Some physical aspects induced by the internal variable in the theory of fields in Finsler spaces

Ikeda, Satoshi

doi:10.1007/bf02721325

Cited by 7 publications

(11 citation statements)

References 14 publications

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“…A non-expert in special directions of differential geometry and geometric mechanics, may not know that beginning E. Cartan (1935) [15] various models of Finsler geometry were developed alternatively by using metric compatible connections which resulted in generalizations to the geometry of Lagrange and Hamilton mechanics and their higher order extensions. Such works and monographs were published by prominent schools and authors on Finsler geometry and generalizations from Romania and Japan [37,38,34,35,39,36,25,26,24,57,84,33,41,42,44,45,10,11] following approaches quite different from the geometry of symplectic mechanics and generalizations [30,31,32,29]. As a matter of principle, all geometric constructions with the Chern and/or simplectic connections can de redefined equivalently for metric compatible geometries, but the philosophy, aims, mathematical formalism and physical consequences are very different for different approaches and the particle physics researches usually are not familiar with such results.…”

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confidence: 99%

Finsler and Lagrange Geometries in Einstein and String Gravity

Vacaru

2008

Int. J. Geom. Methods Mod. Phys.

450

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We review the current status of Finsler-Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kähler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of "orthodox" physicists.Although the bulk of former models of Finsler-Lagrange spaces where elaborated on tangent bundles, the surprising result advocated in our works is that such locally anisotropic structures can be modelled equivalently on Riemann-Cartan spaces, even as exact solutions in Einstein and/or string gravity, if nonholonomic distributions and moving frames of references are introduced into consideration.We also propose a canonical scheme when geometrical objects on a (pseudo) Riemannian space are nonholonomically deformed into generalized Lagrange, or Finsler, configurations on the same manifold. Such canonical transforms are defined by the coefficients of a prime metric and generate target spaces as Lagrange structures, their models of almost Hermitian/ Kähler, or nonholonomic Riemann spaces.Finally, we consider some classes of exact solutions in string and Einstein gravity modelling Lagrange-Finsler structures with solitonic pp-waves and speculate on their physical meaning.

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confidence: 99%

Finsler and Lagrange Geometries in Einstein and String Gravity

Vacaru

2008

Int. J. Geom. Methods Mod. Phys.

450

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“…From a viewpoint of Finslerian field theory, a "fluctuation," "nonlocality," or "anisotropy" is expressed by a variable attached to each point. [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.…”

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

confidence: 99%

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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“…A tangent bundle over M is defined by

T M

, and the local coordinate system on

T M

(x^{α}, y^{α})

. According to the Finsler field theory, the vector

y^{α}

(x^{α}, y^{α})

T M

is regarded as a “fluctuation”, “anisotropy” or “non‐locality” at some microscopic field. The non‐local property of

y^{α}

geometrically effects on a macroscopic field expressed by

(x^{α})

through a reduction process of non‐Riemann space as follows.…”

Section: Differential Geometry Of Fractional‐order Differential Equatmentioning

confidence: 99%

“…Then, the macroscopic field

(x^{α})

non‐localized by the microscopic fluctuation

y^{α} false(x false)

gives the non‐Riemannian structure. Such reduction process

R_{2 (n + 1)} \to F_{n + 1} \to {\overset{R}{true}}_{n + 1}

is represented as osculating conditions from the Finsler space …”

Section: Differential Geometry Of Fractional‐order Differential Equatmentioning

confidence: 99%

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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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Some physical aspects induced by the internal variable in the theory of fields in Finsler spaces

Cited by 7 publications

References 14 publications

Finsler and Lagrange Geometries in Einstein and String Gravity

Finsler and Lagrange Geometries in Einstein and String Gravity

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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