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Lie pseudogroups and mechanics by J. F. Pommaret

By J. F. Pommaret

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For any α, β ∈ R, 1. Φ(β(α x )) = βα, 2. Φ(α x + β x ) = α + β. 1) According to Sec. 7 of Appendix B, mappings like Φ here are called linear isomorphisms and therefore, conceptually, L(O; X), RL(O; X) and R are considered being identical (see Fig. 7). We have already known that the following are equivalent. (1) Only two different points, needless a third one, are enough to determine a unique line. The One-Dimensional Real Vector Space R (or R1 ) 12 O L(O; X) X x αx R L(O; X) [P] Φ P α [X ] 1 [O] 0 0 1 α R Fig.

2) and three-dimensional (Chap. 3) vector spaces will be modeled after the way we have treated here in Chap. 1. 1 Vectorization of a Straight Line: Affine Structure Fix a straight line L. − We provide a directed segment P Q on line L as a (line) vector. If − − P = Q, P Q is called a zero vector, denoted by 0 . On the contrary, P Q is a nonzero vector if P = Q. e. P Q = P Q . ⇔ 1. P Q = P Q (equal in length), 2. “the direction from P to Q (along L)” is the same as “the direction from P to Q ”. 1) We call properties 1 and 2 as the parallel invariance of vector (see Fig.

Notice that “point” is an undefined term without length, width and height. In the physical world, it is reasonable to imagine that there exits two different points. Hence, one has the Postulate line. Any two different points determine one and only one straight A straightened loop, extended beyond any finite limit in both directions, is a lively geometric model of a straight line. Mathematically, pick up two different points O and A on a flat paper, imagining extended beyond any limit in any direction, and then, connect O and A by a ruler.

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