Acta Applicandae Mathematicae

, Volume 99, Issue 1, pp 53–95 | Cite as

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

  • Francesco Bullo
  • Andrew D. Lewis


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.


Geometric mechanics Riemannian geometry Symmetry Reduction Control theory Linearization 

Mathematics Subject Classification (2000)

53B05 70H03 70H33 70Q05 93B18 


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© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of California at Santa BarbaraSanta BarbaraUSA
  2. 2.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada

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