Geometry and Physics Seminar: Francois Gay-Balmaz

Motivated partly by previous work on the zero curvature representation (ZCR) of completely integrable chiral models and partly by the underlying Hamiltonian structures of ideal complex fluids, we derive systems of partial differential equations, called G-strands, that admit a quadratic zero curvature representation for an arbitrary real semisimple Lie algebra. Using the root space decomposition, the G-strand equations can be formulated explicitly for the compact real form and the normal real form of any semisimple Lie algebra. We present several particular examples, including the exceptional group G_2.  We also determine the general form of Hamilton's principles and Hamiltonians for these systems, and analyze the linear stability of their equilibrium solutions.