https://doi.org/10.1140/epjp/s13360-023-03751-6
Regular Article
3D homogeneous potentials generating two-parametric families of orbits on the outside of a material concentration
Department of Physics, Aristotle University of Thessaloniki, 54124, Thessaloniki, University Campus, Greece
Received:
19
December
2022
Accepted:
26
January
2023
Published online:
6
February
2023
We study three-dimensional homogeneous potentials V = V(x, y, z) of degree m which are created outside a finite concentration of matter and they produce a preassigned two-parametric family of spatial regular orbits given in the solved form f(x, y, z) = 
, g(x, y, z) = 
(, 
= {\rm const}). These potentials have to satisfy three linear PDEs; two of them come from the Inverse Problem of Newtonian Dynamics and the last one is the well-known ”Laplace’s equation”. Our aim is to find common solutions for these three PDEs. Besides that we consider that the functions f and g are also homogeneous in the variables 
of any degree and can be represented uniquely by the ”slope functions” 
and 
which are homogeneous of zero degree. Then, we impose three differential conditions on the orbital functions (
). If they are satisfied for a specific value of m, then we can find the potential by quadratures. The values obtained for m so far are consistent with familiar gravitational and electrostatic and quadratic potentials. Finally, pertinent examples are given and cover all the cases.
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