Abstract
The version of the three-dimensional Inverse Problem of Newtonian dynamics considered here is the following: given a family of spatial curves in the form \(f(x,y,z)=k_{1}\) , \(g(x,y,z)=k_{2}\) ( \(k_{1}\) , \(k_{2}=\text{const}\) ), find all the potentials \(V=F(u,w)\) , where \(u=u(x,y,z)\) , \(w=w(x,y,z)\) , which give rise to this two-parametric family of orbits. Each family of orbits is represented uniquely by a pair of ‘‘slope functions’’ \(\alpha=\alpha(x,y,z)\) and \(\beta=\beta(x,y,z)\) . The solution of this problem is based on two linear partial differential equations (PDEs) in the unknown potential function \(V=V(x,y,z)\) . In the present work, we establish three differential conditions for the slope functions ( \(\alpha,\;\beta\) ). If these are satisfied, then they guarantee the existence of such a potential and it is found by quadratures. Pertinent examples are given and the families of straight lines are studied separately.
