Lie symmetry perturbation and adiabatic invariants for dynamical system with non-standard Lagrangians

This paper focuses on studying the Lie symmetry perturbation and the adiabatic invariants for a dynamical system with non-standard Lagrangians. First, the Euler-Lagrange equations for dynamical systems with two kinds of non-standard Lagrangians that are exponential Lagrangian and power-law Lagrangian are given in the undisturbed case and the disturbed cases, respectively. Secondly, the determining equations of Lie symmetry for undisturbed system and disturbed system are established, and the definitions of the Lie symmetry are given. Thirdly, the exact invariants of Noether type and Hojman type led by Lie symmetry for the undisturbed system are given. Fourthly, the adiabatic invariants of Noether type and Hojman type led by Lie symmetry and the conditions that need to be satisfied for the disturbed system are obtained. Finally, some examples are given to illustrate the application of the results.