Rubber rolling (meaning no-slip and no-twist constraints) of a convex body on the plane under the influence of gravity is a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(SE(2)\)</EquationSource> </InlineEquation> 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 [<CitationRef CitationID="CR1">1</CitationRef>, 2008], also found in A. V. Borisov, I. S. Mamaev and I. A. Bizyaev [<CitationRef CitationID="CR2">2</CitationRef>, 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(N(\theta)=\big{(}I_{1}\cos^{2}\theta+I_{3}\sin^{2}\theta+mz_{C}^{2}(\theta)\big{)}^{1/2}\)</EquationSource> </InlineEquation> where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\theta\)</EquationSource> </InlineEquation> is the nutation and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(z_{C}(\theta)\)</EquationSource> </InlineEquation> is the center of mass height. <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(N(\theta)\)</EquationSource> </InlineEquation> appears somewhat miraculously in the process of the almost symplectic reduction. I work in a space frame using the Euler angles <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\phi \text{(yaw)}, \psi \text{ roll and } \theta\)</EquationSource> </InlineEquation>. The reduction to 1 DoF is done in two stages: first, reduction by the group <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(SE(2)=\{(x,y,\phi)\}\)</EquationSource> </InlineEquation> to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(T^{*}S^{2}\)</EquationSource> </InlineEquation> with almost symplectic 2-form <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\Omega_{NH}=dp_{\theta}\wedge d\theta+dp_{\psi}\wedge d\psi+J\cdot K\)</EquationSource> </InlineEquation>. The semibasic term is <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(J\cdot K=-p_{\psi}(d\log\big{(}N(\theta)\big{)}\wedge d\psi\)</EquationSource> </InlineEquation>. It follows that <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(\Omega_{NH}\)</EquationSource> </InlineEquation> is conformally symplectic in the sense that <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(d\left(\frac{1}{N}\Omega_{NH}\right)=0.\)</EquationSource> </InlineEquation>The conserved quantity due to the <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(S^{1}\)</EquationSource> </InlineEquation> symmetry about the body axis is <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(\ell=N(\theta)\sin^{2}\theta\dot{\psi}\)</EquationSource> </InlineEquation>, yielding the desired reduction to <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\((\theta,p_{\theta})\)</EquationSource> </InlineEquation>. Further simplification results by taking the new time <InlineEquation ID="IEq16"> <EquationSource Format="TEX">\(dt=\sqrt{B(\theta)}d\tau, \text{ with } B=I_{1}+m|CP|^{2}\)</EquationSource> </InlineEquation> where <InlineEquation ID="IEq17"> <EquationSource Format="TEX">\(P=(x,y)\)</EquationSource> </InlineEquation> is the point of contact.One gets finally <InlineEquation ID="IEq18"> <EquationSource Format="TEX">\(H=\frac{1}{2}\tilde{p}^{2}_{\theta}+V(\theta),V(\theta)=\ell^{2}/2\sin^{2}\theta+mgz_{C}(\theta) \text{ with } \tilde{p}_{\theta}=p_{\theta}/\sqrt{B}\)</EquationSource> </InlineEquation> and usual symplectic form <InlineEquation ID="IEq19"> <EquationSource Format="TEX">\(d\tilde{p}_{\theta}\wedge d\theta\)</EquationSource> </InlineEquation>. The moments of inertia <InlineEquation ID="IEq20"> <EquationSource Format="TEX">\(I_{1},I_{3}\)</EquationSource> </InlineEquation> reappear in the reconstruction.As an example, very basic observations are presented for the torus.A detailed study wasjust finished by A. Kilin and E. Pivovarova in [<CitationRef CitationID="CR3">3</CitationRef>].

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)

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2025年12月20日

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Jair Koiller

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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 wasjust finished by A. Kilin and E. Pivovarova in [3].