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Morse Theory for Hamiltonian Systems





ISBN 9781584882022
Published March 15, 2001 by Chapman and Hall/CRC
208 Pages - 5 B/W Illustrations

 
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Book Description

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.

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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Reviews

"…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