Index theory for symplectic paths with applications

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This book is derived from my 1993 monograph on Index Theory for Hamiltonian Systems and my lecture notes from various institutions over the past decade. It aims to achieve two main objectives: first, to introduce the index theory for symplectic matrix paths and its iteration theory, which underpin the Morse theoretical study of Hamiltonian systems, along with applications to periodic boundary value problems in nonlinear Hamiltonian systems. The iteration theory specifically addresses the index theory of periodic solutions and symplectic matrix paths. Second, it serves as a reference on these subjects. The book presents an accessible introduction to the index theory for symplectic paths, facilitating the establishment of Morse-type index theory for Hamiltonian systems. I have opted for a relatively elementary approach, utilizing the homotopy classification method of symplectic matrix paths, which relies on linear algebra, point set topology, and fundamental aspects of linear functional analysis. I have aimed to make this section self-contained while encompassing all major results, allowing researchers and students to engage with the material without significant difficulty and to grasp the key findings for potential applications.