Semi-Separable Potentials as Solutions to the 3D Inverse Problem of Newtonian Dynamics

Abstract

We study the motion of a test particle in a conservative force-field. Our aim is to find three-dimensional potentials with symmetrical properties, i.e., V(x,y,z)=P(x,y)+Q(z), or, V(x,y,z)=P(x2+y2)+Q(z) and V(x,y,z)=P(x,y)Q(z), where P and Q are arbitrary C2-functions, which are characterized as semi-separable and they produce a pre-assigned two-parametric family of orbits f(x,y,z) = c1, g(x,y,z) = c2 (c1, c2 = const) in 3D space. There exist two linear PDEs which are the basic equations of the Inverse Problem of Newtonian Dynamics and are satisfied by these potentials. Pertinent examples are presented for all the cases. Two-dimensional potentials are also included into our study. Families of straight lines is a special category of curves in 3D space and are examined separately.