The Hamiltonian and Lagrangian Approaches to the Dynamics of
Nonholonomic Systems
Koon, W. S. and J. E. Marsden
Rep. Math. Phys., 40, 21-62
Abstract:
This paper compares the Hamiltonian approach to systems with nonholonomic
constraints (see Weber [1982], Arnold [1988], and Bates and Sniatycki [1993],
van der Schaft and Maschke [1994] and references therein) with the Lagrangian
approach (see Koiller [1992], Ostrowski [1996] 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 were first found on the
Lagrangian side and are 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.