# Philip Arathoon - Singular reduction of the 2-body problem on the 3-sphere and the 4-dimensional Lagrange top

Dates: | 5 December 2019 |
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Times: | 15:00 - 16:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Current University students |

Speaker: | Philip Arathoon |

Title: Singular reduction of the 2-body problem on the 3-sphere and the 4-dimensional Lagrange top

Abstract:

For the motion of two attracting bodies in Euclidean space, the translational symmetry reduces the system to that of the usual Kepler problem. For two bodies on a sphere this is no longer the case as no such group of translational symmetries existâ€”except for when the sphere in question is the 3-sphere. In this case, both left and right translations act as symmetries, and together generate the action of SO(4). Furthermore, owing to the double cover of SO(4), the dynamics for the 2-body problem on the sphere double cover the dynamics of a symmetric heavy top in 4 dimensions. The top is a generalised Lagrange top possessing an axis of symmetry within the body. The left and right SO(3) reductions, about the line of gravity and the line of symmetry in the top, correspond under the double cover to the left and right translations of the 2-body problem. We investigate the geometry of the reduced manifolds and study their relative equilibria.

### Speaker

Philip Arathoon