Infinite-dimensional nonholonomic and vakonomic systems

TL;DR

This work surveys infinite-dimensional nonholonomic and vakonomic mechanics, clarifying how these dynamics arise as limits of holonomic systems with Rayleigh dissipation and how a unifying geometric framework spans Lie groups, loop groups, and diffeomorphism groups. It highlights key examples such as the skate problem, the Heisenberg chain, and the general Camassa–Holm equation, illustrating vakonomic geodesics as subriemannian flows and nonholonomic dynamics as Lagrange–d'Alembert constraints, with a unifying parameter $\mu$ bridging the two. It develops nonholonomic Moser theory and flows tangent to nonholonomic distributions, connecting them to optimal transport, cortical signal processing, and Burgers-type equations, and extends Goursat geometry to infinite dimensions via snake-like motion models. The paper also analyzes kinematic systems of cars with trailers, their Engel/Goursat structures, and the infinite-dimensional snake limit, offering potential physical realizations, discretization avenues, and applications across mathematical physics and geometric mechanics.

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-Poincaré-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.

Figures (5)

• Figure 1: Skate motion according to the vakonomic equations corresponding to $\mu=0$. Initial conditions are: $x_0=y_0=0$, $\theta_0=\pi/4$, $v_0=1$, $\omega_0=-10$. Left: the trajectory of the center of mass of the skate under gravity ($g =1$) for $0 \leq t \leq 8$. Right: the same motion in the absence of gravity ($g = 0$). (Note that the scale of the two panels is different. The motion on the left starts heading up before changing to the downward drift.)
• Figure 2: Skate motion governed by the Lagrange--d'Alembert equations in the limit $\mu \to +\infty$. Initial conditions are the same as in Figure \ref{['fig:trit2']}. Left: for gravitational acceleration $g = 1$, the trajectory exhibits bounded oscillations forming a cycloidal path kozlov1983realization. Right: in the absence of gravity ($g = 0$), the skate follows a circular trajectory without gravitational drift.
• Figure 3: Skate motion for an intermediate value of the parameter $\mu = 100$. Initial conditions are the same as in Figure \ref{['fig:trit2']}, but the plot scale is different. Left: motion under gravity ($g = 1$). Right: the same dynamics in the absence of gravity ($g = 0$).
• Figure 4: The car position is described by its midpoint $(x,y)$ of the axle, the angle $\theta$ of the car axle with a fixed direction, and the steering angle $\varphi$ of the front wheels, see michor2008topics.
• Figure 5: Illustration of the infinite-dimensional "snake constraint": the string evolves so that the velocity $z_t$ of each point remains collinear to its tangent vector $z_s$, enforcing the nonholonomic skate-like constraint. The evolving curve slides along itself and follows the trajectory of its own head point $z(0,t)$. Blue segments show the shape of the snake at times $t_1 < t_2 < t_3$, each tangent to the common trajectory (dashed green).

Theorems & Definitions (19)

• Definition 3.1
• Remark 3.2
• Proposition 3.3
• Remark 3.4
• Proposition 3.5: misiolek1998grong2015subkhesin2024information
• Proposition 4.1
• Theorem 4.2
• Theorem 5.1: khesin2009nonholonomic
• Remark 5.2
• Remark 5.3