Lie canonical perturbation theory

Since both initial and final states satisfy the same equations of motion, the transformation of Eq. (6) is a natural canonical transformation. This kind of canonical transformation is a basic tool in the so-called Lie canonical perturbation theory for obtaining approximate constants of the motion. 

For nonautonomous systems, an additional term involving the time derivatives of W(p, q s) must be included in Eq. (A.66) [45, 46, 53]. In this Appendix, we have described how Lie transforms provide us with an important breakthrough in the CPT free from any cumbersome mixed-variable generating function as one encounters in the traditional Poincare-Von Zeipel approach. After the breakthrough in CPT by the introduction of the Lie transforms, a few modifications have been established in the late 1970s by Dewar [56] and Drag and Finn [47]. Dewar established the general formulation of Lie canonical perturbation theories for systems in which the transformation is not... 

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