The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems

Abstract

This paper compares the Hamiltonian approach to systems with nonholonomic constraints (see Weber [1982], Arnold [1988], and Bates and Snyatycki [1992], Van der Schaft and Maschke [I9941 and references therein) with the Lagrangian approach (see Koiller [1992], Ostrowski ([996] and Bloch, Krishnaprasad, Marsden and Murray [1996]). There are many differences in the approaches and each has its own advantages; some structures have been discovered on one side and their analogues on the other side are interesting to clarify. For example, the momentum equation and the reconstruction equation was first found on the Lagrangian side and is useful for the control theory of these systems, while the failure of the reduced two form to be closed (i.e., the failure of the Poisson bracket to satisfy the Jacobi identity) was first noticed on the Hamiltonian side. Clarifying the relation between these approaches is important for the future development of the control theory and stability and bifurcation theory for such systems. In addition to this work, we treat, in this unified framework, a simplified model of the bicycle (see Getz [1994] and Getz and Marsden [1995]), which is an important underactuated (nonminimum phase) control system.