Morse Theory for Hamiltonian Systems

Morse Theory for Hamiltonian Systems

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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.
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Product details

  • Paperback | 208 pages
  • 155.4 x 230.6 x 12.4mm | 303.91g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 5 black & white illustrations
  • 1584882026
  • 9781584882022

Review quote

"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
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Table of contents

THE MASLOV INDEX 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 THE RELATIVE MORSE INDEX Commensurable Spaces and Relative Dimension Fredholm Pairs of Subspaces Relative Morse Index of Critical Points Finite Dimensional Reductions Some Bibliography and Further Remarks FUNCTIONAL SETTING Fractional Sobolev Spaces Linear Hamiltonian Systems Nonlinear Hamiltonian systems Linear Lagrangian Systems Nonlinear Lagrangian Systems Some Bibliography and Further Remarks SUPERQUADRATIC HAMILTONIANS Abstract Critical Point Theory Superquadratic Hamiltonians A Birkhoff-Lewis Type Theorem Some Bibliography and Further Remarks ASYMPTOTICALLY LINEAR SYSTEMS Non-Resonant Systems Morse Relations for Autonomous Systems Systems with Resonance at Infinity Some Bibliography and Further Remarks THE ARNOLD CONJECTURES FOR SYMPLECTIC FIXED POINTS The Arnold Conjectures The Arnold Conjectures on the Projective Space Periodic Points on the Torus Some Bibliography and Further Remarks
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