02-101 Yaakov Friedman, Yuriy Gofman

Relativistic Dynamic Equation in Invariant Form (24K, LaTeX 2e) Mar 5, 02

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Abstract. In some dynamical systems boosts and rotations occur synchronically. Usually in analysis of such systems we separate these two types of motion. Unfortunately, the result of such analysis depends on the order in which operations are performed. To avoid the order dependence of the above operations, we propose a new dynamic variable called the symmetric velocity. This new velocity could be calculated directly from the regular velocity and is its relativistic half. The set of all possible symmetric velocities is a three dimensional ball of radius \textit{c} and the Lorentz group acts on this ball via conformal maps. The generators of these maps (elements of the Lie algebra) are second order transformations expressed by a triple product. This triple product is the one corresponding to the Bounded Symmetric Domain of type 4 in Cartan's classification, also called the spin factor. The product is connected with the Geometric (Clifford) product, explaining why use of the geometric product simplifies formulae in several areas of physics.

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