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On Some Aspects of the Dynamics of a Ball in a Rotating Surface of Revolution and of the Kasamawashi Art

Abstract

We study some aspects of the dynamics of the nonholonomic system formed by a heavy homogeneous ball constrained to roll without sliding on a steadily rotating surface of revolution. First, in the case in which the figure axis of the surface is vertical (and hence the system is \(\textrm{SO(3)}\times\textrm{SO(2)}\)-symmetric) and the surface has a (nondegenerate) maximum at its vertex, we show the existence of motions asymptotic to the vertex and rule out the possibility of blowup. This is done by passing to the 5-dimensional SO(3)-reduced system. The SO(3)-symmetry persists when the figure axis of the surface is inclined with respect to the vertical — and the system can be viewed as a simple model for the Japanese kasamawashi (turning umbrella) performance art — and in that case we study the (stability of the) equilibria of the 5-dimensional reduced system.

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Notes

  1. As long as it is regular at the vertex, thus excluding, e. g., the case of a conical surface

  2. From now it is understood that, unless otherwise specified, \(\psi\) and its derivatives are evaluated at \({\textstyle\frac{1}{2}}|x|^{2}\) and \(F\) at \(|x|\).

  3. They form two-parameter families and therefore there are at least two zero eigenvalues of the linearization. But in fact, there are always three zero eigenvalues; this can be explained through the already mentioned fact that the \(\textrm{SO{(3)}}\times\textrm{SO{(2)}}\)-reduced system has a Hamiltonian structure.

  4. Such as the one available at https://www.youtube.com/watch?v=FeDyMdh1JLQ

  5. In the movie, the angle \(\alpha\) is small and the ball sits at a distance from the rotation axis which is approximately two to three times its radius, hence \(x_{1}>1\).

References

  1. Balseiro, P. and Yapu, L. P., Conserved Quantities and Hamiltonization of Nonholonomic Systems, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2021, vol. 38, no. 1, pp. 23–60.

    MathSciNet  Article  Google Scholar 

  2. Balseiro, P. and Sansonetto, N., First Integrals and Symmetries of Nonholonomic Systems, Arch. Ration. Mech. Anal., 2022, vol. 244, no. 2, pp. 343–389.

    MathSciNet  Article  Google Scholar 

  3. Bates, L. and Śniatycki, J., Nonholonomic Reduction, Rep. Math. Phys., 1993, vol. 32, no. 1, pp. 99–115.

    MathSciNet  Article  Google Scholar 

  4. Bloch, A. M., Krishnaprasad, P. S., Marsden, J. E., and Murray, R. M., Nonholonomic Mechanical Systems with Symmetry, Arch. Rational Mech. Anal., 1996, vol. 136, no. 1, pp. 21–99.

    MathSciNet  Article  Google Scholar 

  5. Borisov, A. V., Ivanova, T. B., Karavaev, Yu. L., and Mamaev, I. S., Theoretical and Experimental Investigations of the Rolling of a Ball on a Rotating Plane (Turntable), Eur. J. Phys., 2018, vol. 39, no. 6, 065001, 13 pp.

    Article  Google Scholar 

  6. Borisov, A. V., Ivanova, T. B., Kilin, A. A., and Mamaev, I. S., Nonholonomic Rolling of a Ball on the Surface of a Rotating Cone, Nonlinear Dynam., 2019, vol. 97, no. 2, pp. 1635–1648.

    Article  Google Scholar 

  7. Borisov, A. V., Ivanova, T. B., Kilin, A. A., and Mamaev, I. S., Circular Orbits of a Ball on a Rotating Conical Turntable, Acta Mech., 2020, vol. 231, no. 3, pp. 1021–1028.

    MathSciNet  Article  Google Scholar 

  8. Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Jacobi Integral in Nonholonomic Mechanics, Regul. Chaotic Dyn., 2015, vol. 20, no. 3, pp. 383–400.

    MathSciNet  Article  Google Scholar 

  9. Borisov, A. V., Mamaev, I. S., and Kilin, A. A., The Rolling Motion of a Ball on a Surface: New Integrals and Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 201–219.

    MathSciNet  Article  Google Scholar 

  10. Chicone, C., Ordinary Differential Equations with Applications, 2nd ed., Texts Appl. Math., vol. 34, New York: Springer, 2006.

