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 $\textrm{SO(3)}$-reduced system.
The $\textrm{SO(3)}$-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.
Keywords:
nonholonomic mechanical systems with symmetry, rolling rigid bodies, relative
equilibria, kasamawashi.
Citation:
Francesco Fassò, Nicola Sansonetto, “On Some Aspects of the Dynamics of a Ball in a Rotating
Surface of Revolution and of the Kasamawashi Art”, Regul. Chaotic Dyn., 27:4 (2022), 409–423
\Bibitem{FasSan22}
\by Francesco Fass\`o, Nicola Sansonetto
\paper On Some Aspects of the Dynamics of a Ball in a Rotating
Surface of Revolution and of the Kasamawashi Art
\jour Regul. Chaotic Dyn.
\yr 2022
\vol 27
\issue 4
\pages 409--423
\mathnet{http://mi.mathnet.ru/rcd1172}
\crossref{https://doi.org/10.1134/S1560354722040025}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4462430}
Linking options:
https://www.mathnet.ru/eng/rcd1172
https://www.mathnet.ru/eng/rcd/v27/i4/p409
This publication is cited in the following 3 articles:
Alexander A. Kilin, Elena N. Pivovarova, “Bifurcation analysis of the problem of a “rubber” ellipsoid of revolution rolling on a plane”, Nonlinear Dyn, 2024
Alexander A. Kilin, Elena N. Pivovarova, Tatiana B. Ivanova, “Rolling of a Homogeneous Ball on a Moving Cylinder”, Regul. Chaot. Dyn., 2024
Marco Dalla Via, Francesco Fassò, Nicola Sansonetto, “On the Dynamics of a Heavy Symmetric Ball that Rolls Without Sliding on a Uniformly Rotating Surface of Revolution”, J Nonlinear Sci, 32:6 (2022)
Citation in format AMSBIB
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