Recently I. Mladenov and J. Oprea have investigated a number of surfaces of revolution, and in particular, developed numerical shooting methods to investigate geodesics on those surfaces, which in turn led them to raise some questions concerning closed geodesics on those surfaces. Here we develop explicit formulae, usually in terms of elliptic integrals, that permit us to answer the questions. The computations are based of course on the classical Clairaut's formulae, and a major point is to demonstrate that explicit computations can be made. A closed geodesic on a surface of revolution oscillates $q$ times across the equator of the surface, while winding $p$ times around the axis of rotation. Call the pair $(p,q)$ the type of the geodesic. In particular, the permitted types are explicitly determined for the investigated surfaces.

This work is a review of our research activity during the last ten years concerning the problems of modeling and control of multi-body mechanical systems. Because the treatment of the above topics is quite sensitive with respect to the different parameterizations of the rotation group in three dimensional space $\mathrm{SO}(3)$ and because the properties of the parameterization more or less influence the efficiency of the dynamic model, here the so called vector-parameter is used for parallel considerations. The consideration of the mechanical system in the configurational space of pure vector-parameters with a group structure opens the possibilities for the Lie group theory to be applied in the problems of the dynamics and control. The sections in this paper present independent parts of an unified scientific approach.