Stability Study Of The Elliptic Homographic Solutions In N-body Probiem And The Closed Characteristics On Compact Convex Hypersurfaces

In this dissertation, we will study the stability of periodic orbits of two classical Hamiltonian systems. One is about the stability of elliptic homographic solutions in N-body problem. The other is about the stability of the closed characteristics on compact convex hypersurfaces.N-body problem comes from the study of celestial movement. The research of celestial movement started in ancient Greece. Until the Newton era, it has a big breakthrough because of the development of calculus. In theory, N body can be regarded as N particles. Under the gravity, we study their motion and stability. The most typical system is the system of Sun-Earth-Moon. Naturally, people want to understand the motion of our star system. It’s well know that, the solutions of two body problem have an explicit expression. When N≥3, their equations are not integrable, hence they don’t have explicit solutions. In face, in the study of three body, Poincare found that their movements could be very complicated, hence the study of many-body problem becomes very difficult. The first step in understanding the complex movements of many-body problem is try to study some special movement, like periodic and quasi-periodic movement.The first part of our dissertation is to study the linear stability of peri-odic solutions of central configurations. This type of periodic solutions have a long history. The first two periodic solutions are found by Euler(1767) and La-grange(1772). Now we call them Euler solution and Lagrangian solution. The Euler solution is collinear periodic solution, in which three bodies of any masses move such that they oscillate along a rotating line. The Lagrangian solution is consist of three bodies and they form an equilateral triangle at any instant of the motion and at the same time each body travels along a specific Keplerian elliptic orbit about the center of masses of the system. This two solutions are the special cases of the periodic solutions which come from the center configurations. In fact, Euler solution comes from collinear configuration and Lagrangian solution comes from regular triangle. Moreover, there are n-gon and 1+n-gon configurations. In the beginning, this solution in found by mathematics points. Later, people found that this solutions can be used to describe the real star system. For example, the Lagrangian solution can be used to describe the Sun-Jupiter-Trojan system and the 1+n-gon can be used as an approximate model of Saturn and Saturn’s ring. For this reasons, the study of their stability has a strong physical background.For the Lagrangian solution, all the existing stability results are in the case which requires e or 1 - e small enough. For other cases, there are only numerical results or qualitative results. For more than three bodies, there are only some results in the case e= 0, but for any eccentricity, there are no results. In the first part of our paper, we study the stability of elliptic homographic solutions of some typical center configurations. The main results include the first estimation of the stability and hyperbolic region of Lagrangian solution. Moreover, we will give an analysis about the hyperbolicity of homographic solutions of the 4-gon, 1+3-gon and strong minimal center configuration.The second part of our dissertation is about the stability of the closed charac-teristics on compact convex hypersurfaces. The closed characteristic is produced by some special vector field on the compact convex hypersurfaces. The study of its existence, multiplicity and stability is an classical problem. It promotes the development of index theory and Floer homology theory. In 1892, A.M.Liapounov started to study this problem and got some local results. The first global result is proved by P.Rabinowitz and A.Weinstein independently. Furthermore, the multi-plicity results are got by I.Ekeland.et.al. A big breakthrough is made by Yiming Long and Chaofeng Zhu. They used the common index jump theory to prove that the lower bound of the closed characteristics is at least [n/2]+1. On the other hand, with the results of multiplicity, it’s natural to study the properties of them, such as the stability. The first work about the stability is got by I.Ekeland.et.al, they proved the existence of the elliptic closed characteristics under some symmetric or pinch condition. In 2000, Yiming Long proved that for the compact convex hypersurfaces in R4, if there exists just two closed characteristics, then they are all elliptic. In paper [38], Yiming Long and Chaofeng Zhu further proved that if the total number of geometrically distinct closed characteristics is finite, then there exists at least an elliptic one them, and there exist at least (?)n(Σ) - 1 of them possessing irrational mean indices. If this total number is at most 2(?)n(Σ) - 2, there exist at least two elliptic ones among them. In all the existing results, if we want to get two elliptic characteristics, we need the strictly