We show that the equations of mathematical physics that describe real processes and include the equations of the conservation laws for energy, momentum, moment of momentum, and mass have a peculiar feature. The conservation law equations turn out to be inconsistent. This produces the integrability of the mathematical physics equations and the specific features of their solutions, namely the presence of double solutions: on a nonintegrable coordinate space and on integrable structures. As a consequence, we can describe evolutionary processes, such as the emergence of discrete structures and observable formations (waves, vortices, turbulent pulsations). Since double solutions are defined on different spatial objects, they can be obtained only in two coordinate systems and using simultaneously two methods of equation solving: numerical and analytical. Appropriate results have been derived in the framework of skew-symmetric forms.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. I. Petrova, “Role of skew-symmetric differential forms in mathematics,” (2010); http://arxiv.org/pdf/1007.4757.pdf.
R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer, New York (1982).
M.-A. Tonnelat, Les Principles de la Theorie Electromagnetique et la Relativite, Masson, Paris (1959).
L. Petrova, “Connection between functionals of the field-theory equations and state functionals of the mathematical physics equations,” Journal of Physics: Conference Series, 1051(012025), 1–8 (2018).
L. I. Petrova, “Double solutions of the Euler and Navier–Stokes equations. Process of origination the vorticity and turbulences,” Fluid Mechanics, (2017, Mar. 21), pp. 6–12.
J. F. Clark and M. Machesney, The Dynamics of Real Gases, Butterworths, London (1964).
L. Petrova, “Evolutionary relation of mathematical physics equations. Evolutionary relation as foundation of field theory. Interpretation of the Einstein equation,” Axioms, 10(2), No. 46, 1–10 (2021). https://doi.org/10.3390/axioms10020046.
L. I. Petrova, “Evolutionary relation of mathematical physics equations. Duality of conservation laws and their role in the processes of emergence of physical structures and formations,” Mathematics for Applications, 8, No. 9, 55–70 (2021).
L. Petrova, “Qualitative investigation of Hamiltonian systems by application of skew-symmetric differential forms,” Symmetry, 13, No. 1, 1–7 (2021).
L. I. Petrova, “Spontaneous emergence of physical structures and observable formations: fluctuations, waves, turbulent pulsations and so on,” Journal of Applied Mathematics and Physics, 4, No. 5, 864–870 (2016).
L. Petrova, “Discrete quantum transitions, duality: Emergence of physical structures and occurrence of observed formations (Hidden properties of mathematical physics equations),” Journal of Applied Mathematics and Physics, 8, No. 9, 1911–1921 (2020).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Dinamicheskogo Upravleniya, No. 70, 2022, pp. 29–51.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Petrova, L.I. Hidden Unique Possibilities of Mathematical Physics Equations (The Formalism of Skew-Symmetric Forms). Comput Math Model 33, 121–135 (2022). https://doi.org/10.1007/s10598-023-09562-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10598-023-09562-9