On the Integration Theory of Equations of Nonholonomic Mechanics

TL;DR

This paper explores the integration of nonholonomic mechanical systems, discovering new integrable cases using invariant measures, symmetries, and known integrals, thereby advancing the understanding of their dynamics.

Contribution

It introduces new integrable nonholonomic systems, generalizes classical problems, and proposes innovative methods using invariants and symmetries for solving equations of motion.

Findings

Discovered new integrable nonholonomic systems.

Generalized Chaplygin's rolling ball problem.

Identified conditions for invariant measures with analytical density.

Abstract

The paper deals with the problem of integration of equations of motion in nonholonomic systems. By means of well-known theory of the differential equations with an invariant measure the new integrable systems are discovered. Among them there are the generalization of Chaplygin's problem of rolling nonsymmetric ball in the plane and the Suslov problem of rotation of rigid body with a fixed point. The structure of dynamics of systems on the invariant manifold in the integrable problems is shown. Some new ideas in the theory of integration of the equations in nonholonomic mechanics are suggested. The first of them consists in using known integrals as the constraints. The second is the use of resolvable groups of symmetries in nonholonomic systems. The existence conditions of invariant measure with analytical density for the differential equations of nonholonomic mechanics is given.