Families of Planar Orbits in Polar Coordinates Compatible with Potentials

In light of the planar inverse problem of Newtonian Dynamics, we study the monoparametric family of regular orbits $f(r,\theta )=c$ in polar coordinates (where c is the parameter varying along the family of orbits), which are generated by planar potentials $V=V(r,\theta )$. The corresponding family of orbits can be uniquely represented by the “slope function” $\gamma ={\displaystyle \frac{f_{\theta }}{f_r}}$. By using the basic partial differential equation of the planar inverse problem, which combines families of orbits and potentials, we apply a new methodology in order to find specific potentials, e.g., $V=A\left(r\right)+B\left(\theta \right)$ or $V=H\left(\gamma \right)$ and one-dimensional potentials, e.g., $V=A\left(r\right)$ or $V=G\left(\theta \right)$. In order to determine such potentials, differential conditions on the family of orbits $f(r,\theta )$ = c are imposed. If these conditions are fulfilled, then we can find a potential of the above form analytically. For the given families of curves, such as ellipses, parabolas, Bernoulli’s lemniscates, etc., we find potentials that produce them. We present suitable examples for all cases and refer to the case of straight lines.