Infinite-dimensional nonholonomic and vakonomic systems

TL;DR

This paper explores infinite-dimensional nonholonomic and vakonomic systems, illustrating their properties through classical examples and extending to complex systems like fluid dynamics and multi-trailer vehicles.

Contribution

It introduces new infinite-dimensional examples of nonholonomic and vakonomic systems, connecting finite-dimensional limits to complex systems such as fluid flows and vehicle kinematics.

Findings

Infinite-dimensional systems include subriemannian and Euler-Poincare-Suslov systems.

Connections between finite and infinite-dimensional dynamics are established.

Examples include nonholonomic fluids and infinite-dimensional geometry of a nonholonomic Moser theorem.

Abstract

In this paper, we present a collection of infinite-dimensional systems with nonholonomic constraints. In finite dimensions the two essentially different types of dynamics, nonholonomic or vakonomic ones, are known to be obtained by taking certain limits of holonomic systems with Rayleigh dissipation, as in [Koz83]. We visualize this phenomenon for the classical example of a skate on an inclined plane. The infinite-dimensional examples of nonholonomic and vakonomic systems revisited in the paper include subriemannian and Euler-Poincare-Suslov systems on Lie groups, the Heisenberg chain, the general Camassa-Holm equation, infinite-dimensional geometry of a nonholonomic Moser theorem, subriemannian approximations of an ideal hydrodynamics, parity-breaking nonholonomic fluids, and potential solutions to Burgers-type equations arising in optimal mass transport. Finally, we return to a higher-dimensional analogue of the skate, the kinematics of a car with $n$ trailers, as well as its limit as $n\to \infty$. We show that its infinite-dimensional version is a snake-like motion of the Chaplygin sleigh with a string, and it is subordinated to an infinite-dimensional Goursat distribution.