The Kepler Cone, Maclaurin Duality and Jacobi-Maupertuis metrics

Abstract: The Kepler problem is the special case $\alpha = 1$ of the power law problem: to solve Newton's equations for a central force whose potential is of the form $-\mu/r^{\alpha}$ where $\mu$ is a coupling constant. Associated to such a problem is a two-dimensional cone with cone angle $2 \pi c$ with $c = 1 - \frac{\alpha}{2}$. We construct a transformation taking the geodesics of this cone to the zero energy solutions of the $\alpha$-power law problem. The Kepler Cone' is the cone associated to the Kepler problem. This zero-energy cone transformation is a special case of a transformation discovered by Maclaurin in the 1740s transforming the $\alpha$- power law problem for any energies to aMaclaurin dual' $\gamma$-power law problem where $\gamma = \frac{2 \alpha}{2-\alpha}$ and which, in the process, mixes up the energy of one problem with the coupling constant of the other. We derive Maclaurin duality using the Jacobi-Maupertuis metric reformulation of mechanics. We then use the conical metric to explain properties of Rutherford-type scattering off power law potentials at positive energies. The one possibly new result in the paper concerns ``star-burst curves'' which arise as limits of families negative energy solutions as their angular momentum tends to zero. We relate geodesic scattering on the cone to Rutherford type scattering of beams of solutions in the potential. We describe some history around Maclaurin duality and give two derivations of the Jacobi-Maupertuis metric reformulation of classical mechanics. The piece is expository, aimed at an upper-division undergraduate. Think American Math. Monthly.