This project is a continuation of principal investigator's study of the geometry, dynamics and control of mechanical systems and in particular nonholonomic and quantum systems. The investigator proposes to study the dynamics and control of various mechanical systems with nonholonomic constraints, certain optimal control and navigation problems; and problems in quantum control with particular application to ion traps. In particular the investigator will study nonholonomic systems with symmetries and the relationship of symmetry to dynamics, and nonholonomic integrability and its relationship to measure preservation. The role of the nonholonomic Hamilton-Jacobi equation in integrability will also be studied, as well as the relationship between the inverse theory of Lagrangian systems and nonholonomic systems. The investigator will also consider he dynamics of nonholonomic systems with an infinite number of degrees of freedom with possible applications to the dynamics of elastic rolling solids and sliding flexible blades. He will also study the control of quantum systems, in particular models of coupled oscillator/spin systems which model ion traps. This leads to problems in infinite-dimensional controllability and the study of the control of such systems in the presence of decoherence and dissipation.
The theory of nonholonomic dynamics is the study of mechanical systems subject to constraints imposed on velocities. Such constraints are typical for systems consisting of rigid bodies rolling on surfaces without slipping. Nonholonomic systems occur frequently in practical mechanical problems including wheeled vehicles such as cars, bicycles and robots. A particular problem of interest is navigating robots around obstacles. Nonholonomic dynamics is playing an important role in the development of nonlinear control and mechanical systems theory. Quantum control is closely related to the control of nonholonomic systems because of the type of mathematics involved. Many interesting new issues arise however such as the role of dissipation, decoherence and measurement. Quantum control has become very important recently because of applications to quantum computers. We study here a particular system which has been proposed for computing computing: the ion trap. Quantum control provides a very interesting extension of nonholonomic and nonlinear control and has important mathematical issues associated with it. It is hoped that this research will lead to advances in engineering. The proposer continues to collaborate with many 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 also involve the work of Ph.D students and undergraduates.