    MATH  Google Scholar 

  11. Cushman, R., Duistermaat, H., and Śniatycki, J., Geometry of Nonholonomically Constrained Systems, Adv. Ser. Nonlinear Dynam., vol. 26, Hackensack, N.J.: World Sci., 2010.

    MATH  Google Scholar 

  12. Dalla Via, M., Fassò, F., and Sansonetto, N., On the Dynamics of a Heavy Symmetric Ball That Rolls without Sliding on a Uniformly Rotating Surface of Revolution, arXiv:2109.00236 (2021).

  13. Fassò, F., García-Naranjo, L. C., and Sansonetto, N., Moving Energies As First Integrals of Nonholonomic Systems with Affine Constraints, Nonlinearity, 2018, vol. 31, no. 3, pp. 755–782.

    MathSciNet  Article  Google Scholar 

  14. Fassò, F. and Giacobbe, A., Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System, SIGMA Symmetry Integrability Geom. Methods Appl., 2007, vol. 3, Paper 051, 12 pp.

    MathSciNet  MATH  Google Scholar 

  15. Fassò, F., Giacobbe, A., and Sansonetto, N., Periodic Flows, Rank-Two Poisson Structures, and Nonholonomic Mechanics, Regul. Chaotic Dyn., 2005, vol. 10, no. 3, pp. 267–284.

    MathSciNet  Article  Google Scholar 

  16. Fassò, F. and Sansonetto, N., Conservation of Energy and Momenta in Nonholonomic Systems with Affine Constraints, Regul. Chaotic Dyn., 2015, vol. 20, no. 4, pp. 449–462.

    MathSciNet  Article  Google Scholar 

  17. Fassò, F. and Sansonetto, N., Conservation of “Moving” Energy in Nonholonomic Systems with Affine Constraints and Integrability of Spheres on Rotating Surfaces, J. Nonlinear Sci., 2016, vol. 26, no. 2, pp. 519–544.

    MathSciNet  Article  Google Scholar 

  18. Field, M. J., Equivariant Dynamical Systems, Trans. Amer. Math. Soc., 1980, vol. 259, no. 1, pp. 185–205.

    MathSciNet  Article  Google Scholar 

  19. Golubitsky, M. and Guillemin, V., Stable Mappings and Their Singularities, Grad. Texts in Math., vol. 14, New York: Springer, 1973.

    Book  Google Scholar 

  20. Hermans, J., A Symmetric Sphere Rolling on a Surface, Nonlinearity, 1995, vol. 8, no. 4, pp. 493–515.

    MathSciNet  Article  Google Scholar 

  21. Krupa, M., Bifurcations of Relative Equilibria, SIAM J. Math. Anal., 1990, vol. 21, no. 6, pp. 1453–1486.

    MathSciNet  Article  Google Scholar 

  22. Routh, E. J., The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies: Being Part II of a Treatise on the Whole Subject, 6th ed., New York: Dover, 1955.

    MATH  Google Scholar 

  23. Watanabe, S., Kouda, M., and Kiyohiro, N., Positioning Control of a Rolling Ball on a Rotating Umbrella by a Kasamawashi Robot, Adv. Robot., 1998, vol. 13, no. 3, pp. 339–341.

    Article  Google Scholar 

  24. Whitney, H., Differentiable Even Functions, Duke Math. J., 1943, vol. 10, pp. 159–160.

    MathSciNet  MATH  Google Scholar 

  25. Zenkov, D. V., The Geometry of the Routh Problem, J. Nonlinear Sci., 1995, vol. 5, no. 6, pp. 503–519.

    MathSciNet  Article  Google Scholar 

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ACKNOWLEDGMENTS

The authors would like to thank Prof. Toshiro Iwai for pointing out to one of them the similarity between the dynamics of the ball on a rotating surface and the kasamawashi performance art.

Funding

F. F. has been partially supported by the MIUR-PRIN project 20178CJA2B New Frontiers of Celestial Mechanics: Theory and Applications.

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Correspondence to Francesco Fassò or Nicola Sansonetto.

