Richard Montgomery
Publications:
Montgomery R.
The Hyperbolic Plane, ThreeBody Problems, and Mnëv’s Universality Theorem
2017, vol. 22, no. 6, pp. 688–699
Abstract
We show how to construct the hyperbolic plane with its geodesic flow as the reduction of a threeproblem whose potential is proportional to $I/\Delta^2$ where $I$ is the moment of inertia of this triangle whose vertices are the locations of the three bodies and $\Delta$ is its area. The reduction method follows [11]. Reduction by scaling is only possible because the potential is homogeneous of degree $2$. In trying to extend the assertion of hyperbolicity to the analogous family of planar $N$body problems with threebody interaction potentials we run into Mnëv’s astounding universality theorem which implies that the extended assertion is doomed to fail.

Montgomery R.
MICZKepler: Dynamics on the Cone over $SO(n)$
2013, vol. 18, no. 6, pp. 600607
Abstract
We show that the $n$dimensional MICZKepler system arises from symplectic reduction of the "Kepler problem" on the cone over the rotation group $SO(n)$. As a corollary we derive an elementary formula for the general solution of the MICZKepler problem. The heart of the computation is the observation that the additional MICZKepler potential, $\phi^2/r^2$, agrees with the rotational part of the cone’s kinetic energy.
