Lie-series transformations and applications to construction of analytical solutions

Abstract

In this study, Lie-series transformations including Hori's, Deprit's and Dragt--Finn's are discussed and applied to construction of analytical solutions of invariant manifolds in the circular restricted three-body problem (CRTBP). It shows that three Lie-series transformations lead to the same Kamiltonian as well as the same expressions of analytical solutions, implying that these three transformations are equivalent from the viewpoint of constructing analytical solutions. By taking Lie-series transformations, two types of analytical solutions are formulated for invariant manifolds in terms of action--angle variables as well as motion amplitudes. To validate the Lie-series analytical solutions, analytical trajectories are compared with numerical ones, showing that the analytical solution with a higher order has a higher accuracy. Motivated by the fact that Dragt--Finn's recursion can be derived from Hori's transformation by ignoring the commutation relation of Poisson brackets, we propose a simplified version of Deprit's recursion. It is shown that the simplified Deprit's recursion is equivalent to the other three transformations. From the viewpoint of computational cost, the simplified Deprit's recursion and Dragt--Finn's transformation are preferred in practice because they have much smaller number of Poisson brackets than the other two transformations at high orders. At last, explicit expressions are provided for Dragt--Finn's and the simplified Deprit's recursions up to order 10.