Construction of 3D potentials from a preassigned two-parametric family of orbits

One of the main problems of astrophysics is to determine the mean field potential of galaxies. The astronomical observations, as well as the numerical simulations, lead to the determination of families of star orbits in a galaxy. On this basis, using the tools of the inverse problem of dynamics, it is possible to find the gravitational potential which gives rise to such motions. The problem can be formulated in various ways; we consider here that the particle trajectories are given by a spatial two-parameter family of curves. From these trajectories we obtain certain functions known as orbital functions, and look for potentials of a special form. If the orbital functions satisfy some differential conditions, a step-by-step procedure offers the expression of the potential. Similar problems arise in thermodynamics and nuclear physics, where axially symmetric potentials are used as models for deformed nuclei.