### Abstract

The objective of this paper is to show that the group SE(3) with an imposed Lie-Poisson structure can be used to determine the trajectory in a spatial frame of a rigid body in Euclidean space. Identical results for the trajectory are obtained in spherical and hyperbolic space by scaling the linear displacements appropriately, since the influence of the moments of inertia on the trajectories tend to zero as the scaling factor increases. The semi-direct product of the linear and rotational motions gives the trajectory from a body frame perspective. It is shown that this cannot be used to determine the trajectory in the spatial frame. The body frame trajectory is thus independent of the velocity coupling. In addition, it is shown that the analysis can be greatly simplified by aligning the axes of the spatial frame with the axis of symmetry which is unchanging for a natural system with no forces and rotation about an axis of symmetry.

Original language | English |
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Article number | 467520 |

Number of pages | 21 |

Journal | ISRN Mathematical Physics |

Volume | 2012 |

Early online date | 4 Jun 2012 |

DOIs | |

Publication status | Published - Jul 2012 |

### Keywords

- Rigid body trajectories
- semi-direct product of translations and rotations
- structure constants and equations of motion equations
- imposed Lie-Poisson structure on Special Euclidean group

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## Cite this

Linton, C., Holderbaum, W., & Biggs, J. (2012). Rigid body trajectories in different 6D spaces.

*ISRN Mathematical Physics*,*2012*, [467520]. https://doi.org/10.5402/2012/467520