Morse Theory for Hamiltonian Systems: 1st Edition (Paperback) book cover

Morse Theory for Hamiltonian Systems

1st Edition

By Alberto Abbondandolo

Chapman and Hall/CRC

208 pages | 5 B/W Illus.

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pub: 2001-03-15
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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.

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

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

About the Series

Chapman & Hall/CRC Research Notes in Mathematics Series

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Subject Categories

BISAC Subject Codes/Headings:
MAT007000
MATHEMATICS / Differential Equations
MAT012000
MATHEMATICS / Geometry / General
MAT037000
MATHEMATICS / Functional Analysis