1st Edition

Morse Theory for Hamiltonian Systems

By Alberto Abbondandolo Copyright 2001
    208 Pages 5 B/W Illustrations
    by Chapman & Hall

    208 Pages
    by Chapman & Hall

    This Research Note explores existence and multiplicity questions for periodic solutions of first order, non-convex Hamiltonian systems. It introduces a new Morse (index) theory that is easier to use, less technical, and more flexible than existing theories and features techniques and results that, until now, have appeared only in scattered journals.

    Morse Theory for Hamiltonian Systems provides a detailed description of the Maslov index, introduces the notion of relative Morse index, and describes the functional setup for the variational theory of Hamiltonian systems, including a new proof of the equivalence between the Hamiltonian and the Lagrangian index. It also examines the superquadratic Hamiltonian, proving the existence of periodic orbits that do not necessarily satisfy the Rabinowitz condition, studies asymptotically linear systems in detail, and discusses the Arnold conjectures about the number of fixed points of Hamiltonian diffeomorphisms of compact symplectic manifolds.

    In six succinct chapters, the author provides a self-contained treatment with full proofs. The purely abstract functional aspects have been clearly separated from the applications to Hamiltonian systems, so many of the results can be applied in and other areas of current research, such as wave equations, Chern-Simon functionals, and Lorentzian geometry. Morse Theory for Hamiltonian Systems not only offers clear, well-written prose and a unified account of results and techniques, but it also stimulates curiosity by leading readers into the fascinating world of symplectic topology.

    The Symplectic Group
    The Maslov Index in Dimension 2
    The Maslov Index in Dimension 2N
    The Maslov Index a Linear Hamiltonian System
    The Maslov Index of an Autonomous system
    Some Bibliography and Further Remarks
    Commensurable Spaces and Relative Dimension
    Fredholm Pairs of Subspaces
    Relative Morse Index of Critical Points
    Finite Dimensional Reductions
    Some Bibliography and Further Remarks
    Fractional Sobolev Spaces
    Linear Hamiltonian Systems
    Nonlinear Hamiltonian systems
    Linear Lagrangian Systems
    Nonlinear Lagrangian Systems
    Some Bibliography and Further Remarks
    Abstract Critical Point Theory
    Superquadratic Hamiltonians
    A Birkhoff-Lewis Type Theorem
    Some Bibliography and Further Remarks
    Non-Resonant Systems
    Morse Relations for Autonomous Systems
    Systems with Resonance at Infinity
    Some Bibliography and Further Remarks
    The Arnold Conjectures
    The Arnold Conjectures on the Projective Space
    Periodic Points on the Torus
    Some Bibliography and Further Remarks


    Alberto Abbondandolo

    "…provides an interesting introduction to index theories in the study of periodic solutions of Hamiltonian systems… the author presents some recently published results in the perspective of well-known ones and along the way he discusses several critical point techniques that could be useful in other problems."
    - Mathematical Reviews 2002