Abstract
This chapter and the following two are devoted to kinematics. It starts with the kinematics of a single rigid body, which is a natural continuation of the chapter on Lie groups, especially on the Euclidean group D. Indeed the set S of positions of a rigid body, also called the configuration space of the body, is a principal homogeneous space of D, what is meaning that there is a left-action of D on S with the properties explained below in [4.1]. Some mathematical symmetry properties of the calculations in a Lie group disappear in S, what demonstrates the differences between Lagrangian and Eulerian pictures of mechanics. Fortunately the differential calculus is easily extended from D to S and it leads to nice coordinate free formulas for the velocities and acceleration of a rigid body as well in Eulerian picture as in Lagrangian picture. The relationships between the two pictures is now explicit and clear: it consists in an adjoint transformation in the Lie group D (for more details see article).
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