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 $$N(\theta)=\big{(}I_{1}\cos^{2}\theta+I_{3}\sin^{2}\theta+mz_{C}^{2}(\theta)\big{)}^{1/2}$$ 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 $$\phi \text{(yaw)}, \psi \text{ roll and } \theta$$ . 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 $$dt=\sqrt{B(\theta)}d\tau, \text{ with } B=I_{1}+m|CP|^{2}$$ where $$P=(x,y)$$ is the point of contact. One gets finally $$H=\frac{1}{2}\tilde{p}^{2}_{\theta}+V(\theta),V(\theta)=\ell^{2}/2\sin^{2}\theta+mgz_{C}(\theta) \text{ with } \tilde{p}_{\theta}=p_{\theta}/\sqrt{B}$$ 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].