3D Homogeneous and Axisymmetric Potentials Producing Two-Parametric Families of Orbits - Kotoulas, Thomas; Meletlidou, E. (Discontinuity, Nonlinearity, and Complexity, 2026)
One of the major problems in classical mechanics is to determine the mean field potential in which the motion of a test particle takes place. In the light of the inverse problem of dynamics, we study three-dimensional genuine potentials V=V(x,y,z) producing a set of two-parametric families of regular orbits f(x,y,z)=c_1, g(x,y,z)=c_2 (c_1, c_2=const.). We focus on homogeneous and axisymmetric potentials which have many physical applications. Then, we establish three differential conditions to be fulfilled by the given two-parametric families of orbits (traced in 3D space by a material point) so that these families can result in the presence of such a potential. All possible cases for the ``given’’ family of orbits are studied and several compatible pairs of families and potentials are found. Finally, some potentials of physical interest are also presented.