Finsler geometry of topological singularities for multi‐valued fields: Applications to continuum theory of defects

Yajima, Takahiro; Nagahama, Hiroyuki

doi:10.1002/andp.201600172

Cited by 7 publications

(11 citation statements)

References 60 publications

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“…For example, a dislocation in crystals is related to the torsion tensor [46][47][48][49][50]. Then, a path-dependency of the topological singularity is linked with the torsion and curvature tensors through the integrability conditions (13) and (15) [51]. This relation implies that the two cases of the gimbal locks may be characterized by using the path-dependency of the Euler angles.…”

Section: Discussion (Topological Singularity and Torsion Of The Eulermentioning

confidence: 99%

Non-holonomic geometric structures of rigid body system in Riemann-Cartan space

2018

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A theory of non-Riemannian geometry (Riemann-Cartan geometry) can be applied to a free rotation of a rigid body system. The Euler equations of angular velocities are transformed into equations of the Euler angle. This transformation is geometrically non-holonomic, and the Riemann-Cartan structure is associated with the system of the Euler angles. Then, geometric objects such as torsion and curvature tensors are related to a singularity of the Euler angle. When a pitch angle becomes singular ±π/2, components of the torsion tensor diverge for any shape of the rigid body while components of the curvature tensor do not diverge in case of a symmetric rigid body. Therefore, the torsion tensor is related to the singularity of dynamics of the rigid body rather than the curvature tensor. This means that the divergence of the torsion tensor is interpreted as the occurrence of the gimbal lock. Moreover, attitudes of the rigid body for the singular pitch angles ±π/2 are distinguished by the condition that a path-dependence vector of the Euler angles diverges or converges.

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Section: Discussion (Topological Singularity and Torsion Of The Eulermentioning

confidence: 99%

Non-holonomic geometric structures of rigid body system in Riemann-Cartan space

2018

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“…Then, the multivalued function i has a path dependency. The topological charge is geometrically represented by a "discrepancy" along a closed curve C, [27] which is called non-evanescible circuit. [54] Moreover, the quantities V i jk and W i jkh are related to the continuity conditions:…”

Section: Brief Review Of Integrability Of Multivalued Field and The Pmentioning

confidence: 99%

“…[28][29][30][31][32][65][66][67] For example, when a macroscopic displacement is attached to each point, the geometric framework is described by a first-order vector bundle. [27] However, in the micromechanics, the microrotation is defined at the one more microscopic level than the macroscopic displacement level. This means that the geometric structure of the micropolar continuum should be described in a higher-order space whose connection structure has been discussed in ref.…”

Section: Parallelism and Geometric Objects In Second-order Vector Bundlementioning

confidence: 99%

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Connection Structures of Topological Singularity in Micromechanics from a Viewpoint of Generalized Finsler Space

Yajima

Nagahama

2020

Annalen der Physik

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Geometric structures of Cosserat or micropolar continuum are discussed based on geometric objects in a non-Riemannian space. A microrotation is described in a microscopic level than a macroscopic displacement level. In this case, a microscopic rotation can be expressed as a nonlocal internal variable attached to each point in a generalized Finsler space. Such non-local hierarchy is geometrically realized by using a second-order vector bundle viewpoint. Then, two kinds of torsion tensor in the second-order vector bundle are obtained. One is characterized by the macroscopic displacement. The other is characterized by the microscopic rotation. These torsion tensors are equivalent to nonintegrability conditions for multivalued macroscopic displacement and microscopic rotation. Especially, a path dependency of the displacement and the microscopic rotation is represented by a non-vanishing condition of torsion tensors. Moreover, the concept of non-locality of the Finsler geometry implies that the approach of higher-order geometry is applicable to a finite deformation in nonlinear mechanics. The singularity given by the multivalued function is also described as a boundary value problem. An application of the generalized Finsler geometry to a gradient theory is also discussed.

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“…Then, a torsion tensor arises as a non‐Riemannian geometric object:

T_{β γ}^{α} = false(Γ_{β γ}^{α} - Γ_{γ β}^{α} false) / 2

. From perspective of geometry and topology, the torsion tensor is related to the non‐periodicity and irreversibly of the system . For example, the non‐periodic behavior of a nonlinear dynamical system is represented by a “discrepancy” of solution curve caused by the torsion tensor.…”

Section: Differential Geometry Of Fractional‐order Differential Equatmentioning

confidence: 99%

Geometric Structures of Fractional Dynamical Systems in Non‐Riemannian Space: Applications to Mechanical and Electromechanical Systems

Yajima

Nagahama

2018

Annalen der Physik

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Based on a non-Riemannian treatment of geometric objects, the geometric structures of fractional-order dynamical systems are investigated. A fractional derivative describes non-local effects across a space or a history encoded in memory features of the system. A system of fractional-order differential equations is formulated in film space that includes fictitious forces. Film space is a geometric space whose coordinates comprise time, and the geometric quantities vary in time. Fractional-order torsion tensors that appear are related to the dissipated energy and the energy conversions between subsystems and power of the system. The geometric treatment is then applied to damped-harmonic and fractional oscillators and the hybrid electromechanical Rikitake system. The damped-harmonic oscillator is characterized by two torsion tensors, whereas the fractional oscillator is characterized by one torsion tensor. Herein, the fractional order of the derivative of the metric tensor is used to characterize the damping of the fractional oscillator. The energy conversions between electromechanical subsystems in the Rikitake system are characterized by the torsion tensor. These results suggest that the non-Riemannian geometric objects can represent the non-local properties of fractional-order dynamical systems.

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Finsler geometry of topological singularities for multi‐valued fields: Applications to continuum theory of defects

Cited by 7 publications

References 60 publications

Non-holonomic geometric structures of rigid body system in Riemann-Cartan space

Non-holonomic geometric structures of rigid body system in Riemann-Cartan space

Connection Structures of Topological Singularity in Micromechanics from a Viewpoint of Generalized Finsler Space

Geometric Structures of Fractional Dynamical Systems in Non‐Riemannian Space: Applications to Mechanical and Electromechanical Systems

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