TAILIEUCHUNG - Nonholonomic Mechanical Systems with Symmetry

This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian me- chanics and with a view to control-theoretical applications. The basic methodology is that of geometric mechanics applied to the Lagrange-d’Alembert formulation, generalizing the use of connections and momentum maps associated with a given symmetry group to this case. We begin by formulating the mechanics of nonholo- nomic systems using an Ehresmann connection to model the constraints, and show how the curvature of this connection enters into Lagrange’s equations. Unlike the situationwith standard configuration-space constraints, the presence of symmetries in the nonholonomic case may or may not lead to. | Arch. Rational Mech. Anal. 136 1996 21-99. Springer-Verlag 1996 Nonholonomic Mechanical Systems with Symmetry Anthony M. Bloch P S. Krishnaprasad Jerrold E. Marsden Richard M. Murray Communicated by P. Holmes Table of Contents Abstract. 21 1. Introduction. 22 2. Constraint Distributions and Ehresmann Connections. 30 3. Systems with Symmetry. 38 4. The Momentum Equation. 47 5. A Review of Lagrangian Reduction. 57 6. The Nonholonomic Connection and Reconstruction. 62 7. The Reduced Lagrange-d Alembert Equations. 70 8. Examples . 77 9. Conclusions. 94 References. 95 Abstract This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics and with a view to control-theoretical applications. The basic methodology is that of geometric mechanics applied to the Lagrange-d Alembert formulation generalizing the use of connections and momentum maps associated with a given symmetry group to this case. We begin by formulating the mechanics of nonholo-nomic systems using an Ehresmann connection to model the constraints and show how the curvature of this connection enters into Lagrange s equations. Unlike the situation with standard configuration-space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws. However the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables. This momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates. An alternative description using 22 A. Bloch et al. a body reference frame relates part of the momentum equation to the components of the Euler-Poincare equations along those symmetry directions consistent with the constraints. one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish .

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