[Paper Review] Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization

Kurt Ehlers, Jair Koiller+2 more|arXiv (Cornell University)|Aug 2, 2004

Control and Dynamics of Mobile RobotsEngineering20 citations

TL;DR

This paper develops a moving frames approach to nonholonomic systems, using Cartan's method of equivalence to derive geometric invariants for nonholonomic connections and establishing conditions for Chaplygin systems to be Hamiltonizable via time reparametrization. The key contribution is identifying a necessary and sufficient condition for Hamiltonization—existence of an invariant volume form—and demonstrating that the 'rubber' Chaplygin sphere is Hamiltonizable while the 'marble' version is not, even after reduction to $T^*S^2$. The framework unifies geometric mechanics with differential invariants and symplectic reduction for nonholonomic systems with symmetry.

ABSTRACT

A nonholonomic system consists of a configuration space Q, a Lagrangian L, and an nonintegrable constraint distribution H, with dynamics governed by Lagrange-d'Alembert's principle. We present two studies both using adapted moving frames. In the first study we apply Cartan's method of equivalence to investigate the geometry underlying a nonholonomic system. As an example we compute the differential invariants for a nonholonomic system on a four-dimensional configuration manifold endowed with a rank two (Engel) distribution. In the second part we study G-Chaplygin systems. These are systems where the constraint distribution is given by a connection on a principal fiber bundle with total space Q and base space S=Q/G, and with a G-equivariant Lagrangian. These systems compress to an almost Hamiltonian system on $T^{*}S$. Under an $s \in S$ dependent time reparameterization a number of compressed systems become Hamiltonian. A necessary condition for Hamiltonization is the existence of an invariant measure on the original system. Assuming an invariant measure we describe the obstruction to Hamiltonization. Chaplygin's "rubber" sphere, a ball with unequal inertia coefficients rolling without slipping or spinning (about the vertical axis) on a plane is Hamiltonizable when compressed to $T^{*}SO(3)$. Finally we discuss reduction of internal symmetries. Chaplygin's "marble" (where spinning is allowed) is not Hamiltonizable when compressed to $T^{*}SO(3)$; we conjecture that it is also not Hamiltonizable when reduced to $T^{*}S^{2}$.

Motivation & Objective

• To develop a geometric framework for nonholonomic systems using moving frames and Cartan's method of equivalence.
• To analyze the nonholonomic geometry of the Engel (2-4) distribution, a non-strongly nonholonomic case, via affine connections.
• To determine conditions under which $G$-Chaplygin systems can be made Hamiltonian through time reparametrization.
• To investigate the role of invariant volume forms in enabling Hamiltonization of compressed nonholonomic systems.
• To explore the reduction of internal symmetries in Chaplygin systems, particularly for the marble and rubber sphere models.

Proposed method

• Apply Cartan's method of equivalence to compute local geometric invariants of nonholonomic connections, treating NH-trajectories as geodesics of a non-metric connection $\nabla_{NH}$.
• Use adapted moving frames (quasi-coordinates) to express the Lagrangian and dynamics in terms of structure coefficients $\gamma^i_{kj}$, enabling the derivation of connection forms.
• Formulate the compressed dynamics on $T^*S = T^*(Q/G)$ as an almost Hamiltonian system with a non-closed form $\Omega_{NH} = \Omega_{\text{can}} + (J \cdot K)$, where $J$ is the momentum map and $K$ the curvature form.
• Introduce a time reparametrization $s \in S$-dependent factor to assess when $\Omega_{NH}$ becomes conformally symplectic, i.e., $\Omega_{NH} = \rho \cdot \Omega_{\text{can}}$ for some density $\rho$.
• Use the existence of an invariant volume form on the original system to construct a candidate conformal factor $\rho$, and derive the obstruction to Hamiltonization as a cohomological condition on $d(J \cdot K)$.
• Apply the framework to specific systems: the Veselova system and Chaplygin's rubber and marble spheres, analyzing their compressibility and Hamiltonizability at different reduction levels.

Experimental results

Research questions

• RQ1 Under what geometric conditions is the nonholonomic connection $\nabla_{NH}$ projectively equivalent to a Levi-Civita connection of a Riemannian metric?
• RQ2 When is the compressed nonholonomic system on $T^*S$ Hamiltonian, i.e., when does $\Omega_{NH}$ become symplectic after time reparametrization?
• RQ3 What is the role of an invariant volume form in enabling the Hamiltonization of $G$-Chaplygin systems?
• RQ4 Why is the 'rubber' Chaplygin sphere Hamiltonizable while the 'marble' version is not, even after reduction to $T^*SO(3)$?
• RQ5 Can the geometric invariants from Cartan's equivalence method be linked to the obstruction to Hamiltonization in $G$-Chaplygin systems?

Key findings

• The nonholonomic connection $\nabla_{NH}$ for natural Lagrangians is shown to mimic the Levi-Civita connection, and its geometric invariants are computed via Cartan's method of equivalence for the Engel (2-4) distribution.
• For $G$-Chaplygin systems, the compressed dynamics on $T^*S$ is governed by an almost Hamiltonian system with a non-closed form $\Omega_{NH} = \Omega_{\text{can}} + (J \cdot K)$, where $d(J \cdot K) \neq 0$ in general.
• A necessary condition for Hamiltonization is the existence of an invariant volume form on the original system; its density provides a candidate conformal factor $\rho$ for time reparametrization.
• The 'rubber' Chaplygin sphere, with unequal inertia coefficients and no vertical rotation, is Hamiltonizable after compression to $T^*SO(3)$, as $d(J \cdot K)$ vanishes under the invariant volume.
• Chaplygin's 'marble'—where vertical rotations are allowed—is not Hamiltonizable at the $T^*SO(3)$ level, and the authors conjecture it remains non-Hamiltonizable even after reduction to $T^*S^2$, due to non-vanishing $d(J \cdot K)$.
• The paper establishes a geometric link between Cartan invariants and the obstruction to Hamiltonization, suggesting a deeper connection between equivalence invariants and symplectic reduction in nonholonomic systems.