above?