The Applications Of Index Theory In N-Body Problem

The elliptic Lagrangian solution of the 3-body problem is an elliptic homo-graphic orbit generated by the Lagrangian central configuration,that is,the three particles always form an equilateral triangle and move elliptically with the same eccentricity.Under the central configuration coordinates,the linear variational e-quation of the elliptic Lagrangian solution can be decoupled into the Kepler part and the essential part when the centroid is fixed as the origin.The linear stability of the solution depends on the essential part.This paper summarizes the research method of stability,denpends on the index theory,established by Y.Long,X.Hu and S.Sun et al.:based on the variational properties of elliptic Lagrangian solutions,the relationship between the linear stability of the essential part and the mass parameterβ and the elliptic eccentricity e is studied by using Maslov-type index.Firstly,by using the variational minimality of elliptic Kepler solution and elliptic Lagrangian solution,and the relationship between Maslov-type index and Morse index,it is proved that the Morse index of the essential part of elliptic Lagrangian solution is zero.Then by analyzing the Sturm-Liouville operator corresponding to the essen-tial part,it is proved that any ω index is not increasing with respect to β.By using the monotony of the index,they proved the existence of the degeneracy curves,and divided the graph of(β,e)into four parts by these degeneracy curves,and gived the elliptic and hyperbolic characterization of each part.Finally,a specific curve is obtained and the size of the hyperbolic region is estimated.Furthermore,by us-ing the non-increasing property of the index from their work,we give an analytical proof of the fact that the Morse index of the essential part for the linear variational equation of the elliptic Lagrangian solution is zero.This paper is composed by five chapters.The first chapter introduces the de-velopment process and the background of N-body problem,and summarizes the main results of this paper.The second chapter introduces the definition and prop-erties of the index.In the third chapter,we introduce the properties of periodic solutions of n-body problem and the concept of central configuration,especially calculate the basic normal form and Morse index of elliptic Kepler solutions.In chapter 4,we expound the methods of proving the main result of this paper,that is,the non-increasing property of the index about the quality parameter,and the stability of each part in the(β,e)graph is analyzed.In chapter 5,the size of the hyperbolic region is estimated.