The Wagner Curvature Tensor in Nonholonomic Mechanics

TL;DR

This paper revisits Wagner's classical curvature tensor for nonholonomic manifolds, providing invariant and coordinate formulations, and applies it to mechanical examples including rolling objects to analyze curvature and flatness conditions.

Contribution

It offers a detailed presentation of Wagner's curvature tensor in nonholonomic mechanics with invariant and coordinate approaches, and applies it to specific mechanical systems.

Findings

Wagner curvature tensor formulated for nonholonomic manifolds.

Conditions for flatness in rolling sphere and disc systems identified.

Application of curvature tensor to mechanical examples demonstrated.

Abstract

We present the classical Wagner construction from 1935 of the curvature tensor for completely nonholonomic manifolds in both invariant and coordinate way. The starting point is the Shouten curvature tensor for nonholonomic connection introduced by Vranceanu and Shouten. We illustrate the construction on two mechanical examples: the case of a homogeneous disc rolling without sliding on a horizontal plane and the case of a homogeneous ball rolling without sliding on a fixed sphere. In the second case we study the conditions on the ratio of diameters of the ball and the sphere to obtain a flat space - with the Wagner curvature tensor equal zero.