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The role of single orbits in dynamics

We describe in some detail the structure of the set of the infinitely many monoparametric families f(x,y) = c to which one, two or three single orbits f0(x,y) = 0 (possibly resulting from astronomical observations) may be classified. By a heuristic argument we show how families including more than three preassigned single orbits can be constructed. The motivation for this study stems from the fact that the inverse problem of dynamics relates potentials V (x,y) to monoparametric families of orbits (and not to single orbits). So, as an application of the geometrical considerations of the present study we find a partial differential equation, the solution of which helps us to gain a full family out of one observed orbit in a known potential.