Abstract
The 3D Inverse Problem of Newtonian dynamics is governed by two linear partial differential equations (PDEs) in the unknown potential function \(V=V(x,y,z)\) . The first one is of first-order and the second-one is of second-order. These PDEs combine potentials and two-parametric families of orbits in 3D space. The potential function \(V=V(x,y,z)\) produces all the two-parametric families of orbits given in the form \(f(x,y,z)=d_{1}\) , \(g(x,y,z)=d_{2}\) ( \(d_{1},\;d_{2}=const.\) ) which are compatible with it. Each two-parametric family of orbits is represented uniquely by a pair of ‘‘slope functions’’ \(\alpha=\alpha(x,y,z)\) , \(\beta=\beta(x,y,z)\) . In the present work, we study all the pairs of ‘‘slope functions’’ \(\{\alpha=\alpha(y),\;\beta=\beta(z)\}\) , or, \(\{\alpha=\beta\}\) , or, \(\{\alpha\neq 0,\;\beta=\frac{{z}}{{x}}\}\) , and we find a new second-order PDE for the potential. From this PDE, we can retrieve solutions for the potential function \(V=V(x,y,z)\) . On the other hand, if the potential is given in advance, we study the direct problem of dynamics and we find suitable pair of families compatible with it. Pertinent examples are given and the families of straight lines are studied separately.
