Family boundary curves for holonomic systems with two degrees of freedom

Abstract

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 monoparametric family of curves f(u,v)=c, on the configuration manifold (M2,g) of a conservative holonomic system with n=2 degrees of freedom, can be described as dynamical trajectories (orbits) of the representative 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 only 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 constant energy). If, however, the holonomic system is subject to non-conservative generalized forces, it is shown that we can always find many pairs of such forces (Q1,Q2) giving rise to any family of trajectories lying in any pre-assigned (open or closed) region of the configuration space. Three examples are presented to account both for conservative and non-conservative forces.