3D Homogeneous and Axisymmetric Potentials Producing Two-Parametric Families of Orbits
DOI:
https://doi.org/10.5890/DNC.2023.12.002Abstract
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.References
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