Perturbation to Noether symmetries and adiabatic invariants for nonconservative dynamic systems

The problem of perturbation to Noether symmetry and adiabatic invariant for a nonconservative dynamic system is studied under a dynamic model presented by El-Nabulsi. First of all, the fractional action-like variational problem proposed by El-Nabulsi under the framework of the fractional calculus and based on the definition of the Riemann-Liouville fractional integral is introduced, and the Euler-Lagrange equations of the nonconservative system are given. Secondly, the definition and criterion of the Noether quasi-symmetric transformation are given, the relationship between the Noether symmetry and the invariant is established, and the exact invariant is obtained. Finally, the perturbation to the Noether symmetry of the system after the action of a small disturbance and corresponding adiabatic invariant are proposed and studied, the conditions for the existence of adiabatic invariant and the formulation are given. An example is given to illustrate the application of results.