3D homogeneous potentials generating two-parametric families of orbits on the outside of a material concentration

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) = c(1,) g(x, y, z) = c(2) (c(1), c(2) = {\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 x, y, z of any degree and can be represented uniquely by the slope functions alpha (x, y, z) and beta (x, y, z) which are homogeneous of zero degree. Then, we impose three differential conditions on the orbital functions (alpha, beta). 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.

Automated summary

We study homogeneous potentials V = V(x, y, z) of degree m which are created outside a finite concentration of matter and produce a preassigned two-parametric family of regular orbits. These potentials satisfy three linear PDEs, with two coming from the Inverse Problem of Newtonian Dynamics and the last one being Laplace's equation. By imposing differential conditions on the orbital functions, we can find the potential by quadratures for specific values of m. The obtained values are consistent with familiar gravitational and electrostatic potentials.