The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems
The aim of this paper is to bring together two main approaches to nonholonomic mechanics: the Hamiltonian approach [see, for example, \textit{L. Bates} and \textit{J. Sniatycki}, Rep. Math. Phys. 32, No. 1, 99-115 (1993; Zbl 0798.58026)] and the Lagrangian approach [see \textit{A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden} and \textit{R. M. Murray}, Arch. Ration. Mech. Anal. 136, No. 1, 21-99 (1996; Zbl 0886.70014)]. Consider a configuration space \(Q\), a hyperregular Lagrangian \(L\), and a distribution \({\mathcal D}\) that describes the kinematic nonholonomic constraints. The authors construct a distribution \({\mathcal K}\) on the constraint submanifold \({\mathcal D}\) of \(TQ\). The solution of the constrained dynamics \(X_{\mathcal K}\) is given by the unique solution of equation \(i_{X_{\mathcal K}}\Omega_{K}=dE_{\mathcal K}\), where \(\Omega_{\mathcal K}\) and \(dE_{\mathcal K}\) are respectively the restrictions of the Poincaré-Cartan 2-form \(\Omega_L\) and the energy \(dE_L\) to \({\mathcal K}\) [see, also, \textit{M. de León, J. C. Marrero} and the reviewer, Int. J. Theor. Phys. 36, No. 4, 979-995 (1997; Zbl 0874.70012)]. Then, via the Legendre map, the authors obtain a precise relation between the constructions of the first two references. Now, it is added the hypothesis of symmetry, that is, it is assumed that a Lie group \(G\) acts on \(Q\) leaving the Lagrangian and the distribution invariant. It is proved that the reduced Lagrange-d'Alembert equations established in the first reference are equivalent to the reduced nonholonomic equations given in the second reference. As a nice application, the example of a simplified model of bicycle illustrates the results of the paper.
scientific article; zbMATH DE number 1136528
scientific article