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Rund, H. (1966) The Hamilton-Jacobi Theory in the Calculus of Variations: Its Role in Mathematics and Physics. D. Van Nostrand Co., New York.

has been cited by the following article:

  • TITLE: Physics on the Adiabatically Changed Finslerian Manifold and Cosmology

    AUTHORS: Anton A. Lipovka

    KEYWORDS: Aharonov-Bohm Effect, Electrodynamic, Cosmology, Foundation of Quantum Theory

    JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.5 No.3, March 8, 2017

    ABSTRACT: In present paper, we confirm our previous result [5] that the Planck constant is adiabatic invariant of the electromagnetic field propagating on the adiabatically changed Finslerian manifold. Direct calculation of the Planck constant from the first principles with using of the actually measured cosmological parameters, gives value h = 6×10-27 (erg · s). We also confirm that Planck constant (and hence other fundamental constants which depend on h) is varied on time due to changing of geometry. As an example the variation of the fine structure constant is calculated. Its relative variation ((d/a/dt)/a) consist 1.0×10-18(1/s). We show that on the Finsler manifold characterized by adiabatically changed geometry, the classical free electromagnetic field is quantized geometrically, from the properties of the manifold in such manner that the adiabatic invariant of field is ET = 6×10-27 = h. Equations of electrodynamics on the Finslerian manifold are obtained. It is stressed that quantization naturally appears from these equations and is provoked by adiabatically changed geometry of the manifold. We consider in details two direct consequences of the equations: i) cosmological redshift of photons and ii) effects of Aharonov-Bohm, that immediately follow from obtained equations. It is shown that quantization of systems consists of electromagnetic field and baryonic component (like atoms) is obvious and has clear explanation.