Nonholonomic Dynamics

320 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Introduction Nonholonomic systems are, roughly speaking, mechanical systems with constraints on their velocity that are not derivable from position constraints. They arise, for instance, in mechanical systems that have rolling contact (for example, the rolling of wheels without slipping) or certain kinds of sliding contact (such as the sliding of skates). They are a remarkable generalization of classical Lagrangian and Hamiltonian systems in which one allows position constraints only. There are some fascinating differences between nonholonomic systems and classical Hamiltonian or Lagrangian systems. Among other things: nonholonomic systems are nonvariational—they arise from the Lagrange-d’Alembert principle and not from Hamilton’s principle; whereas energy is preserved for nonholonomic systems, momentum is not always preserved for systems with symmetry (i.e., there is nontrivial dynamics associated with the nonholonomic generalization of Noether’s theorem); nonholonomic systems are almost Poisson but not Poisson (i.e., there is a bracket that together with the energy on the phase space defines the motion, but the bracket generally does not satisfy the Jacobi identity); and finally, unlike the Hamiltonian setting, volume may not be preserved in the phase space, leading to interesting asymptotic stability in some cases, despite energy conservation. The purpose of this article is to engage the reader’s interest by highlighting some of these differences along with some current research in the area. There has been some confusion in the literature for quite some time over issues such as the variational character of nonholonomic systems, so it is appropriate that we begin with a brief review of the history of the subject. Some History The term “nonholonomic system” was coined by Hertz (1894). The oldest publication that addresses the dynamics of a rolling rigid body known to the authors is Euler (1734), in which small oscillations of a rigid body moving without slipping on a horizontal plane were studied. Later, the dynamics of a rigid body rolling on a surface was studied in Routh (1860), Slesser (1861), Vierkandt (1892), and Walker (1896). The derivation of the equations of motion of a nonholonomic system in the form of the EulerLagrange equations corrected by some additional terms to take into account the constraints (but without Lagrange multipliers), was outlined by Ferrers (1872). The formal derivation of this form of equations was performed in Voronetz (1901). In the case in which some of the configuration variables Anthony M. Bloch is professor of mathematics at the University of Michigan, Ann Arbor. His email address is [email protected]. Research partially supported by NSF grants DMS 0103895 and 0305837.