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Nonholonomic Lagrangian systems on Lie algebroids. (English) Zbl 1161.70336
Summary: This paper presents a geometric description on Lie algebroids of Lagrangian systems subject to nonholonomic constraints. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee that the dynamics of the system can be obtained as a suitable projection of the unconstrained dynamics. The proposed novel formalism provides new insights into the geometry of nonholonomic systems, and allows us to treat in a unified way a variety of situations, including systems with symmetry, morphisms, reduction, and nonlinearly constrained systems. Various examples illustrate the results.

MSC:
70F25 Nonholonomic systems related to the dynamics of a system of particles
70H03 Lagrange’s equations
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
37J60 Nonholonomic dynamical systems
53D17 Poisson manifolds; Poisson groupoids and algebroids
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