Persistence Of Invariant Tori For Nearly Integrable Hamiltonian Systems

Hamiltonian equations first appeared in problems of geometric optics and ce-lestial mechanics.Later it was used to describe plenty of systems in classical me-chanics,chemistry,physics,and other domains.In this thesis,we begin with the definition of symplectic manifold,and follow the definition of completely integrable Hamiltonian systems.As shown by Poincare[61]that the dynamical system in its general form was non-integrable and the investigation of a completely integrable Hamiltonian under a small perturbation is a fundamental problem of dynamic-s.The main task of this paper is to study the influence of perturbation to the invariant tori of a completely integrable Hamiltonian system.It is well-known that collision and escape for n-body problem closely relate to the persistence of invariant tori[57].The classical KAM theory(Kolmogorov[38],Arnold[1],Moser[55])deals with the persistence of non-resonant invariant tori for non-degenerate Hamiltonian systems,which solves the 200 year-old problem.How-ever,there are some degenerate Hamiltonian systems in celestial mechanics.To overcome the degeneracy,Arnold introduced 2-scale Hamiltonian systems and showed the persistence of non-resonant invariant tori[2].There are some cases that the degeneracy cannot be removed by 2-scale[14,21,52,53,59,60].Then Han,Li and Yi in-troduced multi-scale Hamiltonian systems.When there is an order relation among scales,they showed the persistence of non-resonant invariant tori[35].For the case that there is no order relation among scales,it is complex to estimate the normal for inverse operator during the process of solving homological equations.We over-came this difficulties and showed the persistence of non-resonant invariant tori for multi-scale Hamiltonian systems that there is no order relation among scales in§2.3.For detail,refer to Theoreml.1.So far,Russmann non-degenerate condition is the weakest condition for the persistence of invariant tori.And is there a family of Lagrange invariant tori without Russmann non-degenerate condition?Arnold,Kozlov and Neishtadt gave an example showing that a small perturbation will destroy the stability of the non-resonant Hamiltonian systems[3].In Chapter two,we improved the classical KAM iteration and show the persistence of Lagrange invariant tori under a non-degenerate condition on perturbation.For details,refer to Theorem1.2.More-over,combining this theorem we gave a weaker condition on the persistence of invariant tori for degenerate systems rising from restricted three-body problems,e-specially,for Lagrange equilibrium position L2 with certain degeneracy.For detail,refer to Example1.7.Resonance does occur in celestial mechanics.For example,the frequency ratio of Saturn and Jupiter is 2:5 and the frequency ratio of Uranus and Neptune is 2:1.A family of tori,which contains resonant torus of all type of resonances,tend to be destroyed under arbitrary generic perturbation and born out a resonance zone consisting of both stochastic trajectories and regular orbits.To character-ize regular orbits in the resonance region,it is essential to study mechanisms of destruction of the resonant tori and the number of resonant torus survived[10,71].For 1-order perturbation,Poincare[61].Treshchev[76],Cong,Kupper,Li and You[20].Li and Yi[43]studied mechanisms of destruction of the resonant tori and the num-ber of resonant torus survived.For general perturbation,there is a long standing conjecture[10,20,23,30,37]:the number of the resonant torus survived should be equal to the number of critical points of smooth functions on the torus.For high or-der degenerate perturbation,using non-linear KAM iteration we gave a positive answer to this conjecture.Moreover,we proved the persistence of frequency and the persistence of frequency ratio on a given energy surface.For details,refer to Theorem1.3.When there is no full dimension invariant torus,it is nature to consider the persistence of lower dimension invariant tori,where Melnikov non-degenerate con-ditions play an important role[47,48].In Chapter four,we give a weaker Melnikov conditions,which are similar to Russmann non-degenerate condition.For detail,refer to Theorem1.5.And under the new Melnikov nondegenerate conditions we prove the persistence of lower dimension invariant tori.