Comments on a Paper about Rubber Rolling by A. V. Borisov, I. S. Mamaev and I. A. Bizyaev (with an Appendix by Luis C. García-Naranjo)

Rubber rolling (meaning no-slip and no-twist constraints) of a convex body on the plane under the influence of gravity is a $SE(2)$ Chaplygin system that reduces to the cotangent bundle of the unit sphere of Poisson vectors. I comment here upon an observation by A. V. Borisov and I. S. Mamaev [1, 2008], also found in A. V. Borisov, I. S. Mamaev and I. A. Bizyaev [2, 2013] that surfaces of revolution are special: the additional integral of motion is elementary, while for marble rolling it requires special functions. I use the term ``Nose function'' to refer to their expression $N(\theta) = \big(I_1\, \cos^2 \theta + I_3\, \sin^2 \theta + m \, z_C^2(\theta)\big)^{1/2} $ where $\theta$ is the nutation and $z_C(\theta)$ is the center of mass height. $N(\theta)$ appears somewhat miraculously in the process of the almost symplectic reduction. I work in a space frame using the Euler angles $ \phi$ (yaw), $\psi$ roll and $\theta$. The reduction to 1 DoF is done in two stages: first, reduction by the group $SE(2) = \{ (x, y, \, \phi) \} $ to $T^* S^2 $ with almost symplectic 2-form $\Omega_{NH} = dp_\theta \, \wedge d\theta + dp_\psi \, \wedge d\psi + J \cdot K$. The semibasic term is $J \cdot K = - p_\psi \, (d \log\big(N(\theta)\big) \wedge d\psi$. It follows that $\Omega_{NH} $ is conformally symplectic in the sense that $d\left(\frac{1}{N}\, \Omega_{NH}\right) = 0. $ The conserved quantity due to the $S^1$ symmetry about the body axis is $\ell = N(\theta)\,\sin^2 \theta\, \dot{\psi}$, yielding the desired reduction to $(\theta, p_\theta)$. Further simplification results by taking the new time $dt = \sqrt{B(\theta)} \, d\tau$, with $B = I_1 + m \, |CP|^2 $ where $P = (x,y)$ is the point of contact. One gets finally $H = \frac{1}{2} \tilde{p}^2_\theta + V(\theta), \, V(\theta) = \ell^2/2 \sin^2 \theta + m g\, z_C(\theta)$ with $ \tilde{p}_\theta = p_\theta/\sqrt{B} $ and usual symplectic form $ d\tilde{p}_\theta\wedge d\theta$. The moments of inertia $I_1, I_3$ reappear in the reconstruction. As an example, very basic observations are presented for the torus. A detailed study was just finished by A. Kilin and E. Pivovarova in [3].