Adiabatic invariants for disturbed fractional Hamiltonian system in terms of Herglotz differential variational principle

Abstract

Based on the differential variational principle of Herglotz type, we reveal the internal relation between the perturbation and the adiabatic invariants for fractional Hamiltonian system with combined Caputo derivatives. First, based on the Herglotz variational problem, the Herglotz differential variational principle for fractional Hamiltonian systems is derived, and the fractional Hamilton canonical equations are given. Second, by introducing the infinitesimals, the transformation of the invariance condition of the Herglotz differential variational principle is established and an exact invariant of the system is derived. Third, the adiabatic invariants of Herglotz type for the disturbed fractional Hamiltonian system is obtained. Finally, the fractional linear damped oscillator of Herglotz type is discussed as an example to demonstrate the results.