Applications Of KAM Theory In Planetary Many Body Problem And Generalized Benjamin-Ono Equation

In this thesis, we mainly investigate the application of KAM theory in both spatial planetary many body problem and generalized Benjamin-Ono equation, and it’s divided into four chapters.In Chapter 1, we introduce the history about planetary many body problem and KAM theory in both finite and infinite dimensional cases, some recent results and our main work.In Chapter 2, we present some preliminaries. Firstly, we introduce the classic Hamil-tonian model for planetary many body problem. Secondly, we introduce a series of fa-mous variables occurred during the study on planetary many body problem, Poincare variables and Regular Planetary Symplectic variables are included. In the lase section, we present an important statement for spatial planetary many body problem.In Chapter 3, we consider the spatial planetary many body problem we manage to investigate the Birkhoff normal form for spatial planetary many body prob-lem by making use of Averaging theorem. Then we apply the degenerate KAM theorem from [8] into spatial planetary many body problem, and derive the desired intermediate dimensional invariant tori. For the measure estimate part, we manage to get through by checking the non-degeneracy of frequency map of planetary many body problem and deriving the measure estimates for resonance regions.In Chapter 4, we consider the generalized Benjamin-Ono equation with periodic boundary condition We manage to derive the partial Birkhoff Normal Form of six order for generalized Benjamin-Ono equation, by choosing specific index, then we apply the KAM theorem established by Liu-Yuan[51] into this hamiltonian dynamical system, and prove the ex-istence of quasi-periodic solutions with two frequencies.