This project is a continuation of the principal investigator's study of the geometry, dynamics and control of mechanical systems including Hamiltonian and Lagrangian systems, nonholonomic systems, and gradient flows. Nonholonomic systems are a generalization of classical Hamiltonian and Lagrangian systems where the system is subject to nonintegrable constraints on the velocities. The investigator proposes to study the dynamics and control of mechanical systems with such constraints, including nonholonomic systems with internal degrees of freedom, nonholonomic systems with controls, systems with nonlinear constraints, the Hamilton Jacobi equation in the nonholonomic setting, discrete systems, infinite-dimensional systems, and certain optimal control problems. He also proposes to study integrable Hamiltonian and nonholonomic systems and their relation to gradient flows. In the course of this work the proposer will study the role that symmetries play in the formulation of the nonholonomic equations of motion and in their integrability, the new dynamics that arises when one has a nonholonomic system interacting with a fluid, and how systems with nonlinear nonholonomic constraints behave, including thermostats (systems interacting with a heat bath). He will also study flexible nonholonomic systems. and the role of Hamiltonization (transformation to Hamiltonian form) in understanding the integrability of a nonholonomic system. In addition he will analyze gradients flows in finite and infinite dimensions including flows on loops groups and other infinite-dimensional spaces, and their relationship to integrability.
The dynamics of mechanical systems is of great importance in technology. The theory of nonholonomic dynamics in particular is the study of mechanical systems subject to constraints imposed on velocities. Such constraints arise for example in systems consisting of rigid bodies rolling on surfaces without slipping Nonholonomic systems occur frequently in many practical systems including wheeled vehicles such as cars (in particular self steering cars) and robots. Control of such systems is an important technological problem. In addition, the mathematics behind the control of nonholonomic systems plays a key role in control of nonlinear systems in general, such as the control of aircraft or underwater vehicles. Also important is how dissipation (or friction) affects the behavior and stability of such systems We are interested in describing the behavior of these systems, how to control and stabilize them and how to simulate the dynamics and control on a computer. It is hoped that this research will lead to advances in engineering and the proposer will collaborate with engineers and physicists. The proposed program has a strong educational impact. Material related to this research will be used in an advanced dynamics class. The research will involve the work of Ph.D students and undergraduates.
