Lagrangian Reduction for Simple Mechanical Systems With Constant Orbit Type

Andrew D. Lewis, Mathematics, University of Warwick

An action of a Lie group $G$ on a manifold $Q$ is said to be of \emph{constant orbit type} if the isotropy group of $q_1\in Q$ is conjugate to the isotropy group of $q_2\in Q$ for each $q_1,q_2\in Q$. In such cases the group orbits are each diffeomorphic to a homogenous space of the group $G$. We thus begin by investigating simple mechanical systems (i.e., those whose Lagrangians are kinetic minus potential energies) whose configuration manifold is a homogeneous space (generalising the Euler-Poincar\'e equations). We then use the structure of these systems to discuss the local geometry of general simple mechanical systems with a symmetry group giving an action of constant orbit type.