RICHARD MONTGOMERY Professor of Mathematics B.A., Sonoma State University Ph.D., University of California, Berkeley My Page  My Courses

Phone: 831-459-4841 Office: Baskin 357B Office Hours: By Appt. Email: rmont_at_ucsc_dot_edu

RESEARCH INTERESTS

Richard Montgomery's work can be characterized as applied differential geometry. Much of it involves applications of gauge theory (fiber bundles with connections) to problems in mechanics and control theory. One such problem is that of a falling cat (see figure) dropped from upside down. The cat flips itself right side up, even though its angular momentum is zero.

It does this by changing its shape. In terms of gauge theory, the shape space of the cat forms the base space of a principal SO(3)-bundle, and the statement “angular momentum equals zero” defines a connection on this bundle. This geometric point of view on the cat's problem gives us a deep understanding and allows us to solve it explicitly for certain model cats.

Most recently, Montgomery has returned to the N-body problem, a problem with a long history. He has been approaching it using modern methods taken from equivariant differential geometry and calculus of variations. The last two publications listed here are representative of this work.

SELECTED PUBLICATIONS

R. Montgomery: The braid group and action-minimizing periodic orbits. Nonlinearity, Vol 11, 363-376 (1998).

R. Montgomery: The connection for a family of completely integrable systems whose holonomy is the classical adiabatic angle (Berry's phase). Comm. Math. Phys. v.120, 269-294 (1998).

M. Kazarain, R. Montgomery and B. Shapiro: Characteristic classes for the degenerations of two-plane fields in four dimensions. Pac. J. Math., v. 179, 2, 355-370 (1997).

R. Montgomery: The geometric phase of the three-body problem. Nonlinearity, v. 9, 1-20 (1996).

H. Berg, K. Ehlers, R. Montgomery and A. Samuel: Do cyanobacteria swim using travelling surfaces waves? Proceedings of the National Academy, Biophysics section, v. 93, no.16, 8340-8343 (1996).

R. Montgomery: Hearing the zero locus of a magnetic field, Communications in Mathematical Physics, v.168, No. 3 (1995).

R. Montgomery: Abnormal minimizers. SIAM J. Control and Optimization, v. 32, no. 6, 1605-1620 (1994).

R. Montgomery: Gauge theory and control theory, nonholonomic motion planning. (J. Canny and Z. Li, editors), Kluwer Acad. Press, 343-378 (1993).

R. Montgomery: Isoholonomic problems and some applications. Comm. Math. Phys. v.128, 565-592 (1990).

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