The planar inverse problem for autonomous systems.
Abstract
The authors study the general version of the inverse problem for planar trajectories and for autonomous dynamical systems possessing three integrals, i.e., for a given threeparametric family of curves f(x,y,a,b) = c they find the potential V(x,y) for which these curves are orbits of a unit mass. All possible cases, depending on the preassigned function f, are classified and in each case the necessary and sufficient conditions for the existence of a solution are established. Among the examples there is the case of the Keplerian conic sections which is studied in detail.
 Publication:

IAU Colloq. 74: Dynamical Trapping and Evolution in the Solar System
 Pub Date:
 1983
 DOI:
 10.1007/9789400972148_36
 Bibcode:
 1983ASSL..106..353X
 Keywords:

 Celestial Mechanics;
 Orbit Calculation;
 Systems Analysis;
 Cartesian Coordinates;
 Dynamic Characteristics;
 Invariance;
 Kepler Laws;
 Partial Differential Equations;
 Astronomy;
 Celestial Mechanics:Potential Theory;
 Potential Theory:Celestial Mechanics