Applied Mathematics Seminar

Symmetries of Hamiltonian systems can be used to reduce the number of variables, or to find adapted coordinates that simplify various structures of interest. Symmetries also introduce new dynamical features, notably relative equilibria, which are trajectories that only move "in a symmetry direction"; an example is a rigid body spinning steadily around one of its principal axes. In this talk, I will introduce symmetric Hamiltonian systems from a modern geometrical point of view. I will then focus on the case where the phase space is a cotangent bundle and present some new geometrical results, including a splitting of the symplectic normal space and a constructive cotangent bundle slice theorem. I will conclude by outlining the relevance of this work to dynamical questions.