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Ruslan Sharipov

Publications and source records attributed to Ruslan Sharipov.

At least 91 records · Page 5Linked to original sources
arXiv2004

Gauge or not gauge?

The analogy of the nonlinear dislocation theory in crystals and the electromagnetism theory is studied. The nature of some quantities is discussed.

cond-mat.mtrl-sci↗
arXiv2004

Course of linear algebra and multidimensional geometry

This is a standard textbook for the course of linear algebra and multidimensional geometry as it was taught in 1991-1998 at Mathematical Department of Bashkir State University. Both coordinate and invariant approaches are used, but invariant approach is preferred.

math.HO↗
arXiv2004

Quick introduction to tensor analysis

I wrote this book in a "do-it-yourself" style so that I give only a draft of tensor theory, which includes formulating definitions and theorems and giving basic ideas and formulas. All other work such as proving consistence of definitions, deriving formulas, proving theorems or completing details to proofs is left to the reader in the form of numerous exercises. I hope that this style makes learning the subject really quick and more effective for understanding and memorizing.

math.HO↗
arXiv2003

Classical Electrodynamics and Theory of Relativity

This book is a manual for the course of electrodynamics and theory of relativity. It is recommended primarily for students of mathematical departments. This defines its style: I use elements of vectorial and tensorial analysis, differential geometry, and theory of distributions in it. Russian version of this book was published in 1997 under the approval by Methodic Commission of Mathematical Department of Bashkir State University.

physics.ed-ph↗
arXiv2003

Note on kinematics, dynamics, and thermodynamics of plastic glassy media

Unified geometric approach to describing kinematics of elastic and plastic deformations of continuous media is suggested. On the base of this approach we study mechanical deformations, viscous flow, and heat transport in glassy plastic media. As a result we derive appropriate differential equations for these phenomena in a form applicable to liquids, elastic solids, and to plastic solid materials as well.

cond-mat.mtrl-sci↗
arXiv2002

On the subset of normality equations describing generalized Legendre transformation

Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. They were first derived in Euclidean geometry, then in Riemannian geometry. Recently they were rederived in more general case, when geometry of manifold is given by generalized Legendre transformation. As appears, in this case some part of normality equations describe generalized Legendre transformation itself irrespective to that Newtonian dynamical system, for which others are written. In present paper this smaller part of normality equations is studied.

math.DG↗
arXiv2002

V-representation for normality equations in geometry of generalized Legendre transformation

Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. These equations were first derived in Euclidean geometry. Then very soon they were rederived in Riemannian and in Finslerian geometry. Recently I have found that normality equations can be derived in geometry given by classical and/or generalized Legendre transformation. However, in this case they appear to be written in p-representation, i. e. in terms of momentum covector and its components. The goal of present paper is to transform normality equations back to v-representation, which is more natural for Newtonian dynamical systems.

math.DG↗
arXiv2002

On the concept of normal shift in non-metric geometry

Theory of Newtonian dynamical systems admitting normal shift of hypersurfaces was first developed for the case of Riemannian manifolds. Recently it was generalized for manifolds geometric equipment of which is given by some regular Lagrangian or, equivalently, by some regular Hamiltonian dynamical system. In present paper we consider further generalization of this theory for the case, when geometry of manifold is given by generalized Legendre transformation.

math.DG↗
arXiv2002

Minimal tori in five-dimensional sphere in $C^3$

Special class of surfaces in five-dimensional sphere in $C^3$ is considered. Immersion equations for minimal tori of that class are shown to be reducible to the equation $u_{z\bar z}=e^u-e^{-2u}$ which is integrable by means of inverse scattering method. Finite-gap minimal tori are constructed.

math.DG↗
arXiv2002

Comparative analysis for pair of dynamical systems, one of which is Lagrangian

It is known that some equations of differential geometry are derived from variational principle in form of Euler-Lagrange equations. The equations of geodesic flow in Riemannian geometry is an example. Conversely, having Lagrangian dynamical system in a manifold, one can consider it as geometric equipment of this manifold. Then properties of other dynamical systems can be studied relatively as compared to this Lagrangian one. This gives fruitful analogies for generalization. In present paper theory of normal shift of hypersurfaces is generalized from Riemannian geometry to the geometry determined by Lagrangian dynamical system. Both weak and additional normality equations for this case are derived.

math.DG↗
arXiv2001

Normal shift in general Lagrangian dynamics

It is well known that Lagrangian dynamical systems naturally arise in describing wave front dynamics in the limit of short waves (which is called pseudoclassical limit or limit of geometrical optics). Wave fronts are the surfaces of constant phase, their points move along lines which are called rays. In non-homogeneous anisotropic media rays are not straight lines. Their shape is determined by modified Lagrange equations. An important observation is that for most usual cases propagating wave fronts are perpendicular to rays in the sense of some Riemannian metric. This happens when Lagrange function is quadratic with respect to components of velocity vector. The goal of paper is to study how this property transforms for the case of general (non-quadratic) Lagrange function.

math.DG↗
arXiv2001

Dynamical systems admitting normal shift and wave equations

High frequency limit for most of wave phenomena is known as quasiclassical limit or ray optics limit. Propagation of waves in this limit is described in terms of wave fronts and rays. Wave front is a surface of constant phase whose points are moving along rays. As it appears, their motion can be described by Hamilton equations being special case for Newton's equations. In simplest cases (e. g. light propagation in isotropic non-homogeneous refracting media) the dynamics of points preserves orthogonality of wave front and rays. This property was generalized in the theory of dynamical systems admitting normal shift of hypersurfaces. In present paper inverse problem is studied, i. e. the problem of reconstructing wave equation corresponding wave front dynamics for which is described by Newtonian dynamical system admitting normal shift of hypersurfaces in Riemannian manifold.

math.DG↗
arXiv2001

A note on Newtonian, Lagrangian and Hamiltonian dynamical systems in Riemannian manifolds

Newtonian, Lagrangian, and Hamiltonian dynamical systems are well formalized mathematically. They give rise to geometric structures describing motion of a point in smooth manifolds. Riemannian metric is a different geometric structure formalizing concepts of length and angle. The interplay of Riemannian metric and its metric connection with mechanical structures produces some features which are absent in the case of general (non-Riemannian) manifolds. The aim of present paper is to discuss these features and develop special language for describing Newtonian, Lagrangian, and Hamiltonian dynamical systems in Riemannian manifolds.

math.DG↗
arXiv2000

On the solutions of weak normality equations in multidimensional case

The system of weak normality equations constitutes a part in the complete system of normality equations. Solutions of each of these two systems of equations are associated with some definite classes of Newtonian dynamical systems in Riemannian manifolds. In this paper for the case of simplest flat Riemannian manifold $M=\Bbb R^n$ with $n\geq 3$ we show that there exist solutions of weak normality equations that do not solve complete system of normality equations in whole. Hence associated classes of Newtonian dynamical systems do not coincide with each other.

math.DG↗