THE FRACTIONAL CALCULUS OF VARIATIONS FROM EXTENDED ERDELYI-KOBER OPERATOR

Fractional calculus of variations (FCV) has recently attracted considerable attention as it is deeply related to the fractional quantization procedure. In this work, the FCV from extended Erdelyi-Kober fractional integral is constructed. Our main goal is to exhibit a general treatment for dissipative systems, in particular the harmonic oscillator (HO) that has time-dependent mass and time-dependent frequency. The general linear equation of damped Erdelyi-Kober harmonic oscillator is constructed from which a time-dependent mass generalized law was derived exhibiting different types of behavior. This relatively new time-dependent mass law permits us to point out several possible cases simultaneously in contrast to many models discussed in the literature and without making use of any types of fractional derivatives. Some results on Hamiltonian part, namely Hamilton equations for the damped HO were obtained and discussed in detail.