Reduction,Linearization, and Stability of Relative Equilibria for Mechanical Systems on Riemannian Manifolds

Consider a Riemannian manifold equipped with an infinitesimal isometry. For this setup, a unified treatment is provided, solely in the language of Riemannian geometry, of techniques in reduction, linearization, and stability of relative equilibria. In particular, for mechanical control systems, an explicit characterization is given for the manner in which reduction by an infinitesimal isometry, and linearization along a controlled trajectory “commute.” As part of the development, relationships are derived between the Jacobi equation of geodesic variation and concepts from reduction theory, such as the curvature of the mechanical connection and the effective potential. As an application of our techniques, fiber and base stability of relative equilibria are studied. The paper also serves as a tutorial of Riemannian geometric methods applicable in the intersection of mechanics and control theory. F. Bullo’s research supported in part by grant CMS 0442041 from the USA National Science Foundation. A.D. Lewis’ research supported in part by grant from the Natural Sciences and Engineering Research Council of Canada.