Dr. Marsden's first lecture, accessible to a general audience, will introduce the role of geometry and symmetry in the mechanics and dynamics of familiar systems. A falling cat is able to right itself through the geometric generation of rotations, while other systems, such as the robotic snake and the snakeboard, generate locomotion. Geometry and symmetry lead to a better understanding of practical engineering problems, such as the control and stability of underwater vehicles. Dr. Marsden will illustrate the basic examples and concepts with concrete systems and videos, as well as trace the mathematical development of geometric mechanics through the works of Euler, Lagrange, Hamilton, Routh, Riemann, Lie, and Poincare.

In this lecture, pitched at the level of a colloquium talk, Dr. Marsden develops the setting of geometric mechanics and gives a survey of some of the progress made in the stability theory of steady motions of mechanical systems, a time honored subject going back to Routh in the last century.

In this talk, a little more specialized in nature, Dr. Marsden presents some recent work with Anthony Bloch, Gloria Sanchez and Naomi Leonard on the stabilization of mechanical systems with symmetry such as a rigid body with an internal rotor, the inverted pendulum on a cart, and underwater vehicles. Starting from a given Lagrangian with a relative equilibrium that is unstable, he introduces a modified Lagrangian whose Euler-Lagrange equations differ from the given ones by terms that can be identified with control forces. Equilibria of the modified Lagrangian can be analyzed using the energy-momentum method or other techniques from mechanics and dynamical systems. One such modification is a Kaluza-Klein construction, whereby the kinetic energy is modified and a second modification is the introduction of symmetry breaking potentials. He will also indicate how these techniques can be used for tracking problems, such as how to make the underwater vehicle follow a desired trajectory, including both rotational and translational motion.