Nonholonomic systems and the geometry of constraints

Abstract

In a recent paper [9] we analyze conservation of volume for a series of examples of mechanical systems with linear, affine and non linear constraints aiming to make evident some geometric aspects related with them. Here, we only consider examples with linear constraints (defined by a constant rank distribution), in which we have conservation of volume. Conservation of volume means, equivalently, that the orthogonal distribution (the metric is defined by the kinetic energy) is minimal (see [15]) and so, if it is integrable, the corresponding foliation has minimal leaves. Properties of the falling penny and of the vertical disc rolling on a horizontal plane without slipping are very special. A dynamically symmetric sphere that rolls without slipping on a given surfaceS⊂ℝ³ conserves volume, and the orthogonal distribution is integrable if, and only if,S isparallel to a surface with a fixed constant mean curvature. Semi-simple Lie groups endowed with suitable metrics have foliations with minimal leaves. Geometric questions related with the kinematics of the rolling motion of two surfaces are also considered.