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Richard Montgomery

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


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.
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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