Petrova L.I. A new mathematical formalism:
Skew-symmetric differential forms in mathematics, mathematical physics and field theory

Contents

Introduction

Skew-symmetric differential forms possess unique capabilities that manifest themselves in various branches of mathematics and mathematical physics. The invariant properties of closed exterior skew-symmetric differential forms lie at the basis of practically all invariant mathematical and physical formalisms. The closed exterior forms, which properties correspond to conservation laws, explicitly or implicitly manifest themselves essentially in all formalisms of field theory. In present paper, firstly, the role of closed exterior skew-symmetric differential forms in mathematics, mathematical physics and field theory is illustrated, and, secondly, it is shown that there exist skew-symmetric differential forms that generate closed exterior differential forms. These skew-symmetric forms are derived from differential equations and possess evolutionary properties (whose existence has been established by the author)проверьте правильность внесения правки. The process of extracting closed exterior forms from evolutionary forms enables one to describe discrete transitions, quantum jumps, the generation of various structures, origination of such formations as waves, vortices and so on. In none of other mathematical formalisms such proceses can be described since their description includes degenerate transformations and transitions from nonintegrable manifolds to integrable.

The unique role played by skew-symmetric differential forms in mathematics and mathematical physics, firstly, is due to the fact that they are differentials and differential expressions, and, therefore, they are suitable for describing invariants and invariant structures. And, secondly, skew-symmetric forms have a structure that combines objects of different nature, namely, the algebraic nature for the form coefficients and the geometric nature of the base. The interaction between these objects enables to describe evolutionary processes, discrete transitions, the realization of conjugacy of operators or objects, the emergence of structures and observable formations, and so.

The present paper is a collection of separate completed articles (the repeated description of the properties of skew-symmetric forms is sometimes possible).

Two first articles (Chapters 1, 2) contain general principles related to the properties of skew-symmertic forms and specific features of their mathematical apparatus. The properties of closed exterior forms (invariance, conjugacy, duality, and so on), which manifest themselves in various branches of mathematics such as algebra, differential geometry, the theory of functions of complex variables, tensor analysis, differential and integral calculus, are demonstrated. This discloses an internal connection between various mathematical formalisms. It has been shown that there exist skew-symmetric forms that are obtained from differential equations describing any processes. Such skew-symmetric forms, which are evolutionary ones and are based on a nontraditional mathematical apparatus (such as nonidentical relations, degenerate transformations, the transition from nonintegrable manifold to integrable one, and others), possess an unique property, namely, they generate closed exterior forms whose invariant properties lie at the basis of many mathematical formalisms. The process of obtaining closed exterior forms, which discloses a realization of conjugacy and an emergence of invariant structures, has been shown.

In the second article (Chapter 2) it has been shown that the properties of skew-symmetric forms correspond to conservation laws. Thus, closed exterior forms account for the properties of coservation laws for physical fields, whereas evolutionary skew-symmetric forms disclose the properties of conservation laws for energy, momentum, angular momentum, and mass, which are conservation laws for material systems such as thermodynamic, gas-dynamic, cosmological and other ones. It has been shown that from conservation laws for material systems one derives the evolutionary skew-symmetric differential forms from which the closed exterir forms corresponding to conservation laws for physical fields are obtained. This discloses a relation between physical fields and material systems and demonstrates a relation between the field-theory equations and the equations of conservation laws for material systems. Such a dependence discloses the properties and peculiarities of the field-theory equations.

In Chapters (3–10) of the collected articles the below listed articles, in which various problems and specific features of mathematical physics are studied with the help of skew-symmetric forms, are presented.

Conservation laws. Generation of physical structures. Principles of field theories.

The effect of noncommutativity of the conservation laws on the development of thermodynamic and gas-dynamic instability.

The development of gas dynamical instability. Mechanism of origination of vorticity and turbulence.

Qualitative investigation of Hamiltonian systems by application of skew-symmetric differential forms.

Electromagnetic field.

Formation of physical fields and manifolds.

Skew-symmetric forms: On integrability of the equations of mathematical physics.

Foundations of general field theory.

Specific features of differential equations of mathematical physics.

Conclusion

It is shown that the skew-symmetric differential forms play an unique role in mathematics and mathematical physics.

The invariant properties of exterior skew-symmetric differential forms lie at the basis of practically all invariant mathematical and physical formalisms. The closed exterior forms, which properties correspond to conservation laws, explicitly or implicitly manifest themselves essentially in all formalisms of field theory.

The unique role of evolutionary skew-symmetric differential forms, which were outlined in present work, relates to the fact that they generate the closed exterior forms possessing invariant properties. This is of fundamental importance for mathematical physics and field theories. The process of generation of closed exterior forms disclose the mechanism of evolutionary processes in material media, emergence of physical structures formatting physical fields and the determinacy of these processes.

Due their properties and peculiarities the closed exterior forms and evolutionary forms enable one to see the internal connection between various branches of mathematics.

Many foundations of the mathematical apparatus of evolutionary forms may occur to be of great importance for development of mathematics and mathematical physics. The nonidentical relations, degenerate transformations, transitions from nonidentical relations to identical ones, transitions from one frame of reference to another (nonequivalent) frame, the generation of closed inexact exterior forms and invariant structures, formatting fields and manifolds, the transitions between closed inexact exterior differential forms and exact forms and other phenomena may find many applications in such branches of mathematics as the qualitative theory of differential and integral equations, differential geometry and topology, the theory of functions, the theory of series, the theory of numbers, and others.

The evolutionary skew-symmetric differential forms may become a new branch in mathematics. They possess the possibilities that are contained in none of mathematical formalisms.

About avtor

Candidate of Physical and Mathematical Sciences, Senior researcher at The Moscow State University, Department of Computational Mathematics and Cybernetics, the Chair of Mathematial Physics. The field of scientific interests: Mechanics of Continuous Media, Differential Equations, Differential Geometry, Field Theory, etc. The main contributions to the problems of turbulence, stability and integrability of differential equations. In the works the properties of skew-symmetric differential forms, which lie at the basis of many formalisms of mathematics and mathetical physics, are used.

El autor

Lyudmila Ivanovna PETROVA

PhD in Mathematical Physics, Senior Fellow of the Department of Computational Mathematics and Cybernetics at MSU, Honoured Fellow of Moscow State University. Fields of scientific interest: Mechanics of Continuous Media, Differential Equations, Differential Geometry, Field Theory, etc. Has made an important contribution to the turbulence problem, and the problem of stability and integrability of differential equations. In her research, she widely applies the formalism of skew-symmetric differential forms. She is the author of more than 100 publications.