The Ricci decomposition of the inertia tensor for a rigid body in arbitrary spatial dimensions. (English) Zbl 1541.70005
Summary: The rotations of rigid bodies in Euclidean space are characterized by their instantaneous angular velocity and angular momentum. In an arbitrary number of spatial dimensions, these quantities are represented by bivectors (antisymmetric rank-2 tensors), and they are related by a rank-4 inertia tensor. Remarkably, this inertia tensor belongs to a well-studied class of algebraic curvature tensors that have the same index symmetries as the Riemann curvature tensor field used in general relativity. Any algebraic curvature tensor can be decomposed into irreducible representations of the orthogonal group via the Ricci decomposition. We calculate the Ricci decomposition of the inertia tensor for a rigid body in any number of dimensions, and we find that (unlike for the Riemann curvature tensor field) its traceless Weyl tensor is always zero, so the inertia tensor is completely characterized by its (rank-2) Ricci contraction. So unlike in general relativity, the traceless Weyl tensor does not cause any qualitatively new phenomenology for rigid-body dynamics in \(n \geq 4\) dimensions.
MSC:
| 70E15 | Free motion of a rigid body |
| 70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |
| 53Z05 | Applications of differential geometry to physics |
| 15A75 | Exterior algebra, Grassmann algebras |
Keywords:
angular velocity; angular momentum; higher dimension; antisymmetric rank-2 tensor; exterior algebra; algebraic curvature tensor; Weyl tensorSoftware:
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