PERTURBATION TO LIE SYMMETRIES FOR GENERALIZED BIRKHOFFIAN SYSTEMS ON TIME SCALES

In recent years, the study of symmetry and conserved quantity under time-scale framework has increasingly attracted attention. Nevertheless, as this field is still in the exploratory stage, the reliability of its research results is still under further investigation, and the study of symmetry perturbations and adiabatic invariants is also based on this work. Therefore, in-depth exploration of this field is of great significance. First of all, under time-scale framework, the definitions of exact invariants and adiabatic invariants are given. For both undisturbed and disturbed generalized Birkhoffian systems, the determining equations and structural equations for Lie symmetry and its perturbations are established, respectively. Based on these, the exact invariants caused by Lie symmetry and the adiabatic invariants caused by perturbations to Lie symmetry in the both systems are obtained, and the corresponding proofs are given. Secondly, regarding the Birkhoffian systems under constrained conditions on time scales, we explore the conditions under which the Lie symmetry of the constrained Birkhoffian system and its corresponding free Birkhoffian system leads to exact invariants, as well as the conditions under which the perturbation to Lie symmetry of the corresponding disturbed systems leads to adiabatic invariants. At the end of the corresponding sections, examples are provided and numerical simulations are conducted on the obtained conserved quantities, which intuitively verified the effectiveness of the conclusions in this paper. Taking the time scale as the set of real numbers and integer, all conclusions in this paper can be degraded to classical continuous and discrete dynamical systems. The methods and research results of this paper have certain reference and guiding significance for the study of symmetry and perturbation theory of dynamical systems on time scales, and can be applied and extended to nonshifted systems, dual systems, the combination of fractional and time-scale systems, and nabla derivative cases, and so on.