Nonlinear Dynamics

, Volume 97, Issue 2, pp 1635–1648 | Cite as

Nonholonomic rolling of a ball on the surface of a rotating cone

  • Alexey V. BorisovEmail author
  • Tatiana B. Ivanova
  • Alexander A. Kilin
  • Ivan S. Mamaev
Original paper


This paper investigates the rolling without slipping of a homogeneous heavy ball on the surface of a rotating cone in two settings: without dissipation in a nonholonomic setting and with rolling friction torque which is proportional to the angular velocity of the ball. In the nonholonomic setting, the resulting system of five differential equations on the level set of first integrals is reduced to quadratures. A bifurcation analysis of the above system is carried out to determine the possible types of motion. In the second case, it is shown that there are not only trajectories emanating from the lower point of the cone (its vertex), but also trajectories to the vertex of the cone (fall). An analysis of the dependence of the type of terminal motion of the center of mass of the ball on initial conditions is carried out.


Rotating surface Cone Nonholonomic constraint Rolling friction Jacobi integral Bifurcation analysis 



The work of A. V. Borisov and T. B. Ivanova (Sect. 2) was supported by Grant 15-12-20035 of the Russian Science Foundation. The work of I. S. Mamaev and A. A. Kilin (Sect. 3) was carried out at MIPT within the framework of Project 5-100 for state support for leading universities of the Russian Federation.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  2. 2.Udmurt State UniversityIzhevskRussia

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