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Fractional oscillator: phenomenological and theoretical issues. (English) Zbl 1179.26029

Summary: The fractional derivative is a convenient tool to model physical processes in the complex. media. A general trend is to consider fractional generalization of an equation of physics and study its consequences. For the case of a fractional oscillator we provide justifications for this procedure.
In a media with low-level of fractionality the order of the fractional derivative is close to a positive integer. And one can use perturbation theory to treat the deviation from the integer case. Using this method we study fractional oscillator in a media with low-level fractionality. We find expressions for the position, momenta, energy and the intrinsic damping force for this system. We show that the energy is a monotonously decreasing function of time.
We propose a new model Hamiltonian for this system. And we make some remarks on non-casual fractional oscillator as well.

MSC:

26A33 Fractional derivatives and integrals
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