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Cosserat Elasticity
Published in Lev Steinberg, Roman Kvasov, Cosserat Plate Theory, 2023
where γ is called the Cosserat strain tensor and χ is called the torsion tensor. These tensors can be written in the component form as the Cosserat strain-displacement and torsion-microrotation relations [55]: γji=ui,j+εijkϕk,χji=ϕi,j.
Analysis
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
If g is any metric tensor field, the above identity implies that R(ij)kℓ=∇[k∇ℓ]gij-12∇mgijSkℓm $$ R_{{(ij)k\ell }} = \nabla _{{[k}} \nabla _{{\ell ]}} g_{{ij}} - \frac{1}{2}\nabla _{m} g_{{ij}} S^{m} _{{k\ell }} $$ The torsion tensor S and curvature tensor R satisfy the following identities:
Exact Methods for ODEs
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
Garrido and Masoliver [474] determine the conditions under which a Fokker–Planck equation can be transformed into the Ornstein–Uhlenbeck class by a change of variables, and which therefore has an exact solution. Necessary and sufficient conditions are that the curvature tensor and the torsion tensor associated with the diffusion matrix are zero, and the transformed drift is linear. In one dimension, the Ornstein–Uhlenbeck equation is dx=−θxdt+σdw.
Micropolar continua as projective space of Skyrmions
Published in Philosophical Magazine, 2022
The Riemann curvature tensor is defined in terms of the general affine connection And a torsion tensor is defined by We define a contortion tensor by a difference between the general affine connection and a metric compatible connection , also known as the Christoffel symbol The contortion satisfies the antisymmetric property . From this, a dislocation density tensor K is defined by [24–26], The vanishing Riemann curvature (11) and its related measures imply the theory is in the regime of elasticity. And a set of partial differential equations may lead to an integrability condition, which is sometimes called the compatibility condition. In [27], a universal expression of the compatibility condition is studied under the setting of the vanishing curvature tensor in three dimensions. This yields two distinct classes of compatibility conditions, one for the vanishing torsion and another for the non-vanishing torsion. The former is well known by Vallée's classical result [28]. This result states that the vanishing Riemann curvature tensor in the deformed body yields the compatibility conditions equivalent to the Saint-Venant compatibility conditions [29–34] where Λ is a matrix defined by the stretches U and its derivatives. We defined and .
Implications of causality for quantum biology – I: topology change
Published in Molecular Physics, 2018
The Hodge norm of a quantity is invariant with respect to coordinate transformations because of the Hodge-⋆ and integration. It is invariant with respect to matrix-valued gauge transformations because of the trace. The trace converts the gauge group-valued quantity to a real scalar, the * above includes complex conjugation. This Hodge norm can be used to define a least action principle. In fact, for a Abelian gauge field with 1D representations K = F, we have for an action principle of electrodynamics: . The scalar term within the integral is called a Lagrangian density. For non-Abelian fields, using the Hodge norm of the gauge curvature, we have an action principle = extremum; in terms of the gauge curvature scalar where: Here ΓR is the Rth representation of the gauge symmetry group. The condition = extremum states the curvature of the QMST is to be an extremum. If one solves for and sets , it is possible to associate the non-Abelian gauge curvature with a matrix-valued Abelianised gauge field and the contraction of a matrix-valued torsion tensor . This shows the gauge curvature can be replaced by a torsion, so the dynamics can be described over a geometrically flat space (i.e. with vanishing Riemann curvature but non-vanishing torsion). The action is solely a functional of the currents . From such an action, one can use the proof of a theorem of Montesinos and Flores [28] to verify the following result for a symmetric stress-energy tensor:
Homoclinic orbits and Jacobi stability on the orbits of Maxwell–Bloch system
Published in Applicable Analysis, 2022
Yongjian Liu, Haimei Chen, Xiaoting Lu, Chunsheng Feng, Aimin Liu
For a system of second-order differential equations (SODES) where , , each function is in a neighborhood of some initial conditions in Ω. The KCC-covariant differential of a vector field on Ω is defined by where is the coefficients of the nonlinear connection defined by . The first KCC-invariant of (12) is defined by For the disturbed trajectory of the system (12), where is a small parameter, and is the components of a contravariant vector field defined along the path . The deviation equation is The second KCC-invariant of (12) is defined by associated with Berwald connection coefficients for . The third, fourth and fifth invariant of system (12) are The second KCC-invariant stands for the deviation curvature tensor. The third invariant can be interpreted geometrically as a torsion tensor. The fourth invariant and the fifth invariants are called the Riemann-Christoffel curvature tensor and the Douglas tensor, respectively. These tensors always exist in a Berwald space.