Inverse and Direct Problem of the Dynamics of Central Motions

For a given monoparametric family of orbits, a Szebehely-type inverse problem is solved i.e. a linear partial differential equation of the first order is written giving the radial component F = F(r, θ) of a central force (in general not conservative) creating the family. It is shown how this equation can be used also for direct problem considerations and that, in this case, it reads as second order nonlinear partial differential equation of the Monge-Ampère type. The equation is also used to provide conditions so that a preassigned monoparametric or two-parametric family of orbits can be generated by a conservative central force F = F(r). The force F(r) as well as the expressions for the angular momentum and the total energy dependence on the given family are found.