Multibody System Dynamics

, Volume 25, Issue 4, pp 411–427 | Cite as

The d’Alembert–Lagrange equation exploited on a Riemannian manifold

  • Xiaobo Liu
  • R. L. Huston
  • C. Q. Liu


In this paper, we present a geometric exploitation of the d’Alembert–Lagrange equation (or alternatively, Lagrange form of the d’Alembert’s principle) on a Riemannian manifold. We develop the d’Alembert–Lagrange equation in a geometric form, as well as an explicit analytic form with respect to an arbitrary frame in a coordinate neighborhood on the configuration manifold. We provide a procedure to determine the governing dynamic equations of motion. Examples are given to illustrate the new formulation of dynamic equations and their relations to alternative ones. The objective is to provide a generalized perspective of governing equations of motion and its suitability for studying complex dynamic systems subject to nonholonomic constraints.


d’Alembert–Lagrange equation Nonholonomic systems Riemannian manifold 


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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Mechanical, Industrial and Nuclear EngineeringUniversity of CincinnatiCincinnatiUSA
  2. 2.General MotorsWarrenUSA

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