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Absorption of torsion also plays a fundamental role in the study of G-structures and Cartan's equivalence method. Torsion is also useful in the study of unparametrized families of geodesics, via the associated projective connection. In relativity theory, such ideas have been implemented in the form of Einstein–Cartan theory.
The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections.Thus, for example, a vielbein together with a spin connection, when subject to the condition of vanishing torsion, gives a description of Einstein gravity.
Torsion of a square section bar Example of torsion mechanics. In the field of solid mechanics, torsion is the twisting of an object due to an applied torque [1] [2].Torsion could be defined as strain [3] [4] or angular deformation [5], and is measured by the angle a chosen section is rotated from its equilibrium position [6].
Geometric relevance: The torsion τ(s) measures the turnaround of the binormal vector. The larger the torsion is, the faster the binormal vector rotates around the axis given by the tangent vector (see graphical illustrations). In the animated figure the rotation of the binormal vector is clearly visible at the peaks of the torsion function.
The field equations of Einstein–Cartan theory come from exactly the same approach, except that a general asymmetric affine connection is assumed rather than the symmetric Levi-Civita connection (i.e., spacetime is assumed to have torsion in addition to curvature), and then the metric and torsion are varied independently.
The torsion constant or torsion coefficient is a geometrical property of a bar's cross-section. It is involved in the relationship between angle of twist and applied torque along the axis of the bar, for a homogeneous linear elastic bar. The torsion constant, together with material properties and length, describes a bar's torsional stiffness.
To find the torsion coefficient of the wire, Cavendish measured the natural resonant oscillation period T of the torsion balance: T = 2 π I κ {\displaystyle T=2\pi {\sqrt {\frac {I}{\kappa }}}} Assuming the mass of the torsion beam itself is negligible, the moment of inertia of the balance is just due to the small balls.
The other two 1-forms in the Cartan structural equations are given by θ 1 = β and θ 2 = γ. The structural equations themselves are just the Maurer–Cartan equations. In other words; The Cartan structural equations for SO(3)/SO(2) reduce to the Maurer–Cartan equations for the left invariant 1-forms on SO(3). Since α is the connection form,