Based on El-Nabulsi dynamical model for a non-conservative system, the problem of perturbation to Noether symmetries and adiabatic invariants of a Birkhoffian system under the action of a small disturbance is proposed and studied. Firstly, the El-Nabulsi-Pfaff variational problem from extended exponentially fractional integral is presented and the El-Nabulsi-Birkhoff equations are established. Secondly, the definitions and the criterions criteria of the Noether symmetric transformations and quasisymmetric transformations of the Birkhoffian system are given, and the Noether theorems of the system are established, which reveal the inner relationship between the Noether symmetries and the conserved quantities. Thirdly, the perturbation of Noether symmetries under a small disturbance is studied, and corresponding adiabatic invariants are obtained. As special cases, the deductions in nonconservative Hamiltonian system and nonconservative Lagrangian system and standard Birkhoffian system are given. At the end of the paper, the case known as Hojman-Urrutia problem is discussed to investigate the Noether symmetries and the adiabatic invariants, the perturbation to Noether symmetries and the adiabatic invariants under El-Nabulsi dynamical model.

In 1927, Birkhoff [

The fractional variational problems combine the calculus of variations with the fractional calculus through inserting the fractional derivatives in the variational integrals. The fractional calculus of variations was introduced by Riewe in [

For a dynamical system, there exists intimate relation between the integrability of the system and the variations in its symmetry and invariant under the action of small disturbance [

The paper is organized in the following way. In Section

To study dynamics modeling of a nonconservative system, El-Nabulsi proposed the variational problem from extended exponentially fractional integral [

Find an extreme value problem of the integral functional

The above problem can be called the El-Nabulsi-Pfaff variational problem from extended exponentially fractional integral. Functional (

If functional (

Equations (

Introduce the infinitesimal transformations of

The Noether symmetry of the El-Nabulsi-Pfaff variational problem is an invariance of the El-Nabulsi-Pfaff action (

If the El-Nabulsi-Pfaff action (

According to Definition

If the infinitesimal transformations (

Condition (

Under the El-Nabulsi dynamics model, the Noether conserved quantity for a Birkhoffian system can be directly led by the Noether symmetry of the system; we have the following theorem.

For the Birkhoffian system (

The conserved quantities (

Theorem

For the classical Birkhoffian system (

Secondly, let us discuss a nonconservative Hamiltonian system under the El-Nabulsi dynamics model. Theorem

For the nonconservative Hamiltonian system (

Again, let us discuss a nonconservative Lagrange system under the El-Nabulsi dynamics model. Theorem

For the nonconservative Lagrangian system (

For the Birkhoffian system (

For the Birkhoffian system (

Taking the derivative of

According to (

Now we give several special cases of Theorem

For the classical Birkhoffian system (

Secondly, let us discuss a nonconservative Hamiltonian system under the El-Nabulsi dynamics model. Theorem

For the nonconservative Hamiltonian system (

Again, let us discuss a nonconservative Lagrange system under the El-Nabulsi dynamics model. Theorem

For the nonconservative Lagrange system (

As an example, let us consider the problem of Hojman-Urrutia [

According to the results given by Santilli [

Now let us study the adiabatic invariants of the system. Suppose that the system is disturbed by the following small forces of perturbation:

Symmetries are important and common properties of a dynamical system, and the perturbation to symmetries and corresponding invariants under the action of small forces of perturbation have an intimate relationship with the integrability of the system. An adiabatic invariant comes from the essential characteristics of the system, and it is more than just products of a Hamiltonian system. In this paper, we extended the El-Nabulsi dynamics model which is based on extended exponentially fractional integral to a Birkhoffian system, established the Noether theory of the Birkhoffian system under the El-Nabulsi model, and studied the problem of the perturbation to Noether symmetries and the adiabatic invariants for the disturbed Birkhoffian system. The methods and results of this paper are of universal significance. The corresponding consequences of classical Birkhoffian system are special cases of this paper, as well as the nonconservative Hamiltonian system and the nonconservative Lagrangian system under El-Nabulsi dynamics models. Further work could include the extension and the application of El-Nabulsi dynamics models to various types of constrained mechanical systems and the study of geometrical aspects and symplectic manifolds as well as some application to quantum mechanics and field theory.

The author declares that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (Grants nos. 10972151 and 11272227).