The Hamiltonization of Nonholonomic Systems and its Applications

[摘要] 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 theFrobenius sense). Although such mechanical systems are manifestly not Hamiltonian(their mechanics are described by the Lagrange-d;;Alembert principle, not Hamilton;;sprinciple), one can nevertheless attempt to formulate the mechanics of certain classesof nonholonomic systems as almost-Hamiltonian. In this dissertation we study variousmethods of so-called Hamiltonization of nonholonomic systems and discuss theirapplication to optimal control and the quantization of nonholonomic systems.We begin by constructing second-order associated systems for a class of nonholonomicsystems and solving the Inverse Problem of the Calculus of Variations toderive Hamiltonians whose canonical equations, when restricted to certain invariantsubmanifolds, 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, andshow that these systems give a straightforward Hamiltonization for certain classes ofsystems.Lastly, we extend a classical theorem of S.A. Chaplygin, which allows a larger classof nonholonomic systems to be Hamiltonized by reparameterizing time, to higherdimensions. Moreover, in some cases we show that the requirement that the originalsystem possess an invariant measure can be removed.The results are then applied to show that under certain conditions the equationsof motion of nonholonomic systems can be derived by considering an associatedrst-order optimal control problem, similar to the situation in holonomic systems.Moreover, the methods are illustrated throughout by various well known examplesof nonholonomic systems. Several future directions based on the research presentedare also discussed, among them the relatively new problem of quantizing a nonholonomicallyconstrained system. With the advent of nanomachines we expect theimportance of subatomic motions in wheeled robots to raise interest in the classicalquantumequations of motion governing these nonholonomic vehicles. Although thereis currently no accepted quantum mechanical treatment of nonholonomic mechanics,we discuss the application of the results of the Hamiltonizations obtained herein tothe quantization of a well known nonholonomic mechanical system.

[发布机构] University of Michigan

[效力级别] Geometric Mechanics

[关键词] Nonholonomic Mechanics;Geometric Mechanics;Mathematics;Science;Applied and Interdisciplinary Mathematics