### Abstract

Following parts I and II of this series, the geometry of steady states for a general convex axisymmetric rigid body spinning on a horizontal table is analysed. A general relationship between the pedal curve of the cross-section of the body and the height of its centre-of-mass above the table is obtained which allows for a straightforward determination of static equilibria. It is shown, in particular, that there exist convex axisymmetric bodies having arbitrarily many static equilibria. Four basic categories of non-isolated fixed-point branches (i.e. steady states) are identified in the general case. Depending on the geometry of the spinning body and its dynamical properties (i.e. position of centre-of-mass and inertia tensor), these elementary branches are differently interconnected in the six-dimensional system phase space and form a complex global structure. The geometry of such structures is analysed and topologically distinct classes of configurations are identified. Detailed analysis is presented for a spheroid with displaced centre-of-mass and for the tippe-top. In particular, it is shown that the fixed-point structure of the flip-symmetric spheroid, discussed in part I, represents a degenerate configuration whose degeneracy is destroyed by breaking the symmetry. For the spheroid, there are in general nine distinct classes of fixed-point structures and for the tippe-top there are three such structures. Bifurcations between these classes are identified in the parameter space of the system.

Original language | English |
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Pages (from-to) | 371-390 |

Number of pages | 20 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 462 |

Issue number | 2066 |

DOIs | |

Publication status | Published - 2006 Jan 1 |

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### Keywords

- Dynamical systems
- Non-isolated fixed-points
- Rigid body dynamics
- Spinning bodies

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)
- Physics and Astronomy(all)

### Cite this

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*,

*462*(2066), 371-390. https://doi.org/10.1098/rspa.2005.1586