Conformally symplectic Chaplygin reduction in rubber rolling of surfaces of revolution over the plane

TL;DR

Rubber rolling on a plane forms a nonholonomic SE(2) Chaplygin system that axial-reduces to the sphere of Poisson vectors $T^*S^2$; the paper bridges Borisov–Mamaev’s results for surfaces of revolution with modern almost-symplectic reduction, showing a conformally symplectic structure via $d\left(\frac{1}{N}\Omega_{NH}\right)=0$ and a conserved quantity $\ell=\frac{p_\psi}{N(\theta)}$. The central technical feat is the emergence of the Nose function $N(\theta)$, which yields an elementary integral and enables a clean 1-DOF reduction to a Hamiltonian system in $(\theta,p_\theta)$ with $V(\theta)=\frac{\ell^2}{2\sin^2\theta}+mgz_C(\theta)$. The approach applies to revolution bodies (e.g., a torus) and reveals independence from the axial inertia $I_3$ in the reduced dynamics, while reconstruction reintroduces inertia through angular variables. The work also connects to the $\Phi$-simple Chaplygin framework, suggesting potential analytic solutions, dissipation considerations, and control-chaos analyses for nonholonomic tires-like systems.

Abstract

Rubber rolling (no-slip and no-twist) of a convex body on the plane under the influence of gravity is a SE(2) Chaplygin system, that reduces to the sphere of Poisson vectors. I comment upon an observation by A.V Borisov and I.S. Mamaev (Regular and Chaotic Dynamics, 13(5):443-490, 2008) for the case of surfaces of revolution [also in A. V. Borisov, I. S. Mamaev and I. A. Bizyaev (Regular and Chaotic Dynamics, 18(3):277-328, 2013)]. They show that this case is quite special: the additional integral of motion is elementary, while for marble rolling it is not elementary. I connect this finding with recent work about Chaplygin reduced systems that are conformally symplectic (Luis Garcia Naranjo and Juan C. Marrero. The geometry of nonholonomic Chaplygin systems revisited. Nonlinearity, 33(3):1297, 2020).

Figures (8)

• Figure 1: Alexey Borisov and Ivan Mamaev in Rio, hosted by Stefanella Boatto and Jair and a few days later in Recife, hosted by Hildeberto Cabral, 2011.
• Figure 2: The standard position.
• Figure 3: Euler angles for Lagrange's top. Taken from Arnold's Arnoldbook (Fig. 126 pg. 149). The angle $\phi$ from $e_x$ to $e_N = (\cos \phi, \sin \phi, 0) \equiv e^{i \phi}$ (a choice in the nodal line) produces an isometry between the horizontal plane and plane $e_1-e_2$. A choice between $\phi$ and $\phi + \pi$ is needed. Note that $e_N^\perp = + i \, e_N$ is always in the opposite direction to the projection of $e_3$.
• Figure 4: Space frame viewpoint (compare with Fig. 2 \ref{['fig:figdata']}). The vector $e_N = (\cos \phi, \sin \phi, 0)$ is sticking out of the page and $e_N^\perp = (-\sin \phi, \cos \phi)$ points in the direction opposite of the projection of $e_3$ (see (\ref{['Rmatrix']}). Then $z_C(\theta) = h(\theta)\, \sin \theta + f^*(\theta) \, \cos \theta,\,\, f^*(\theta) = f_o - f(\theta)$ and $PT = \Lambda(\theta) \, e_N^{\perp}$ with $\Lambda(\theta) = h(\theta)\, \cos \theta - f^*(\theta) \, \sin \theta.$ A bit surprisingly $dz_C/d\theta = \Lambda(\theta)$. Spoiler: this will be important for the "miracle".
• Figure 5: The torus example. In this picture the symmetry axis $e_3$ is pointing to the sky, so $0 < \theta < \pi/2$. We keep the convention: $e_N^\perp$ is opposite to the projection of $e_3$. Left. Coordinates $\phi$ and $\psi$ are ignored in this picture (they are ignorable). The vector $e_N$ emerges out of the picture. Right. When $\psi$ increases the torus rolls in the direction opposite to $e_N$; when $\theta$ increases it rolls in the direction opposite to $e_N^\perp$.

Theorems & Definitions (33)

• Theorem
• Definition 1
• Proposition 1
• Remark 1
• Proposition 2
• Definition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6