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The Energy-Momentum Method for the Stability of Nonholonomic Systems

Zenkov, Dmitry V. and Bloch, Anthony M. and Marsden, Jerrold E. (1997) The Energy-Momentum Method for the Stability of Nonholonomic Systems. California Institute of Technology , Pasadena, CA. (Unpublished) http://resolver.caltech.edu/CaltechCDSTR:1997.010

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Abstract

In this paper we analyze the stability of relative equilibria of nonholonomic systems (that is, mechanical systems with nonintegrable constraints such as rolling constraints). In the absence of external dissipation, such systems conserve energy, but nonetheless can exhibit both neutrally stable and asymptotically stable, as well as linearly unstable relative equilibria. To carry out the stability analysis, we use a generalization of the energy-momentum method combined with the Lyapunov-Malkin Theorem and the center manifold theorem. While this approach is consistent with the energy-momentum method for holonomic systems, it extends it in substantial ways. The theory is illustrated with several examples, including the the rolling disk, the roller racer, and the rattleback top.


Item Type:Report or Paper (Technical Report)
Additional Information:Research [D.V.Z.] partially supported by National Science Foundation PYI grant DMS-91-57556 and AFOSR grant F49620-96-1-0100. Research [A.M.B.] partially supported by National Science Foundation PYI grant DMS-91-57556, AFOSR grant F49620-96-1-0100, a Guggenheim Fellowship and the Institute for Advanced Study. Research [J.E.M.] partially supported by DOE contract DEFG0395-ER25251.
Group:Control and Dynamical Systems Technical Reports
Record Number:CaltechCDSTR:1997.010
Persistent URL:http://resolver.caltech.edu/CaltechCDSTR:1997.010
Official Citation:Zenkov, Dmitry V., Bloch, Anthony M. and Marsden, Jerrold E. (1997) The Energy-Momentum Method for the Stability of Nonholonomic Systems. Technical Report. California Institute of Technology, Pasadena, CA. [http://resolver.caltech.edu/CaltechCDSTR:1997.010]
Usage Policy:You are granted permission for individual, educational, research and non-commercial reproduction, distribution, display and performance of this work in any format.
ID Code:28133
Collection:CaltechCDSTR
Deposited By: Imported from CaltechCDSTR
Deposited On:19 Dec 2007
Last Modified:26 Dec 2012 14:31

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