The Hamiltonian and Lagrangian Approaches to the Dynamics ofudNonholonomic Systems

摘要

This paper compares the Hamiltonian approach to systems with nonholonomic constraintsud(see Weber [1982], Arnold [1988], and Bates and Sniatycki [1993], van der Schaft and Maschkeud[1994] and references therein) with the Lagrangian approach (see Koiller [1992], Ostrowski [1996]udand Bloch, Krishnaprasad, Marsden and Murray [1996]). There are many differences in theudapproaches and each has its own advantages; some structures have been discovered on one sideudand their analogues on the other side are interesting to clarify. For example, the momentumudequation and the reconstruction equation were first found on the Lagrangian side and are usefuludfor the control theory of these systems, while the failure of the reduced two form to be closedud(i.e., the failure of the Poisson bracket to satisfy the Jacobi identity) was first noticed on theudHamiltonian side. Clarifying the relation between these approaches is important for the futureuddevelopment of the control theory and stability and bifurcation theory for such systems. Inudaddition to this work, we treat, in this unified framework, a simplified model of the bicycle (seeudGetz [1994] and Getz and Marsden [1995]), which is an important underactuated (nonminimumudphase) control system.