Yakov Itin

Finsler geometry usually describes an extension of Riemannian geometry into a direction-dependent geometric structure. Historically, the well-known Riemann quartic length element example served as the inspiration for this construction. Surprisingly, the same quartic expression emerges as a fundamental dispersion relationcovariant Fresnel equationin solid-state electrodynamics. As a result, it is possible to conceive of the Riemann quartic length expression as a mathematical representation of a well-known physical phenomenon. This paper provides a number of Riemann quartic examples that show Finsler geometry to be overly constrictive for many applications, even when the signature space is positive definite in the Euclidean sense. The strong axioms of Finsler geometry are broken down on many more singular hypersurfaces for the spaces having an indefinite (Minkowski) signature. We suggest a more flexible definition of a Finsler structure that only has to hold for open subsets of a manifold’s tangent bundle. We demonstrate the distinctive singular hypersurfaces connected to the Riemann quartic and discuss the potential physics explanations for them. As an illustration of the pseudo-Riemannian quartic, we took into consideration the dispersion relation that appears in electromagnetic wave propagation in uniaxial crystal. Our analysis suggests that the signature of the Finsler measure may be altered for large anisotropy factors.

The Cauchy relations distinguish between rari- and multi-constant linear elasticity theories. These relations are treated in this paper in a form that is invariant under two groups of transformations: indices permutation and general linear transformations of the basis. The irreducible decomposition induced by the permutation group is outlined. The Cauchy relations are then formulated as a requirement of nullification of an invariant subspace. A successive decomposition under rotation group allows to define the partial Cauchy relations and two types of elastic materials. We explore several applications of the full and partial Cauchy relations in physics of materials. The structure's deviation from the basic physical assumptions of Cauchy's model is defined in an invariant form. The Cauchy and non-Cauchy contributions to Hooke's law and elasticity energy are explained. We identify wave velocities and polarization vectors that are independent of the non-Cauchy part for acoustic wave propagation. Several bounds are derived for the elasticity invariant parameters.

Several definitions of the energy–momentum tensor in classical field models on differential manifolds are examined in this work. We specifically talk about the connection between - Hilbert’s symmetric and Noether’s asymmetric energy–momentum tensors. We consider a matter field Lagrangian on a manifold endowed with a teleparallel (coframe) structure. The associated 3-form of coframe conserved current can be viewed as a link between Noether’s and Hilbert’s energy–momentum tensors. In these circumstances, we are able to show the complete equivalence of Hilbert’s and Noether’s currents for the matter field.

Finsler’s geometry usually describes an extension of Riemmann’s geometry into a direction-dependent geometric structure. Historically, the well-known Riemann’s quartic length element example served as the inspiration for this construction. Surprisingly, the covariant Fresnel equation—a fundamental dispersion relation in solid-state electrodynamics—emerges as the exact same quartic expression. As a result, Riemann’s quartic length expression can be regarded of as a mathematical representation of a well-known physical phenomenon. In this study, we offer numerous Riemann’s quartic examples that show Finsler’s geometry, even in the situation of a positive definite Euclidean signature space, is too restrictive for many applications. The strong axioms of Finsler’s geometry are violated in a substantially greater number of distinctive subsets for the spaces having an indefinite (Minkowski) signature. We suggest a weaker characterization of Finsler’s structure based on explicitly calculated two-dimensional examples. In tangential vector space, this concept permits singular subsets. Only open subsets of a manifold’s tangent bundle are required to satisfy the strong axioms of Finsler’s geometry. We demonstrate the distinctive unique subsets of the Riemann’s quartic in two dimensions and briefly discuss their possible physical origins.

Third-order tensors are widely used as a mathematical tool for modeling the physical properties of media in solid-state physics. In most cases, they arise as constitutive tensors of proportionality between basic physical quantities. The constitutive tensor can be considered the complete set of physical parameters of a medium. The algebraic features of the constitutive tensor can be used as a tool for proper identification of natural materials, such as crystals, and for designing artificial nanomaterials with prescribed properties. In this paper, we study the algebraic properties of a general asymmetric third-order tensor relative to its invariant decomposition. In correspondence with different groups acting on the basic vector space, we present the hierarchy of different types of tensor decomposition into invariant subtensors. In particular, we discuss the problem of non-uniqueness and reducibility of high-order tensor decomposition. For a general asymmetric third-order tensor, these features are described explicitly. In the case of special tensors with a prescribed symmetry, the decomposition is demonstrated to be irreducible and unique. We present the explicit results for two physically interesting models: the piezoelectric tensor as an example of pair symmetry and the Hall tensor as an example of pair skew-symmetry.

In linear elasticity, a fourth-order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties elastic constants of a material. The main goal of the current work is to derive a compact matrix representation of the elasticity tensor that correlates with its intrinsic algebraic properties. Such representation can be useful in design of artificial materials. Owing to Voigt, the elasticity tensor is conventionally represented by a (6 × 6) symmetric matrix. In this paper, we construct two alternative matrix representations that conform with the irreducible decomposition of the elasticity tensor. The 3 × 7 matrix representation is in correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by two scalars and three 3 × 3 matrices is suitable to describe the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation.

The premetric formalism is an alternative representation of a classical field theory in which the field equations are formulated without the spacetime metric. Only the constitutive relations between the basic field variables can involve the metric of the underlying manifold. In this paper, we present a brief pedagogical review of the premetric formalism in mechanics, electromagnetism, and gravity.