Perturbation To Noether Symmetries And Adiabatic Invariants Based On El-Nabulsi Dynamics Model

Aimed at nonconservative dynamics model, El-Nabulsi proposed a new nonconservative dynamics based on the definition of Riemann-Liouville fractional integral under the fraction calculus framework in 2005, we called El-Nabulsi fractional model. In this method, the fractional integral about time only need one parameter, the resulting fractional Euler-Lagrange equations are similar to the classical ones and no fractional derivatives appearing, the novelty of El-Nabulsi dynamics model is the presence of the fractional generalized external force acting on the system. This paper studying the perturbation to Noether symmetries and the adiabatic invariants for constrain mechanic systems(nonconservative systems in phase space, Birkhoff systems and nonholonomic systems), which are based on one fractional models presented by El-Nabulsi.In the first part, the El-Nabulsi fractional models for nonconservative systems in phase space is established, which extended Riemann-Liouville fractional integral definition, and the exact invariant led directly by the Noether symmetry is obtained; then Noether symmetry that after the action of small disturbance lead to the adiabatic invariants are studied, and we proved the conditions and its formulation for one-order to high-order adiabatic invariants.In the second part, the El-Nabulsi fractional models for Birkhoff systems is established, which extended Riemann-Liouville fractional integral definition, and the exact invariant led directly by the Noether symmetry is obtained; then Noether symmetry that after the action of small disturbance lead to the adiabatic invariants are studied, and we proved the conditions and its formulation for one-order to high-order adiabatic invariants.In the third part, the El-Nabulsi fractional models for linear constrain nonholonomic systems is established, which extended Riemann-Liouville fractional integral definition, and the exact invariant led directly by the Noether symmetry is given; then Noether symmetry that after the action of small disturbance lead to the adiabatic invariants are studied, and we proved the conditions and its formulation for one-order to high-order adiabatic invariants.In the fourth part, the El-Nabulsi fractional models for non-linear constrain nonholonomic systems is established, which extended Riemann-Liouville fractional integral definition, and the exact invariant led directly by the Noether symmetry is given; then Noether symmetry that after the action of small disturbance lead to the adiabatic invariants are studied, and we proved the conditions and its formulation for one-order to high-order adiabatic invariants.