A Generalization of Chetaev's Principle for a Class of Higher Order Non-holonomic Constraints

TL;DR

This paper extends Chetaev's principle to higher order non-holonomic systems by introducing independent kinematic and variational constraints, broadening the modeling scope beyond traditional D'Alembert's principle.

Contribution

It generalizes Chetaev's principle for higher order systems, allowing independent treatment of kinematic and variational constraints in non-holonomic mechanics.

Findings

Models systems where D'Alembert's principle fails

Includes elastic rolling bodies and pneumatic tires

Provides a geometric, coordinate-free framework

Abstract

The constraint distribution in non-holonomic mechanics has a double role. On one hand, it is a kinematic constraint, that is, it is a restriction on the motion itself. On the other hand, it is also a restriction on the allowed variations when using D'Alembert's Principle to derive the equations of motion. We will show that many systems of physical interest where D'Alembert's Principle does not apply can be conveniently modeled within the general idea of the Principle of Virtual Work by the introduction of both kinematic constraints and variational constraints as being independent entities. This includes, for example, elastic rolling bodies and pneumatic tires. Also, D'Alembert's Principle and Chetaev's Principle fall into this scheme. We emphasize the geometric point of view, avoiding the use of local coordinates, which is the appropriate setting for dealing with questions of global nature, like reduction.