Fractional symmetrical perturbation method of finding adiabatic invariants of disturbed dynamical systems

For a disturbed dynamical system that can be transformed into fractional Birkhoffian representation with all dynamical information, under a more general fractional infinitesimal transformation of Lie group with high time extension and fractional extension, a new kind of fractional Mei symmetrical perturbation method which is most universal significance is presented and it is found that, using the new method, we can find a new kind of non-Noether adiabatic invariant; as the special cases of new method, we, respectively, reveal an autonomous disturbed fractional Birkhoffian system possesses more adiabatic invariants, a new kind of non-Noether exact invariant directly led by fractional Mei symmetry and a new kind of non-Noether exact and adiabatic invariant of integer Birkhoffian systems. Also, as the new method's applications to nonlinear dynamical problems, we, respectively, explore the symmetrical perturbation and adiabatic invariant of a disturbed fractional general relativistic Buchdahl model and a disturbed fractional Emden model. It is worth pointing out that, for a disturbed dynamical system, this work reveals intrinsic relation between the fractional symmetrical perturbation and the adiabatic invariant, and provides a general method for finding adiabatic invariants of an actual disturbed fractional dynamical system that is related to science and engineering.