FAMILY BOUNDARY CURVES FOR HOLONOMIC SYSTEMS WITH 2 DEGREES-OF-FREEDOM

The notion of the family boundary curves (FBC) is introduced for that version of the inverse problem of Lagrangian dynamics which deals with the determination of the potential V(u, v) under which a given monopa rametric family of curves f(u, v) = c, on the configuration manifold ( M(2), g) of a conservative holonomic system with n = 2 degrees of free dom, can be described as dynamical trajectories (orbits) of the repres entative point. It is shown that, in general, curves of the family f(u , v) = c generated by a class of potentials V(u,v) are actual orbits o nly in a subregion of the region where they are defined as geometrical entities. (In general, the FBC are distinct from the well known zero velocity curves (ZVC), the later referring to orbits of the same const ant energy). If, however, the holonomic system is subject to non-conse rvative generalized forces, it is shown that we can always find many p airs of such forces {Q(1), Q(2)} giving rise to any family of trajecto ries lying in any pre-assigned (open or closed) region of the configur ation space. Three examples are presented to account both for conserva tive and non-conservative forces.