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MSC2010

37J60, 70E18, 70F25, 70E50

APPENDIX. THE EQUATIONS OF MOTION OF THE SYSTEM

The equations of motion of the system can be determined in various routine ways which, however, as often happens with nonholonomic systems, involve some tedious computations. Here we follow the approach of [12].

Reference [12] employs a known form of the equations of motion of mechanical nonholonomic systems as the restriction to the constraint manifold of Lagrange equations with the nonholonomic reaction forces, writing them, however, in a way that allows for the use of quasi-velocities (Proposition 16 in the Appendix of [12]). Of course, one might just specialize those formulas to the present case, and this would indeed be the straightest — though somewhat laborious — approach. However, since the computations are there already made for the case \(\alpha=0\), in order to keep the length of this article to a minimum we prefer here to indicate how to modify that deduction to allow for \(\alpha\not=0\). There are in fact three other minor differences. One is technically irrelevant: reference [12] assumes that the domain \(I=(-R,R)\) of the profile function is the entire real axis, so that \(D=\mathbb{R}^{2}\). In addition, the derivation of the equations of motion in the Appendix of [12] uses the profile function \(f\), not \(\psi\), and a different parametrization of \(M_{8}\), which excludes the vertex and uses polar coordinates, namely, \((r,\theta,v_{r},v_{\theta},\mathcal{R},\omega_{\mathrm{z}})\in\mathbb{R}_{+}\times S^{1}\times\mathbb{R}\times\mathbb{R}\times\textrm{SO{(3)}}\times\mathbb{R}=:M_{8}^{\mathrm{pol}}\) with \(x_{1}=r\cos\theta\), \(x_{2}=r\sin\theta\), \(v_{r}=\dot{r}\), \(v_{\theta}=\dot{\theta}\). We thus indicate how to modify such a derivation.

First, the inclination of the surface has the only effect of changing the potential energy of the weight force: instead of \(g\mathrm{z}|_{M_{8}^{\mathrm{pol}}}=a\hat{g}f(r)\), it becomes \(g(\mathrm{z}\cos\alpha+\mathrm{x}\sin\alpha)|_{M_{8}^{\mathrm{pol}}}=a\hat{g}(f(r)\cos\alpha+r\sin\alpha\cos\theta)\). This has the consequence that the nonholonomic reaction force \(R\), given in formula (46) within the proof of Proposition 17 of [12], gets the following changes: in its \(\dot{r}\)-component the term \(\mu\hat{g}f^{\prime}\) has to be replaced with \(\mu\hat{g}(f^{\prime}\cos\alpha+\sin\alpha\cos\theta)\), its \(\dot{\theta}\)-component acquires a term \(-\mu\hat{g}r^{-1}\sin\alpha\sin\theta\) and its \(\omega_{\mathrm{z}}\)-component acquires a term \(-\mu\hat{g}F^{-1}f^{\prime}\sin\alpha\sin\theta\). These changes propagate to the equations for \(\dot{v}_{r}\), \(\dot{v}_{\theta}\) and \(\dot{\omega}_{\mathrm{z}}\) as given in Proposition 17 of [12] after multiplication by the appropriate entries of the inverse of the kinetic matrix (namely, \(F^{-2}\), \(r^{-2}\) and \(k^{-1}\), respectively).

Second, the equations for \(\dot{v}_{r}\) and \(\dot{v}_{\theta}\) can be transformed into equations for \(\dot{v}_{1}\) and \(\dot{v}_{2}\) by using the kinematical identities \(\dot{v}_{1}=\big{(}\frac{\dot{v}_{r}}{r}-v_{\theta}^{2}\big{)}x_{1}-\big{(}\dot{v}_{\theta}+2\frac{v_{r}v_{\theta}}{r}\big{)}x_{2}\) and \(\dot{v}_{2}=\big{(}\frac{\dot{v}_{r}}{r}-v_{\theta}^{2}\big{)}x_{2}+\big{(}\dot{v}_{\theta}+2\frac{v_{r}v_{\theta}}{r}\big{)}x_{1}\) and making the obvious substitutions \(r\to|x|\), \(\sin\theta\to\frac{x_{2}}{r}\), \(\cos\theta\to\frac{x_{1}}{r}\), \(v_{r}\to x\cdot v\), \(v_{\theta}\to\frac{x_{1}v_{2}-x_{2}v_{1}}{r}\). This leads to the equations \(\dot{x}_{1}=v_{1}\), \(\dot{x}_{2}=v_{2}\), \(\dot{\mathcal{R}}=\mathcal{R}^{T}\omega\) and

