Chapman & Hall/CRC Research Notes in Mathematics - 425: Morse Theory for Hamiltonian Systems

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This Research Note delves into existence and multiplicity questions for periodic solutions of first-order, non-convex Hamiltonian systems. It presents a new Morse (index) theory that is more user-friendly, less technical, and more adaptable than previous theories, showcasing techniques and results previously found only in scattered journals. The work offers a comprehensive overview of the Maslov index, introduces the concept of relative Morse index, and outlines the functional framework for the variational theory of Hamiltonian systems, including a novel proof of the equivalence between Hamiltonian and Lagrangian indices. It investigates superquadratic Hamiltonians, demonstrating the existence of periodic orbits that may not meet the Rabinowitz condition, and provides an in-depth study of asymptotically linear systems. The author also addresses Arnold's conjectures regarding the fixed points of Hamiltonian diffeomorphisms on compact symplectic manifolds. In six concise chapters, the text delivers a self-contained treatment with complete proofs, clearly distinguishing abstract functional aspects from practical applications. Many findings are applicable to other research areas, such as wave equations, Chern-Simon functionals, and Lorentzian geometry. The work combines clear prose with a unified presentation of results and techniques, inviting readers into the intriguing realm of symplectic topology.