任意空间维度下刚体惯性张量的Ricci分解
The Ricci decomposition of the inertia tensor for a rigid body in arbitrary spatial dimensions
AI总结：
该研究针对任意空间维度下的刚体惯性张量，利用其与Riemann曲率张量共有的指标对称性，通过Ricci分解发现其Weyl张量恒为零，完全由二阶Ricci缩并表征，明确高维刚体动力学无Weyl张量带来的新定性现象。
摘要：
欧几里得空间中刚体的转动由其瞬时角速度和角动量表征。在任意空间维度下，这些量由双矢量（反对称二阶张量）表示，且通过四阶惯性张量相互关联。值得注意的是，该惯性张量属于一类已被充分研究的代数曲率张量，这类张量具有与广义相对论中所用Riemann曲率张量相同的指标对称性。任何代数曲率张量都可通过Ricci分解分解为正交群的不可约表示。我们计算了任意维度下刚体惯性张量的Ricci分解，发现（与Riemann曲率张量不同）其Weyl张量始终为零，因此该惯性张量完全由其（二阶）Ricci缩并表征。因此与广义相对论不同，在$n \geq 4$维的刚体动力学中，Weyl张量不会引发任何定性上的新现象。
英文摘要：
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 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) its Weyl tensor is always zero, so the inertia tensor is completely characterized by its (rank-2) Ricci contraction. So unlike in general relativity, the Weyl tensor does not cause any qualitatively new phenomenology for rigid-body dynamics in $n \geq 4$ dimensions.
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