Hamiltonization of Elementary Nonholonomic Systems

TL;DR

This paper introduces a method to convert certain nonholonomic systems into conformally Hamiltonian form, simplifying their analysis by reducing them to classical problems like mass points on a plane or sphere.

Contribution

It develops the Chaplygin reducing multiplier method and applies it to three nonholonomic systems, providing new Hamiltonian representations and insights into their dynamics.

Findings

Conformal Hamiltonian forms obtained for three nonholonomic systems.

Reduction of oscillator and Chaplygin sleigh to mass point problems.

Identification of a nonholonomic system not amenable to the method.

Abstract

In this paper, we develop the Chaplygin reducing multiplier method; using this method, we obtain a conformally Hamiltonian representation for three nonholonomic systems, namely, for the nonholonomic oscillator, for the Heisenberg system, and for the Chaplygin sleigh. Furthermore, in the case of an oscillator and the nonholonomic Chaplygin sleigh, we show that the problem reduces to the study of motion of a mass point (in a potential field) on a plane and, in the case of the Heisenberg system, on the sphere. Moreover, we consider an example of a nonholonomic system (suggested by Blackall) to which one cannot apply the reducing multiplier method.