Abstract
We apply general tensor calculus to arbitrary nonrelativistic classical Lagrangian systems and derive general relationships between the torsion tensor and other quantities associated with the coordinate configuration space, such as the metric, the Christoffel symbols, the Euler-Lagrange equations, the affine connections, and the curvature tensor. Using Euler angle metric spaces as examples of spaces with nonzero torsion, we calculate these quantities for nonrelativistic rigid rotators of arbitrary structure. For free rotations, we show that the Euler-Lagrange equations agree with the manifestly correct Euler equations.
Highlights
In 1922, Élie Cartan included torsion [1] in Einstein’s general relativity [2] [3]
Using Euler angle metric spaces as examples of spaces with nonzero torsion, we calculate these quantities for nonrelativistic rigid rotators of arbitrary structure
The results allow the following conclusions: 1) In any affinely connected metric space, the symmetrized affine connections are equal to the corresponding Christoffel symbols if and only if the covariant torsion tensor is completely antisymmetric or zero (see Equation (13)); 2) Any Euler angle 3-space is a flat space with torsion
Summary
In 1922, Élie Cartan included torsion [1] in Einstein’s general relativity [2] [3]. Several authors (including Einstein) investigated the torsion concept in an attempt to unify electromagnetism with gravity [8] [9] [10]. Trautman [11] rewrote the Einstein field equations to include nonzero torsion and solved the system for a model consisting of a spinning cloud of dust previous to the big bang. Shapiro [12] investigated effects of including a torsion background in quantized fields. Despite these and many other investigations, the physical role of torsion is still unclear, and modern general relativity is still torsion-free
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