Richard Montgomery
Publications:
Knauf A., Montgomery R.
Compactification of the Energy Surfaces for $n$ Bodies
2023, vol. 28, nos. 45, pp. 628658
Abstract
For $n$ bodies moving in Euclidean $d$space under the influence of a homogeneous pair interaction we compactify every center of mass energy surface, obtaining a $\big(2d(n1)1\big)$dimensional manifold with corners in the sense of Melrose. After a time change, the flow on this manifold is globally defined and nontrivial on the boundary.

Montgomery R.
Brake Orbits Fill the Nbody Hill Region
2023, vol. 28, nos. 45, pp. 374394
Abstract
A brake orbit for the Nbody problem is a solution for which, at some instant, all
velocities of all bodies are zero. We reprove two “lost theorems” regarding brake orbits and
use them to establish some surprising properties of the completion of the Jacobi – Maupertuis
metric for the Nbody problem at negative energies.

BravoDoddoli A., Montgomery R.
Geodesics in Jet Space
2022, vol. 27, no. 2, pp. 151182
Abstract
The space $J^k$ of $k$jets of a real function of one real variable $x$ admits the structure of Carnot group type. As such, $J^k$ admits
a submetry (subRiemannian submersion) onto the Euclidean plane. Horizontal lifts of Euclidean lines (which
are the left translates of horizontal oneparameter subgroups) are thus globally minimizing geodesics on $J^k$.
All $J^k$geodesics, minimizing or not, are constructed from degree $k$ polynomials in $x$ according to [7–9],
reviewed here.
The constant polynomials correspond to the horizontal lifts of lines. Which other polynomials yield globally minimizers and what
do these minimizers look like? We give a partial answer. Our methods include constructing
an intermediate threedimensional ``magnetic'' subRiemannian space lying between the jet space and the plane, solving a Hamilton – Jacobi (eikonal) equations on this space, and analyzing period asymptotics associated to period degenerations arising from twoparameter families of these polynomials. Along the way, we conjecture the independence of the cut time of any geodesic on jet space from the starting location on that geodesic.

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.
