The Hamiltonization of Nonholonomic Systems and its Applications

Abstract

A nonholonomic mechanical system is a pair (L,D), where L is a mechanical Lagrangian and D is a distribution which is non-integrable (in the Frobenius sense). Although such mechanical systems are manifestly not Hamiltonian (their mechanics are described by the Lagrange-d'Alembert principle, not Hamilton's principle), one can nevertheless attempt to formulate the mechanics of certain classes of nonholonomic systems as almost-Hamiltonian. In this dissertation we study various methods of so-called Hamiltonization of nonholonomic systems and discuss their application to optimal control and the quantization of nonholonomic systems. We begin by constructing second-order associated systems for a class of nonholonomic systems and solving the Inverse Problem of the Calculus of Variations to derive Hamiltonians whose canonical equations, when restricted to certain invariant submanifolds, reproduce the original nonholonomic mechanics.

We also introduce the idea of conditionally variational nonholonomic systems, which arise from a comparison with the variational nonholonomic equations, and show that these systems give a straightforward Hamiltonization for certain classes of systems.

Lastly, we extend a classical theorem of S.A. Chaplygin, which allows a larger class of nonholonomic systems to be Hamiltonized by reparameterizing time, to higher dimensions. Moreover, in some cases we show that the requirement that the original system possess an invariant measure can be removed.

The results are then applied to show that under certain conditions the equations of motion of nonholonomic systems can be derived by considering an associated rst-order optimal control problem, similar to the situation in holonomic systems. Moreover, the methods are illustrated throughout by various well known examples of nonholonomic systems. Several future directions based on the research presented are also discussed, among them the relatively new problem of quantizing a nonholonomically constrained system. With the advent of nanomachines we expect the importance of subatomic motions in wheeled robots to raise interest in the classicalquantum equations of motion governing these nonholonomic vehicles. Although there is currently no accepted quantum mechanical treatment of nonholonomic mechanics, we discuss the application of the results of the Hamiltonizations obtained herein to the quantization of a well known nonholonomic mechanical system.