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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 467520, 21 pages
http://dx.doi.org/10.5402/2012/467520
Research Article

Rigid Body Trajectories in Different 6D Spaces

1School of Systems Engineering, University of Reading, Reading RG6 6AY, UK
2Department of Mechanical Engineering, University of Strathclyde, Glasgow G1 1XQ, UK

Received 26 April 2012; Accepted 4 June 2012

Academic Editors: G. Goldin, D. Ida, and J. A. Nieto

Copyright © 2012 Carol Linton et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The objective of this paper is to show that the group 𝑆 𝐸 ( 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 tends to zero as the scaling factor increases. The semidirect 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.