Perturbation to Noether symmetry and adiabatic invariants for Birkhoffian system,Hamiltonian system and Lagrangian system with delta derivative are investigated,respectively. Firstly,the definition and some related properties of time scale calculus are listed simply as preliminaries. Secondly,the Birkhoffian system with delta derivative is studied. Based on the differential equation of motion as well as Noether symmetry and conserved quantity,perturbation to Noether symmetry and adiabatic invariant are investigated. Thirdly,adiabatic invariants for the Hamiltonian system and the Lagrangian system are presented through some transformations. And finally,an example is given to illustrate the methods and results.

supported by the National Natural Science Foundations of China (Nos. 11802193, 11572212); the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 18KJB130005); the Jiangsu Government Scholarship for Overseas Studies, the Science Research Foundation of Suzhou University of Science and Technology (No.331812137); Natural Science Foundation of Suzhou University of Science and Technology;



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