What makes nonholonomic integrators work?

TL;DR

Evidence is given that reversibility is the main mechanism behind near conservation of first integrals for nonholonomic integrators and perturbed test problems that are integrable but no longer reversible (with respect to the standard reversibility map).

Abstract

A nonholonomic system is a mechanical system with velocity constraints not originating from position constraints; rolling without slipping is the typical example. A nonholonomic integrator is a numerical method specifically designed for nonholonomic systems. It has been observed numerically that many nonholonomic integrators exhibit excellent long-time behaviour when applied to various test problems. The excellent performance is often attributed to some underlying discrete version of the Lagrange--d'Alembert principle. Instead, in this paper, we give evidence that reversibility is behind the observed behaviour. Indeed, we show that many standard nonholonomic test problems have the structure of being foliated over reversible integrable systems. As most nonholonomic integrators preserve the foliation and the reversible structure, near conservation of the first integrals is a consequence of reversible KAM theory. Therefore, to fully evaluate nonholonomic integrators one has to consider also non-reversible nonholonomic systems. To this end we construct perturbed test problems that are integrable but no longer reversible (with respect to the standard reversibility map). Applying various nonholonomic integrators from the literature to these problems we observe that no method performs well on all problems. This further indicates that reversibility is the main mechanism behind near conservation of first integrals for nonholonomic integrators. A list of relevant open problems is given.

Figures (9)

• Figure 1: Illustration of the principle of the continuous variable transmission (CVT) gearbox. The driving subsystem $(\xi,\dot{\xi})$ determines the location of the belt which in turn determines the gear ration between the shafts. A nonholonomic system describing the motion is given in \ref{['sec:cvt']}.
• Figure 2: Illustration of the knife edge system \ref{['eq:knife_edge']}. The contact point of the knife edge, or skate, is sliding under gravity on the inclined plane. The direction of sliding is determined by the angle $\xi$; one may think of a "one-legged skater", changing direction of his skate according to the driving system.
• Figure 3: Illustration of the vertical disk, or rolling penny, given by \ref{['eq:vert_disk']}. The rotation of the penny is described by $x_3$ (measured from the $z$-axis) and the position of the contact point by $(x_1,x_2)$. The directional angle is described by $\xi$. Because of conservation of angular momentum, we have $\ddot\xi = 0$.
• Figure 4: Evolution of the error $\lvert H-H(0)\rvert$ of the energy integral for 5 methods applied to the knife edge. The data are convoluted by a running mean with a window of about 3 time units. For all methods except DLA${}^{0.4}$, the error for the unperturbed system ($\varepsilon=0$) is bounded in time. For the perturned system ($\varepsilon=0.1$) all methods except DD show energy drift.
• Figure 5: Phase diagrams for the $(\xi,\dot{\xi})$ subsystem of the CVT problem, with $\varepsilon=0$ (left) and $\varepsilon=0.5$ (right). The circular (oscillating driver) and upper (rotating driver) paths correspond, respectively, to the low ($H_0 = 2.8$) and high ($H_0=5.0$) energy levels. Both diagrams are symmetric under the standard reversibility map $\dot{\xi} \mapsto -\dot{\xi}$. Notice that the left diagram also is symmetric under the 'non-physical' reversibility map $\xi\mapsto -\xi$, whereas the right diagram does not have this symmetry (due to the perturbation).

Theorems & Definitions (20)

• definition 1
• remark 1
• remark 2
• remark 3
• definition 2
• theorem 1
• proof
• definition 3
• theorem 2
• proof