$$ \begin{aligned} \dot v_1 = & - \frac\gamma{F^2} \bigg( \frac{x_1}{|x|}f'\cos\alpha + \Big(1 + \frac{x_2^2}{|x|^2} f'^2 \Big)\sin\alpha\bigg) + \frac\mu F \Big( \frac{x_1}{|x|^3} x\!\cdot\!Jv f' + \frac{x_2}{|x|^2} x\!\cdot\!v f'' \Big)\omega_z \\ & - \frac\mu{F^2} \frac{v_1}{|x|} x\!\cdot\!v f' f'' - \frac{f'}{(1+k)F^2} \frac{x_1}{|x|^4} \Big( (\!x\!\cdot\!Jv\!)^2f'+ |x|(\!x\!\cdot\!v\!)^2f''\Big) \\ & - \Omega \mu \bigg(v_2 + \frac1F \frac{x_1}{|x|^3} x\!\cdot\!Jv f' + \frac{x_2}{|x|^2}\frac{x\!\cdot\!v}{F^2} f'' \big(F + |x|f'\big)\bigg) \\ \dot v_2 = & - \frac\gamma{F^2} \frac{x_2}{|x|} f' \Big(\cos\alpha - \frac{x_1}{|x|}f'\sin\alpha\Big) + \frac\mu F \Big(\frac{x_2}{|x|^3} x\!\cdot\!Jv f' - \frac{x_1}{|x|^2} x\!\cdot\!v f''\Big) \omega_z \\ & - \frac\mu {F^2} \frac{v_2}{|x|} x\!\cdot\!v f' f'' - \frac{f'}{(1+k)F^2}\frac{x_2}{|x|^4} \Big( (\!x\!\cdot\!Jv)^2 f' + |x|(\!x\!\cdot\!v)^2f''\Big) \\ & + \Omega \mu \bigg( v_1- \frac1F\frac{x_2}{|x|^3}x\!\cdot\!Jv f' + \frac{x_1}{|x|^2} \frac{x\!\cdot\!v}{F^2}f''\big(F+|x|f'\big) \bigg) \\ \dot \omega_\mathrm{z} = & -\frac\gamma F \frac{x_2}{|x|}f'\sin\alpha - \frac{f'f''}{(1+k)F^3} \frac{x\!\cdot\!v}{|x|^2} \big(|x|F\omega_\mathrm{z}- x\!\cdot\!Jv f' \big) \\ & + \Omega \frac{f'}{(1+k)F} \frac{x\!\cdot\!v}{|x|} \Big(1+ \frac{f''}F +|x|\frac{f'f''}{F^2} \Big) \end{aligned} $$
with \(J=\left(\begin{matrix}0&1\\ -1&0\end{matrix}\right)\). After replacing \(f^{\prime}\) with \(|x|\psi^{\prime}\) and \(f^{\prime\prime}\) with \(\psi^{\prime}+|x|^{2}\psi^{\prime\prime}\), see (2.1), these equations take the form (2.7).

In this way we have proven that (2.7) are the equations of motion of the system in the subset of the phase space \(M_{8}\) where \(x\not=0\). Therefore, their right-hand side defines a vector field \(Y\) in \(M_{8}\setminus\{x=0\}\) which coincides with the restriction to such a set of the dynamical vector field of the system. But since the latter is known (from the general theory) to exist in all of \(M_{8}\), \(M_{8}\setminus\{x=0\}\) is dense in \(M_{8}\) and \(Y\) has a continuous extension to \(M_{8}\), the extension of \(Y\) is the dynamical vector field of the system in all of \(M_{8}\).

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Fassò, F., Sansonetto, N. On Some Aspects of the Dynamics of a Ball in a Rotating Surface of Revolution and of the Kasamawashi Art. Regul. Chaot. Dyn. 27, 409–423 (2022). https://doi.org/10.1134/S1560354722040025

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Keywords

  • nonholonomic mechanical systems with symmetry
  • rolling rigid bodies
  • relative equilibria
  • kasamawashi