Infinite-dimensional nonholonomic and vakonomic systems

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]. After visualizing this phenomenon for the classical example of a skate on an inclined plane, we discuss its higher-dimensional analogue, 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. Other examples of nonholonomic and vakonomic systems include subriemannian and Euler-Poincar\'e-Suslov systems on infinite-dimensional Lie groups, the Heisenberg chain, the general Camassa-Holm equation, infinite-dimensional geometry of a nonholonomic Moser theorem, parity-breaking nonholonomic fluids, and potential solutions to Burgers-type equations arising in optimal mass transport.