Understanding rigid body motion in arbitrary dimensions

• Published in: arXiv.org
• Main Author: Leyvraz, Francois
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 25.03.2015
• Subjects: Angular velocity; Economic models; Equations of motion; Euler-Lagrange equation; Inertia; Mathematical analysis; Matrix methods; Moments of inertia; Physics - Classical Physics; Rigid structures; Rigid-body dynamics; Tensors; Three dimensional bodies; Three dimensional motion
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1407.8155

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 specifically three-dimensional constructs is actually easier to grasp than the traditional one 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 equation of motion for two bodies linked by a spherical joint is derived to display the simplicity and the power of the method.