Isoenergetic families of regular orbits on a given surface

Thomas Kotoulas
In the light of inverse problem of dynamics, we study homogeneous potentials of the form V (u, v)= u m R(v u) of any degree m which produce a mono-parametric family of regular orbits f (u, v)=c on a certain surface. Especially, we are interested in isoenergetic orbits, i.e. orbits which are described by a material point of unit mass with constant energy E=E 0. For any given family of regular orbits on a smooth surface, we offer the criteria which must be fulfilled by the given family of orbits
more » ... n family of orbits so that it can be created by such a potential of degree m. Consequently, Mertens' PDE ([13]) is simplified and can be solved completely and uniquely. Special cases are also studied and several examples are given. M.S.C. 2000: 70B05, 70F17, 70M20, 53A05, 37N05, 35Q72.