The Persistence Of Invariant Tori For Degenerate Hamiltonian Systems

In this thesis,we study the persistence of invariant tori for degenerate Hamiltonian systems.For non-degenerate nearly integrable Hamiltonian systems,the classical KAM theory asserts that most of the unperturbed invariant tori persist under any sufficiently small perturbations.However,degenerate Hamiltonian systems exactly exist in celestial mechanics,and the persistence of its invariant tori has been widely concerned.In this thesis,we will give some conditions to prove the persistence of invariant tori for degenerate Hamiltonian systems.The main research of this thesis is divided into three parts:(a)the persistence of full dimensional invariant tori for degenerate Hamiltonian systems with Holder continuous parameters;(b)the persistence of low dimensional invariant tori for degenerate Hamiltonian systems with high co-dimension;(c)the persistence of low dimensional invariant tori for infinite-dimensional Hamiltonian systems with finite number of zeros among normal frequencies.This thesis consists of four chapters,the main contents are as follows:In Chapter 1,we describe the background of our research,including the origin of Hamiltonian systems,the classical KAM theory,the Melnikov persistence theorem,and the KAM-type theory of infinite-dimensional Hamiltonian systems.In addition,we present main results and framework of the thesis.In Chapter 2,we study the Kolmogorov’s theorem for degenerate Hamiltonian systems with Holder continuous parameters,i.e.,the persistence of full dimensional iso-frequency invariant tori(see Theorem 1.5.1).First of all,we give some sufficient conditions.Secondly,we apply the quasi-linear iterative scheme,and show the detailed construction and estimates for one cycle of KAM steps.It should be emphasized that in the usual KAM iteration process,the regularity of the frequency mapping about the parameters is at least Lipschitz continuous to ensure that the parameter domain is not hollowed-out.When the regularity is weaker than Lipschitz continuous,the classical digging parameter method is no longer applicable.So,we use a new translation parameter method.Finally,we prove an iteration lemma and verify the convergence of iteration sequences,thus we complete the proof of the theorem.In Chapter 3,we study the Melnikov’s persistence about lower-dimensional invariant tori for degenerate Hamiltonian systems with high co-dimension,i.e.,the persistence of lower dimensional invariant tori(see Theorem 1.5.4).The similar proof line to Chapter 2 will be used to prove this theorem.The main difficult is to eliminate the first order terms generated by the normal direction during the iteration process,which are difficult to be eliminated for degenerate Hamiltonian systems.We overcome these difficulties and prove the persistence of low dimensional invariant tori.It should be pointed out that the co-dimension can be high dimensional.Hence it is a result about the persistence of lower-dimensional invariant tori in the normal degeneracy of high co-dimension.At present,there is relatively little research on this.In Chapter 4,for infinite-dimensional Hamiltonian systems with finite number of zeros among normal frequencies,we prove the persistence of lower dimensional invariant tori(see Theorem 1.5.5).What different from the finite-dimensional case in Chapter 3 is we need to give similar sufficient conditions by the weighted phase space norm in the corresponding infinite-dimensional space to eliminate the first-order terms in the direction of zero normal frequencies during the iteration process.Generally,when dealing with the KAM theory of infinite-dimensional Hamiltonian systems,the infinitedimensional normal directions can be seen as the whole and the convergence speed is equal.For this problem,since there are finite zero normal frequencies,we need to make the zero normal frequency directions and the remaining other directions have different convergence rates.