From Brake to Syzygy in the Three-Body Problem

IAM-PIMS Distinguished Colloquium
March 26, 2012 10:00 pm

Speaker:  Prof. Richard Montgomery, Department of Mathematics, University of California at Santa Cruz, California

URL for Speaker:

Location:  LSK 301

Intended Audience:  Public

A brake orbit for the Newtonian three-body problem is a solution for which all three velocities are zero at some instant: the brake instant. If we follow such an orbit there will be a later instant at which the three bodies become colinear: the instant of syzygy. In this manner we can define a flow-induced “Poincare map” from brake initial conditions to syzygy configurations. Appropriately viewed, this brake-to-syzygy map is a map between planar domains. Understanding its image destroyed certain myths that the speaker had regarding action-minimizing orbits. The map fits in towards a possible global understanding of the planar three-body problem which we will explain. Key is a viewpoint on the planar three-body problem in which the configuration of all three bodies is represented as a single point in 3-space (its “shape”) and in which Newton’s equations become a mechanical system on this 3-space. Some movies of Paul Klee-like periodic brake orbits inspired by this work will be shown.

Richard Montgomery got undergraduate degrees in mathematics and physics from Sonoma State in Northern California in 1980. He got his PhD under Jerry Marsden at Berkeley in 1986 and after that had a Moore Instructorship at MIT for two years, then two years of postdoc in Berkeley. His research fields are geometric mechanics, celestial mechanics, control theory, and differential geometry. He is perhaps best known for his rediscovery, with Alain Chenciner, of Cris Moore’s figure eight solution to the three-body problem, which led to a slew of new ‘choreography’ solutions. He also established the existence of the first-known abnormal minimizer in subRiemannian geometry (in control lingo this is an abnormal extremal for a problem linear in controls, with control quadratic cost function), and is known for investigations using gauge-theoretic ideas of how a falling cat lands on its feet. He has written one book on subRiemannian geometry. In addition to mathematics and mechanics, he is a minor slowly fading legend in the kayaking world of California for first descents done in the early 1980s. He has two daughters, is married, and lives and works in Santa Cruz, CA.