Lie-series transformations and applications to construction of analytical solution

In this study, Lie-series transformations including Hori’s, Deprit’s and Dragt–Finn’s are discussed and applied to construction of analytical solution of invariant manifolds in the circular restricted three-body problem (CRTBP). It shows that Dragt–Finn’s transformation holds much smaller number of Poisson brackets than the other ones especially at high orders, showing that Dragt–Finn’s approach is more efficient in terms of computational cost. Applications to the CRTBP indicate that three Lie-series transformations lead to the same analytical expression, meaning that they are equivalent from the viewpoint of constructing analytical solution. In particular, two types of analytical solutions are formulated for invariant manifolds and they are compared with Lindstedt–Poincaré series solutions and numerical results, showing good agreement among them.