arXiv subjects

Richard Montgomery

Publications and source records attributed to Richard Montgomery.

98 records · Page 6Linked to original sources

arXiv2004

Fitting Hyperbolic Pants to a Three-Body Problem

Consider the three-body problem with an attractive $1/r^2$ potential. Modulo symmetries, the dynamics of the bounded zero-angular momentum solutions is equivalent to a geodesic flow on the thrice-punctured sphere, or ``pair of pants''. The sphere is the shape sphere. The punctures are the binary collisions. The metric generating the geodesics is the Jacobi-Maupertuis metric. The metric is complete, has infinite area, and its ends, the neighborhoods of the punctures, are asymptotically cylindrical. Our main result is that when the three masses are equal then the metric has negative curvature everywhere except at two points (the Lagrange points). A corollary of this negativity is the uniqueness of the $1/r^2$ figure eight, a complete symbolic dynamics for encoding the collision-free solutions, and the fact that collision solutions are dense within the bound solutions.

math.DS↗

arXiv2003

Convexity in the Figure Eight Solution to the Three-Body Problem

The figure eight is a remarkable solution to the Newtonian three-body problem in which the three equal masses chase each around a planar curve having the qualitative shape and symmetries of a figure eight. Here we prove that each lobe of this eight is convex.

math.DS↗

arXiv2001

Infinitely Many Eclipses

We show that any bounded zero-angular momentum solution for the Newtonian three-body problem must suffer infinitely many eclipses, or collinearities, provided that it does not suffer a triple collision. Motivation for the result comes from the dream of building a symbolic dynamics for the three-body problem, one whose symbols 1, 2, 3 representing the three types of eclipses. The proof involves the conformal geometry of the shape sphere.

math.DS↗

arXiv2000

A remarkable periodic solution of the three-body problem in the case of equal masses

Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry pattern. Its most surprising feature is that the three bodies chase each other around a fixed eight-shaped curve. Setting aside collinear motions, the only other known motion along a fixed curve in the inertial plane is the ``Lagrange relative equilibrium" in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle. Our orbit visits in turns every ``Euler configuration" in which one of the bodies sits at the midpoint of the segment defined by the other two (Figure 1). Numerical computations by Carles Simó, to be published elsewhere, indicate that the orbit is ``stable" (i.e. completely elliptic with torsion). Moreover, they show that the moment of inertia I(t) with respect to the center of mass and the potential U(t) as functions of time are almost constant.

math.DS↗

arXiv1997

Deformations of Nonholonomic Two-plane Fields in Four Dimensions

An Engel structure is a maximally non-integrable field of two-planes tangent to a four-manifold. Any two such structures are locally diffeomorphic. We investigate the space of global deformations of canonical Engel structures arising out of contact three-manifolds. The main tool is Cartan's method of prolongation and deprolongation which lets us pass back and forth between certain Engel four-manifolds and contact three-manifolds. Every Engel manifold inherits a natural one-dimensional foliation. Its leaves are the fibers of the map from Engel to contact manifold, when this map exists. The foliation has a transverse contact structure and tangential real projective structure. As an application of our investigations, we show that a canonical Engel structure on real projective three-space times an interval corresponds to geodesic flow on the two-sphere, and that a subspace of its Engel deformations corresponds to the space of Zoll metrics on the two-sphere.

dg-ga↗

arXiv1997

Characteristic Classes for the Degenerations of Two-Plane Fields in Four Dimensions

There is a remarkable type of field of two-planes special to four dimensions known as an Engel distributions. They are the only stable regular distributions besides the contact, quasi-contact and line fields. If an arbitrary two-plane field on a four-manifold is slightly perturbed then it will be Engel at generic points. On the other hand, if a manifold admits an oriented Engel structure then the manifold must be parallelizable and consequently the alleged Engel distribution must have a degeneration loci -- a point set where the Engel conditions fails. By a theorem of Zhitomirskii this locus is a finite union of surfaces. We prove that these surfaces represent Chern classes associated to the distribution.

dg-ga↗

arXiv1997

Geometric Quantization and No Go Theorems

A geometric quantization of a Kähler manifold, viewed as a symplectic manifold, depends on the complex structure compatible with the symplectic form. The quantizations form a vector bundle over the space of such complex structures. Having a canonical quantization would amount to finding a natural (projectively) flat connection on this vector bundle. We prove that for a broad class of manifolds, including symplectic homogeneous spaces (e.g., the sphere), such connection does not exist. This is a consequence of a ``no go'' theorem claiming that the entire Lie algebra of smooth functions on a compact symplectic manifold cannot be quantized, i.e., it has no essentially nontrivial finite-dimensional representations.

dg-ga↗

arXiv1995

The Geometric Phase in the Three-Body Problem

Suppose that the initial triangle formed by the three moving masses of the three-body problem is similar to the triangle formed at some later time. We derive a simple integral formula for the overall rotation relating the two triangles. The formula is based on the fact that the space of similarity classes of triangles forms a two-sphere which we call the shape sphere. The formula consists of a ``dynamic'' and ``geometric'' term. The geometric term is the integral of a universal two-form on a``reduced configuration space''. This space is a two-sphere bundle over the shape sphere. The fibering spheres are instantaneous versions of the angular momentum sphere appearing in rigid body motion. Our derivation of the formula is similar in spirit to our earlier reconstruction formula for the rigid body motion.

dg-ga↗