NOETHER-TYPE ADIABATIC INVARIANTS FOR CONSTRAINED MECHANICS SYSTEMS ON TIME SCALES

Time scale calculus is one of the research hotspots in recent years. It not only unifies continuous analysis and discrete analysis, but also assists in modeling more complex dynamics systems. The Noether symmetry method is a modern integration method that reveals the potential relationship between the conserved quantity of a mechanics system and its inherent dynamics symmetry. Perturbation to Noether symmetry, as well as adiabatic invariants, are also closely related to the integrability of the system. Although the problems of symmetry for constrained mechanics systems on time scales have been studied by scholars, the depth and accuracy of research results need to be explored due to the immaturity of time scale calculus theory. The focus of this article is to explore the perturbation to Noether symmetry, and the adiabatic invariants for the constrained mechanics systems on time scales, including Lagrangian system, Hamiltonian system and Birkhoffian system. Firstly, for the three perturbed systems, we discuss the changes of the Noether symmetry, and present the corresponding adiabatic invariants. Then, we provide the exact invariants of the three systems under disturbance free conditions. The exact invariant obtained of the Lagrangian system is consistent with the original result, while the exact invariants obtained of the Hamiltonian and Birkhoffian systems are new. Secondly, there are two derivatives on time scales, namely, the delta derivative and the nabla one, and the systems composed of the two derivatives are dual. Based on the results obtained under the delta derivative in this article, the adiabatic invariants and exact invariants of the three dual spaces are given using the method of dual principle. Thirdly, at the end of the article, the Noether type adiabatic invariants of the Kepler problem and the Hojman-Urrutia problem on time scales were discussed respectively, to illustrate the results and methods presented in the three systems of this article by examples.