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Richard Montgomery

Richard  Montgomery
Santa Cruz, CA, USA
Dept. of Mathematics, University of California


Montgomery R.
The Hyperbolic Plane, Three-Body Problems, and Mnëv’s Universality Theorem
2017, vol. 22, no. 6, pp.  688–699
We show how to construct the hyperbolic plane with its geodesic flow as the reduction of a three-problem 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 three-body interaction potentials we run into Mnëv’s astounding universality theorem which implies that the extended assertion is doomed to fail.
Keywords: Jacobi–Maupertuis metric, reduction, Mnev’s Universality Theorem, three-body forces, Hyperbolic metrics
Citation: Montgomery R.,  The Hyperbolic Plane, Three-Body Problems, and Mnëv’s Universality Theorem, Regular and Chaotic Dynamics, 2017, vol. 22, no. 6, pp. 688–699
Montgomery R.
MICZ-Kepler: Dynamics on the Cone over $SO(n)$
2013, vol. 18, no. 6, pp.  600-607
We show that the $n$-dimensional MICZ-Kepler 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 MICZ-Kepler problem. The heart of the computation is the observation that the additional MICZ-Kepler potential, $|\phi|^2/r^2$, agrees with the rotational part of the cone’s kinetic energy.
Keywords: Kepler problem, MICZ-K system, co-adjoint orbit, Sternberg phase space, symplectic reduction, superintegrable systems
Citation: Montgomery R.,  MICZ-Kepler: Dynamics on the Cone over $SO(n)$, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 600-607

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