B.Sc. (University of New Brunswick)
M.Sc. (California Institute of Technology)
Ph.D. (California Institute of Technology)
My work is in the area of mechanics and control theory, both from a differential geometric perspective. When dealing with problems in classical mechanics, one typically must make a choice whether one wishes to work with a Hamiltonian or a Lagrangian framework. I choose to work in the Lagrangian framework, and concentrate on systems with Lagrangians which are kinetic energy with respect to a Riemannian metric, and which possibly also include a potential energy term.
It turns out that the geometry of such systems is nicely encapsulated in an affine connection, and this affine connection provides a powerful tool for studying these systems. By understanding the role of the affine connection, one may investigate fundamental controllability problems, and use results here as a basis for designing a class of control laws which do things like steer a system from one position to another, stabilise a system to a desired position, track a desired trajectory, etc. I am also beginning to look at optimal control theory for these systems, and the results here promise to be very beautiful indeed.
On the control theory front, I am beginning a collaboration in the department with Professor Ron Hirschorn, and the aim here is to use the structure of mechanical systems to provide a certain type of output tracking for systems with singularities (for example, the problem of controlling a ball rolling on a beam). A control lab is available for testing design and analysis ideas on real mechanical systems.
Hand in hand with the control theoretic investigation of mechanical systems is a related program concerning fundamentals of mechanics per se. Here I am concerned with various levels of generality of basic formulation, but with an emphasis again on the Lagrangian setting. In particular, I am interested in understanding the geometry of mechanical systems with nonholonomic constraints. For systems which have Lagrangians of the kinetic minus potential energy form, affine connections again play an interesting role. For more general systems, jet bundle and tangent bundle geometry provide the tools for the appropriate Lagrangian description of constrained mechanics.