Hill-type Formula For Hamiltonian Systems And Its Applications

The Hill-type formula for Hamiltonian systems is derived from Hill's study on the 3-body problem.The Hill-type formula connects the infinite determi-nant of the Hessian of the action functional with the determinant of matri-ces which depend on the monodromy matrix and boundary conditions.The Krein-type trace formula deduced from this can use to study the stability of the periodic solution of the Hamiltonian systems.For example,we can estimate the stable region and hyperbolic region of the elliptic Lagrangian solutions.Firstly,this article summarizes the Hill-type formula and Krein-type trace for-mula for Hamiltonian systems with Lagrangian boundary conditions,and such a kind of boundary conditions comes from the N-reversible symmetry periodic orbits in the n-body problem naturally,where N is an anti-symplectic orthog-onal matrix.Then,considering the eigenvalue problem of the Sturm-Liouville system,the trace formula can be used to obtain some infinite series identities.Finally,we generalize the Hill-type formula for linear Hamiltonian systems and Sturm-Liouville systems with any self-adjoint boundary conditions.This article is divided into five chapters:Chapter One introduces the background and development of Hill-type formula;Chapter Two uses the rele-vant properties of the conditional Fredholm determinant to derive the Hill-type formula for Hamiltonian systems with Lagrangian boundary conditions;Chap-ter 3 introduces the specific process of deriving the Krein-type trace formula from the Hill-type formula.And we show its connection with the Hill-type formula with periodic boundary conditions and Lagrangian boundary condi-tions Chapter 4 gives the Hill-type formula and Krein-type trace formula for Sturm-Liouville systems and derive some identities;Chapter 5 introduces Hill-type formula and Krein-type trace formula with any self-adjoint boundary conditions.