论文标题
从模块上衍生的螺钉理论
The theory of screws derived from a module over the dual numbers
论文作者
论文摘要
螺钉理论阐明了机械中显然无关的概念之间的许多类比，包括力和角速度之间的二元性。众所周知，螺钉的真实6维空间可以赋予操作员E，E^2 = 0，将其转换为双重数字上的秩3个自由模块。在本文中，我们证明了相反的情况，即，在双重数字上有一个排名3的自由模块，并具有方向和合适的标量产品（D模块几何形状），我们表明，可以以规范的方式定义一个欧几里得空间，以使模块的每个元素都由螺杆矢量场代表。新方法具有运动演算的有效性，而独立于任何还原点。它通过表明仿射空间几何形状基本上是双数数字上的矢量空间几何形状来洞悉转移原理。然后，使用此观点恢复了螺丝理论的主要结果。
The theory of screws clarifies many analogies between apparently unrelated notions in mechanics, including the duality between forces and angular velocities. It is known that the real 6-dimensional space of screws can be endowed with an operator E, E^2 = 0, that converts it into a rank 3 free module over the dual numbers. In this paper we prove the converse, namely, given a rank 3 free module over the dual numbers, endowed with orientation and a suitable scalar product (D-module geometry), we show that it is possible to define, in a canonical way, a Euclidean space so that each element of the module is represented by a screw vector field over it. The new approach has the effectiveness of motor calculus while being independent of any reduction point. It gives insights into the transference principle by showing that affine space geometry is basically vector space geometry over the dual numbers. The main results of screw theory are then recovered by using this point of view.
