Understanding rigid body motion in arbitrary dimensions

• Published in: European journal of physics Vol. 36; no. 3; pp. 35021 - 17
• Main Author: Leyvraz, Francois
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.05.2015
• Subjects: Angular velocity; classical mechanics; Construction specifications; Equations of motion; Lagrangian formalism; Mathematical analysis; Moments of inertia; rigid body; Rigid-body dynamics; Symmetry; Tensors; Three dimensional
• ISSN: 0143-0807; 1361-6404
• DOI: 10.1088/0143-0807/36/3/035021

Why would anyone wish to generalize the already unappetizing subject of rigid body motion to an arbitrary number of dimensions? At first sight, the subject seems to be both repellent and superfluous. The author will try to argue that an approach involving no specific three-dimensional constructs is actually easier to grasp than the traditional approach and might thus be generally useful to understand rigid body motion both in three dimensions and in the general case. Specific differences between the viewpoint suggested here and the usual one include the following: here angular velocities are systematically treated as antisymmetric matrices, a symmetric tensor I quite different from the moment of inertia tensor plays a central role, whereas the latter is shown to be a far more complex object, namely a tensor of rank four. A straightforward way to define it is given. The Euler equation is derived and the use of Noether's theorem to obtain conserved quantities is illustrated. Finally the equations of motion for a heavy top as well as for two bodies linked by a spherical joint are derived to display the simplicity and the power of the method.