Pacific Institute for the Mathematical Sciences - PIMS

The geometric approach to mechanics serves as the theoretical
underpinning of innovative control methodologies in geometric control
theory.  These techniques allow the attitude of satellites to be
controlled using changes in its shape, as opposed to chemical
propulsion, and are the basis for understanding the ability of a falling
cat to always land on its feet, even when released in an inverted
orientation.  We will discuss the application of Lie group variational
integrators to the optimal control of mechanical systems.  These methods
are based on a discretization of Hamilton's principle that preserves
the Lie group structure of the configuration space, and yield
minimum-dimensional global representations of the dynamics.  Recent
extensions to homogeneous spaces yield intrinsic methods for Hamiltonian
flows on the sphere, and have potential applications to the simulation
of geometrically exact rods, structures and mechanisms.  Extensions to
Hamiltonian PDEs and uncertainty propagation on Lie groups using
noncommutative harmonic analysis techniques will also be discussed.  In
particular, this allows one to develop attitude state estimation methods
for satellite dynamics without assuming that the measurements are
frequent, and that the uncertainty distribution is localized.

2 PM, WMAX 216