自由落体和刚体力学中的仿射张量
Affine Tensors in Mechanics of Freely Falling Particles and Rigid Bodies
摘要：
在以前的论文中，我们提出了一种方法的基础上使用仿射张量的3D机构和壳的动力学。这一新的理论框架是非常大的，其应用并不局限于连续介质力学。在本文中，我们展示了它如何也可以应用于描述的自由落体粒子和刚体的力学。质量、线动量和角动量被构造成一个单一的物体，称为torsor。排除所有度量的概念，我们定义的torsors作为反对称双线性映射上的仿射函数的线性空间。Torsors是仿射张量的一个特殊家族。在此基础上，我们定义了一个内在的微分算子称为仿射协变导数。接下来，我们声称表征自由落体粒子行为的torsor是无仿射协变导数的，这使得可以恢复线性和角动量定律。最后，它示出了如何刚体的运动可以在这个框架内描述。
英文摘要：
In a previous paper, we proposed an approach for the dynamics of 3D bodies and shells based on the use of affine tensors. This new theoretical frame is very large and the applications are not limited to the mechanics of continua. In the present paper, we show how it can be also applied to the description of the mechanics of freely falling particles and rigid bodies. The mass, the linear and angular momenta are structured as a single object called torsor. Excluding all metric notions, we define the torsors as skew-symmetric bilinear mappings operating on the linear space of the affine functions. Torsors are a particular family of affine tensors. On this ground, we define an intrinsic differential operator called the affine covariant derivative. Next, we claim that the torsor characterizing the behavior of a freely falling particle is affine covariant derivative free, that allows recovering both laws of linear and angular momentum. Finally, it is shown how the motion of rigid bodies can be describe within this frame.
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