Circular orbits of a ball on a rotating conical turntable

Borisov, Alexey V.; Ivanova, Tatiana B.; Kilin, Alexander A.; Mamaev, Ivan S.

doi:10.1007/s00707-019-02556-y

Cited by 1 publication

(7 citation statements)

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“…Its first studies go back to Routh, and there has recently been a renew of interest, see e.g. [19,24,9,14,13,16,8,5,6,7,2,1,11]. A rather general study of the dynamics of the system has been the object of the very recent article [11], which is the basis for the present study.…”

Section: Introductionmentioning

confidence: 93%

“…and the equations of motion of the reduced system are Eqs. (7). These equations define a vector field on M 5 .…”

Section: 2mentioning

confidence: 99%

“…Proof. At an equilibrium, v 1 = v 2 = 0 and the vanishing of v1 , v2 and ωz in (7) gives the three conditions…”

Section: 2mentioning

confidence: 99%

“…3.2 Linearization. Since in the SO(3)-reduced equations of motion (7) the coordinate ω z is always multiplied by either v 1 or v 2 , the last column of the Jacobian matrix of the SO(3)-reduced vector field vanishes at the equilibria. Therefore, the linearization at a reduced equilibrium has always an eigenvalue 0.…”

Section: 2mentioning

confidence: 99%

“…This allowed to study and classify, for instance, its equilibria [11]. Numerical investigations of the reduced dynamics in the particular case of a rotating conical surface are given in [7].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Fassò

Sansonetto

2022

Regul. Chaot. Dyn.

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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 SO(3) × 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 blow up. This is done 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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