For two masses interacting via a central potential $V(r)$, the center-of-mass motion separates, leaving a one-body problem with reduced mass $\mu$. The relative motion obeys $\mu\ddot{\mathbf{r}}=-\nabla V(r)$ and angular momentum conservation confines the orbit to a plane.

Writing the radial dynamics in terms of the effective potential $V_{\mathrm{eff}}(r)=V(r)+\frac{L^2}{2\mu r^2}$ reveals turning points, circular orbits, and stability. For $V(r)=-k/r$, one obtains the Kepler problem, whose bounded solutions are conic sections with the force center at a focus.

The Kepler problem is superintegrable: it possesses more conserved quantities than degrees of freedom, including the Laplace-Runge-Lenz vector. This extra symmetry explains the closed orbits and the high degree of solvability. The problem remains a cornerstone for studying perturbations, resonances, and near-integrable motion.