May 01, 2018 / by George Council / In news

Affine Nonholonomic Systems

Nonholonomic constraints are used to represent limitations an object has in the directions of motion it can directly move in. For example, a car is not able to slide sideways - its velocity must be in-line with the wheels. However, cars can parallel park, so they CAN move in any direction, but through a combination of twisting and turning, rather than ‘strait line’ motion. Nonholonomic constraints are the generalization of this idea. Nonholonomic systems can exhibit strange behavior - e.g. the famous ‘snakeboard’ that can accumulate momentum from a resting position merely by wiggling. The paper shown here demonstrates conserved quantities DO exist for systems that have affine constraints, much like the way energy is conserved for linear constraints. By using a time-dependent change of coordinates, the authors are able to transform affine constraints into linear ones. By using this change of coordinates, it greatly simplifies the analysis one could do to understand the dynamical properties of the system. E.g, one can classify if there are periodic solutions, consider stability, or more, and the change of coordinates makes this easier.

Link to Paper Authors : Authors : Francesco Fasso, Nicola Sansonetto

Abstract

Energy is in general not conserved for mechanical nonholonomic systems with affine constraints. In this article we point out that, nevertheless, in certain cases, there is a modification of the energy that is conserved. Such a function coincides with the energy of the system relative to a different reference frame, in which the constraint is linear. After giving sufficient conditions for this to happen, we point out the role of symmetry in this mechanism. Lastly, we apply these ideas to prove that the motions of a heavy homogeneous solid sphere that rolls inside a convex surface of revolution in uniform rotation about its vertical figure axis, are (at least for certain parameter values and in open regions of the phase space) quasi-periodic on tori of dimension up to three.