The Index Theory And Calculation For Unbounded Orbits Of Hamiltonian Systems

The theory of Hamiltonian systems is based on the movement of objects in nature,which plays outstanding role in the studies of celestial mechanics and aerospace.While the index theory is a useful tool in the study of orbits of Hamiltonian systems.In this thesis,we will mainly study the index theory on halfclinic orbits.There are two parts as followings:First,on the one hand,we will use a geometric approach to assign the Maslov index to any unbounded orbits of a Hamiltonian system.Roughly speaking,its counts algebraically the intersections between a curve of Lagrangian subspaces and a transversally oriented singular Maslov cycle of a fixed Lagrangian subspace.And then we gave the calculation method of Maslov index.On the other hand,we also study another well-konwn topological invariant termed spectral flow of paths of the selfadjoint Fredholm operators.It counts the net number of eigenvalues which pass through the zero in the positive direction when the parameter travels along the whole interval.Therefore we will show that these two indices coincide on unbounded orbits.The first crucial work in this thesis is the whole proof for index theory on halfclinic orbits.Second,based on the application of halfclinic orbits index theory on homographic motion in planar N-body problem,we will give a numerical method to compute the halfclinic index of Euler and Lagrangian.This thesis is organized as follows.Section 1 is devoted to give a brief introduction of the index theory and our main result.Then we set out some basic definitions and properties of Masolv index,spectral flow and relative Morse index in Section 2 and 3 Section 4 is dedicated to describe the index theory on unbounded orbits.Section 5,which is the core of this thesis,is devoted to prove the index theory on halfclinic orbits Finally,we introduce the halfclinic index theory application background in the study of the stability problem of homographic motion in planar N-body problem in Section 6.And the other part of Section 6 is we give the calculation method,and we calculate the Maslov index of Euler and Lagrangian as well as general orbits as examples.