Mir Books

Advances in Science and Technology in the USSR
Mathematics and Mechanics Series

This is a collection of articles by Soviet scientists on current issues of building macroscopic models of matter and fields. Based on thermodynamics concepts the papers develop general variational techniques of modeling material continuous media and fields allowing for their interactions in reversible and irreversible processes. The book is intended for researchers, engi­neers, graduate and postgraduate students interested in the mechanics of continuous media.

Translated from the Russian by Eugene Yankovsky

You can get the book here and here.

Twitter: @MirTitles
Mastodon: @mirtitles@mastodon.world
Mastodon: @mirtitles@mastodon.social
Bluesky: mirtitles.bsky.social

Contents
Preface, L. I. Sedov 7
A Thermodynamic Approach to the Basic Variational Equation for Building Models of Continuous Media, L. I. Sedov 19
Applying the Basic Variational Equation for Building Models of Matter and Fields, L. I. Sedov 43

Introduction 43
Definitions 43
Variations of Tensors for Which Scalar Invariants Retain Their Form 46
Special Types of Tensor Components Qlj 48
Defining Variations and Their Interrelationship in the Comoving and the Observer’s Reference Frame 50
Auxiliary Formulas for Variations 55
Given Scalar and Tensor Parameters Characterizing Models of Material Media and Fields 56
The Determining Parameters in the Characteristics of a Continuous Medium as a Whole and the Characteristics of Individual World Lines 60
The Basic Variational Equation and Identities Following from the Scalar Nature of the Lagrangian Density 62
The Euler Equations for the Basic Variational Equation (2.8.1) 66
The Conditions at Strong Discontinuities 71
On Models of Fluids 74
An Elastic-Body Model 79
Constructing Models of Fields 81
A Model of Interacting Material Medium and Electromagnetic Field 83
Examples 90
Transition from Relativistic to Newtonian Mechanics in the Presence of Irreversible Processes, L. T. Chernyi 98
The Basic Vibrational Equation 98
The Euler Equations and Conditions on Discontinuities 102
Transition to Newtonian Mechanics 106
Irreversible Processes 108
Conclusion 114
Models of Ferromagnetic Continuous Media with Magnetic Hysteresis, L. T. Chernyi 116

Introduction 116
The Determining Parameters 118
The Variational Principle and the Main Equations 121
A Phenomenological Theory of Irreversible Processes 126
Some Corollaries of the General Theory 130
Examples of Models of Magnetizable Media 137
Magnetizable and Polarizable Media with Microstructure, V. A. Zhelnorovich 141

The Determining Parameters of Magnetizable and Polarizable Media with Microstructure 141
Relaxation Models of Magnetizable and Polarizable Media Without Microstructure 150
Models of Magnetizable Liquids with Intrinsic Moment of Momentum 156
Couette Flow of an Incompressible Viscous Magnetizable Liquid 156
Poiseuille Flow in Cylindrical Channel 157
Magnetoacoustic Waves in Magnetizable Liquids 160
On Exact Solutions for Interacting Gravitational and Electromagnetic Fields, G. A. Alekseev 168

Introduction 168
The Einstein-Maxwell Equations in Matrix Form 169
Building the Associated Linear System and the Reduction Conditions 172
Soliton Solutions of the Einstein-Maxwell Equations 176
One-Soliton Solutions with Minkowski’s Space-Time as Background 180
Interaction of Solitons with a Uniform Electromagnetic Field 184
Neutrino Fields in General Relativity, N. R. Sibgatullin 187

Introduction 187
Canonical Equations of Neutrino Fields and Waves 188
On the Infinite Dimensional Algebra and the Lie Group of Neutrino Vacuum Equations 199
Exact Solutions of Neutrino Vacuum Equations 208
Rotation of the Polarization Vector of Gravitational Waves in a Burst of Neutrino Radiation 220
Tensor Representation of Spinor Fields, V. A. Zhelnorovich 224

Introduction 224
Dirac Matrices 224
The Spinor Representation of the Lorentz Group 226
Spinors in Four-Dimensional Pseudo-Euclidean Vector Space 231
Conjugate Spinors 233
The Relation Between Even-Rank Spinors and Tensors 234
The Relation Between First-Rank Spinors and Systems of Complex Tensors 234
Real-Valued Tensors Determined by a Spinor 238
Rotations in Four-Dimensional Space and Spinors 240
Invariant Spinor Subspaces 243
Spinors in Three-Dimensional Euclidean Space 244
Tensor Representation of Spinors in Three-Dimensional Euclidean Space 246
Rotations in Three-Dimensional Space and Spinors 248
Tensor Representation of Differential Spinor Equations in the Minkowski Space 250
Some Solutions of Differential Equations for Relativistic Models of Magnetizable Fluids with Intrinsic Angular Momentum in an Electromagnetic Field 254
Index 26