Monoparametric Families of Orbits Produced by Planar Potentials

We study the motion of a test particle on the xy-plane. The particle trajectories are given by a one-parameter family of orbits f(x,y) = c, where c = const. By using the tools of the 2D inverse problem of Newtonian dynamics, we find two-dimensional potentials that produce a pre-assigned monoparametric family of regular orbits f(x,y)=c that can be represented by the "slope function" ?=f(y)f(x) uniquely. We apply a new methodology in order to find potentials depending on specific arguments, i.e., potentials of the form V(x,y)=P(u) where u=x(2)+y(2),xy,x(3)-y(3),xy (x,y?0). Then, we establish one differential condition for the family of orbits f(x,y) = c. If it is satisfied, it guarantees the existence of such a potential, generating the above family of planar orbits. Then, the potential function V=V(x,y) is found by quadratures. For known families of curves, e.g., ellipse, the logarithmic spiral, the lemniscate of Bernoulli, and circles, we find homogeneous and polynomial potentials that are compatible with this family of orbits. We offer pertinent examples that cover all of the cases, and we examine which of these potentials are integrable. We also study one-dimensional potentials. The families of straight lines in 2D space are also examined.