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An Introduction to Lie Groups

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Book Id: WPLBN0000654338
Format Type: PDF eBook
File Size: 435.88 KB
Reproduction Date: 2005
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Title: An Introduction to Lie Groups  
Language: English
Subject: Science., Mathematics, Logic
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An Introduction to Lie Groups. (n.d.). An Introduction to Lie Groups. Retrieved from

Mathematics document containing theorems and formulas.

Excerpt: Rigid Body. Consider a free rigid body rotating about its center of mass, taken to be the origin. ?Free? means that there are no external forces, and ?rigid? means that the distance between any two points of the body is unchanged during the motion. Consider a point X of the body at time t = 0, and denote its position at time t by f(X, t). Rigidity of the body and the assumption of a smooth motion imply that f(X, t) = A(t)X, where A(t) is a proper rotation, that is, A(t) E SO(3), the proper rotation group of R3, the 3 ? 3 orthogonal matrices with determinant 1. The set SO(3) will be shown to be a three-dimensional Lie group, and since it describes any possible position of the body, it serves as the configuration space. The group SO(3) also plays a dual role of a symmetry group, since the same physical motion is described if we rotate our coordinate axes. Used as a symmetry group, SO(3) leads to conservation of angular momentum.


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