From Kepler's Law To The KAM Theorem

This thesis introduces Kepler's Law and the KAM Theorem which related to the motion of the solar system.In inertial reference and using Cartesian coordinate,the model of the motion of the solar system is Newton equation,Kepler's Law claims that earth has a ellipse orbit surrounding the sun,and the sun lies on one focus point of the orbit.In action-angle variables,the model of the motion of the solar system is Integral Hamilton equation when ignoring the affection of the moon.The KAM theorem claims that the trajectories can be period trajectories or quasi-periodic trajectories with different initial conditions,and they are on invariant torus.The model of the motion of the solar system is nearly Integral Hamilton equation when caring the affection of the moon.The KAM theorem claims that quasi-periodic trajectories can be persisted.At first,this thesis introduces some backgrounds and a simplified 2-body prob-lem:earth is moving surrounding the sun.This thesis introduces Kepler's Law and presents the main steps of the proof.Secondly,this thesis introduces a simple ex-ample of the KAM theorem:diffeomorphisms of a circle are conjugate to rotations.This thesis also presents the key idea of the KAM theorem and the main steps of the proof.The proof contains linearization and iteration.Finally,this thesis intro-duces a usual example of the KAM theorem:The quasi-periodic trajectories of the non-degenerate nearly Integral Hamilton equation can be persisted.This thesis also presents the main steps of the proof.