Abstract
Problem statement: This study is a contribution to the general program of describing complex dynamical systems using the tool of fractional calculus of variations. Approach: Following our previous work, fractional quantum field theory based on the fractional actionlike variational approach supported by Saxena-Kumbhat fractional integrals functionals, fractional derivative of order (α, β) and dynamical fractional exponent on multi-fractal sets is considered. Results: In order to build the required theory, we introduce the Saxena-Kumbhat hypergeometric fractional functionals determined on the functions on a multifractal sets. We prove, developing the corresponding fractional calculus of variations, that a hierarchy of differential equations can be developed from the extended fractional Lagrangian formalism. Besides, a generalization of the resulting Hamiltonian and Lagrangian dynamics on the complex plane is addressed. Conclusion: The new complexified dynamics guides to a new dynamics which may differ totally from the classical mechanics cardinally and may bring new appealing consequences. Some additional interesting results are explored and discussed in some details.
Highlights
Definition of the fractional order derivative and integral are not unique where several definitions exist, e.g., Fractional dynamics is the study of complex dynamical systems that can be cast in terms of solutions to fractional differential equations to which the fractional calculus can be correctly applied
Fractional calculus has been studied for over 300 years, it has been regarded principally as a mathematical curiosity until about 1992, where fractional dynamical equations were pretty much restricted to the realm of mathematics and Grunwald-Letnikov, Caputo, Weyl, Feller, ErdelyiKober, Riesz, Saxena, Kumbhat, Kiryakova, Srivastava and Raina
A theme of present strong research concerns the study fractional Lagrangian and Hamiltonian dynamics systems based on the fractional problems of the Calculus Of Variations (COV) (Ali et al, 2009)
Summary
Definition of the fractional order derivative and integral are not unique where several definitions exist, e.g., Fractional dynamics is the study of complex dynamical systems that can be cast in terms of solutions to fractional differential equations to which the fractional calculus can be correctly applied. In order to stimulate more interest in the subject and to show its utility, this contribution is devoted to a new generalization of the FALVA by introducing mainly inside the fractional action-like integral the Saxena-Kumbhat hypergeometric fractional operator, augmented by a fractional derivative and a dynamical fractional exponent defined on multifractal time and space sets, in particular when the fractional dimensions of time and space are dynamical, e.g., di = 1 + ε(xi ). This notion is based on the Mandelbrot ideas of the fractal geometry of nature and is expected to work on a small multifractal intervals set Si which is build from multifractal subsets S(xi ) (Al-Daoud, 2008; Kobolev, 2000). One expects a complex generalization of Hamiltonian dynamics which certainly will guides to a new dynamics which may differ completely from the classical mechanics cardinally and may bring interesting consequences, e.g. quantum field theory, control theory and we expect that exotic solutions will be physical
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