This paper introduces a new evolutionary system which is uniquely suitable for the description of nonconservative systems in field theories, including quantum mechanics, but is not limited to it only. This paper also introduces a new exact method of solution for such nonconservative systems. These are significant contributions because the vast majority of nonconservative systems with several independent variables do not have self-adjoint Frechet derivatives and because of that can not benefit from the exact methods of the classical calculus of variations. The new evolutionary system is rigorously mathematically derived and the new method for solution is mathematically proved to be applicable to systems of PDEs of second order for nonconservative process. As examples of applications, the method is applied to several nonconservative systems: the propagation of electromagnetic fields in a conductive medium, the nonlinear Schrodinger equation with electromagnetic interactions, and others.
References
[1]
Georgieva, B., Guenther, R. and Bodurov, T. (2003) Generalized Variational Principle of Herglotz for Several Independent Variables. First Noether-Type Theorem. JournalofMathematicalPhysics, 44, 3911-3927. https://doi.org/10.1063/1.1597419
[2]
Georgieva, B. and Guenther, R.B. (2002) First Noether-Type Theorem for the Generalized Variational Principle of Herglotz. TopologicalMethodsinNonlinearAnalysis, 20, 261-273. https://doi.org/10.12775/tmna.2002.036
[3]
Georgieva, B. and Guenther, R.B. (2005) Second Noether-Type Theorem for the Generalized Variational Principle of Herglotz. TopologicalMethodsinNonlinearAnalysis, 26, 307-314. https://doi.org/10.12775/tmna.2005.034
[4]
Georgieva, B. (2011) Symmetries of the Generalized Variational Functional of Herglotz for Several Independent Variables. ZeitschriftfürAnalysisundihreAnwendungen, 30, 253-268. https://doi.org/10.4171/zaa/1434
[5]
Georgieva, B. (2010) Symmetries of the Herglotz Variational Principle in the Case of One Independent Variable. Annual of Sofia University “St. Kliment Ohridski”, 100, 113-122.
[6]
Georgieva, B. and Bodurov, T. (2013) Identities from Infinite-Dimensional Symmetries of Herglotz Variational Functional. JournalofMathematicalPhysics, 54, Article ID: 062901. https://doi.org/10.1063/1.4807728
[7]
Guenther, R., Gottsch, A. and Guenther, C. (1996) The Herglotz Lectures on Contact Trans-Formations and Hamiltonian Systems. Juliusz Center for Nonlinear Studies.
[8]
Herglotz, G. (1979) Gesammelte Schriften, Göttingen. Vandenhoeck & Ruprecht.
[9]
Herglotz, G. (1930) Berührungstransformationen. Lectures at the University of Göttingen.
[10]
Lie, S. (1927) Gesammelte Abhandlungen, Vol. 6. Benedictus Gotthelf Teubner, 649-663.
[11]
Lie, S. (1897) Die Theorie der Integralinvarianten ist ein Korollar der Theorie der Differentialinvarianten. Berichte, 49, 342-357.
[12]
Furta, K., Sano, A. and Atherton, D. (1988) State Variable Methods in Automatic Control. John Wiley, New York.
[13]
Caratheodory, C. (1989) Calculus of Variations and Partial Differential Equations of the First Order. 2nd Edition, Chelsea Publishing.
[14]
Mrugala, R. (1980) Contact Transformations and Brackets in Classical Thermodynamics. Acta Physica Polonica, A58, 19-29.
[15]
Georgieva, B. (2010) The Variational Principle of Herglotz and Related Results. Proceedings of the Twelfth International Conference on Geometry, Integrability and Quantization, Varna, 4-9 June 2010, 214-225.
[16]
Bodurov, T. and Georgieva, B. (2000) Complex Hamiltonian Evolution Equations. International Journal of Differential Equations and Applications, 1A, 15-22.
[17]
Eisenhart, L. (1933) Continuous Groups of Transformations. Princeton University Press.
[18]
Killing, W. (1892) Ueber die Grundlagen der Geometrie. CRLL, 1892, 121-186. https://doi.org/10.1515/crll.1892.109.121
[19]
Logan, J. (1977) Invariant Variational Principles. Academic Press.
[20]
Noether, E. (1918) Invariante Variationsprobleme. Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 235-257.
[21]
Noether, E. (1918) Invarianten beliebiger Differentialausdrücke. Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 37-44.
