Richard Montgomery

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


Knauf A., Montgomery R.
Compactification of the Energy Surfaces for $n$ Bodies
2023, vol. 28, nos. 4-5, pp.  628-658
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 $\big(2d(n-1)-1\big)$-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.
Keywords: regularization, scattering, collision
Citation: Knauf A., Montgomery R.,  Compactification of the Energy Surfaces for $n$ Bodies, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 628-658
Montgomery R.
Brake Orbits Fill the N-body Hill Region
2023, vol. 28, nos. 4-5, pp.  374-394
A brake orbit for the N-body 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 N-body problem at negative energies.
Keywords: N-body problem, Hill region, Jacobi – Maupertuis metric, brake orbit, zero velocity surface
Citation: Montgomery R.,  Brake Orbits Fill the N-body Hill Region, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 374-394
Bravo-Doddoli A., Montgomery R.
Geodesics in Jet Space
2022, vol. 27, no. 2, pp.  151-182
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 (sub-Riemannian submersion) onto the Euclidean plane. Horizontal lifts of Euclidean lines (which are the left translates of horizontal one-parameter 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 three-dimensional ``magnetic'' sub-Riemannian 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 two-parameter 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.
Keywords: Carnot group, Jet space, minimizing geodesic, integrable system, Goursat distribution, sub-Riemannian geometry, Hamilton – Jacobi, period asymptotics
Citation: Bravo-Doddoli A., Montgomery R.,  Geodesics in Jet Space, Regular and Chaotic Dynamics, 2022, vol. 27, no. 2, pp. 151-182
Montgomery R.
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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