upper bound condi-tion. But we know nothing about this upper bound condition except the case in R4. In our paper, we prove the existence of two elliptic closed characteristics only under the finiteness condition, we do not need the strictly upper bound condi-tion. Moreover we get irrational properties about the ratio of mean index. This dissertation consists of three chapters.In the first chapter, we introduce the periodic orbits and its linear stabil-ity in Hamiltonian system. Especially, we introduce the historical background, significance and some recent results of the periodic orbits in two typical Hamil-tonian system, i.e. the homographic solutions in N-body problem and the closed characteristics on compact convex hypersurfaces. At the same time, we point out the difficulties of this problems and the limitation of the previous methods. We also introduce our breakthrough of this problems. This chapter contains three sections. In the first section, we give a brief review of the periodic orbits and its linear stability in Hamiltonian system. This raises two problems in our study. In the second section, we introduce the knowledge of the homographic solutions in N-body problems and give our main results about the stability of homographic solutions. The closed characteristics problem will be introduced in the third sec-tion, it also contain our main results on the stability of closed characteristics problem.In the second chapter, we study the stability of the homographic solutions in N-body problems. It contains the stability analysis of elliptic Lagrangian so-lutions and the homographic solutions of 4-gon,1+3-gon and strong minimal center configuration. The linear stability of Lagrangian solutions depend on two parameters, one is the mass parameter β ∈ [0,9], another is the eccentricity e ∈ [0,1). In this part, based on the Maslov-type index theory and the recently developed trace formulas for linear Hamiltonian system, we first give a quanti-tative estimation of the linear stability region and hyperbolicity region for the elliptic Lagrangian solutions. Moreover, when the mass parameter is in some rang, we develop a positive technique for evaluating the hyperbolicity of elliptic Lagrangian solution for any eccentricity. In real phenomenon, this results tells us that when the masses of the three bodies are in some range, the periodic orbits of this three bodies is unstable for any eccentricity. At the same time, we built up the relation between the 4-gon homographic solutions and the Lagrangian solutions by using the second order differential operator theory. Based on this relation, we will prove that the 4-gon homographic solutions is hyperbolic for any eccentricity. Using the similar estimation methods, we can prove that the homo-graphic solution of minimal center configuration is hyperbolic for any eccentricity e ∈ [0,1). As a corollary, we get that if the center mass u ∈ [0,(?)/24|), then the 1+3-gon homographic solution is hyperbolic for any eccentricity. This is the first results about the hyperbolicity of homographic solutions on four bodies for any eccentricity. This chapter contains four sections. The first section is about our main results of homographic solutions. The research tools is introduced in the second section. The applications of trace formulas in the elliptic Lagrangian so-lutions will be discussed in the third section. In the fourth section, we introduce our positive estimation technique which can be used to analysis the hyperbolicity of homographic solutions for any eccentricity.In the third chapter, we study the stability of closed characteristics on the compact convex hypersurfaces. This closed orbits are generated by some special vector field on the compact convex hypersurfaces. The existence, multiplicity and stability of closed characteristics are classical problems in Hamiltonian system. The study of this problem has promoted the development of variational methods and index theory. In this part, we use Long-Zhu index jump theory [38] to built up some new index inequalities. Base on this inequalities, we prove that if the surface has finitely geometrically distinct closed characteristics, then there are at least two geometrically distinct elliptic closed characteristics. Moreover, there exist at least (?)n(Σ) (?)n(Σ) ≥ [n/2]+1) geometrically distinct closed characteristics such that for any two elements among them, the ratio of their mean indices is irrational number. This results improve the stability and irrational results in the paper of Yiming Long and Chaofeng Zhu [38]. It is also the first result about two elliptic closed characteristics for any dimension under finitely condition. This chapter contains four sections. The first section is about our main results of the stability of closed characteristics problem. In the second section, we review the Long-Zhu jump index theory and give a further discussion, then we get some new index inequalities. In the third section, we will proof our main results. The appendix is about the index iteration theory which will be introduced in the last section.