Abstract
Lie symmetry of a dynamical system is the invariance of differential equations of motion under the infinitesimal transformations of a group and it can lead to invariants under certain conditions. Firstly, the Lie symmetry of Birkhoffian system on time scales is studied when there is no disturbance, the determining equations of Lie symmetry are established, and the exact invariants led by the Lie symmetry are given. Secondly, the perturbation to Lie symmetry and adiabatic invariants are studied when the system is subjected to small disturbance, and the determining equations of Lie symmetry of the disturbed system are established, and the condition of the Lie symmetry leading to adiabatic invariants and the form of adiabatic invariants are given. As an application of the results, we give the Lie symmetry theorems of Hamiltonian system on time scales. The results contain the exact invariants and adiabatic invariants of Lie symmetry for the classical continuous systems and discrete systems as their special cases. Two examples are given to illustrate the application of the results.
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