Morse Theory for Hamiltonian Systems

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Alberto Abbondandolo (Scuola Normale Superiore, Pisa, Italy) Alan Jeffrey (University of Newcastle upon Tyne, UK)

Morse Theory for Hamiltonian Systems

Alberto Abbondandolo (Scuola Normale Superiore, Pisa, Italy), Alan Jeffrey (University of Newcastle upon Tyne, UK), Haim Brezis , Ronald G. Douglas (Texas A & M University)

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English

Chapman & Hall/CRC
15 March 2001

Calculus & mathematical analysis; Functional analysis & transforms; Differential calculus & equations; Geometry; Analytic topology

Series: Chapman & Hall/CRC Research Notes in Mathematics Series

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.

By: Alberto Abbondandolo (Scuola Normale Superiore Pisa Italy)
Series edited by: Alan Jeffrey (University of Newcastle upon Tyne UK), Haim Brezis, Ronald G. Douglas (Texas A & M University)
Imprint: Chapman & Hall/CRC
Country of Publication: United States
Volume: 425
Dimensions: Height: 234mm, Width: 156mm, Spine: 12mm
Weight: 280g
ISBN: 9781584882022
ISBN 10: 1584882026
Series: Chapman & Hall/CRC Research Notes in Mathematics Series
Pages: 208
Publication Date: 15 March 2001
Audience: Professional and scholarly , General/trade , Undergraduate , Undergraduate , ELT Advanced
Format: Paperback
Publisher's Status: Active

The Morse Index. The Relative Morse Index. Functional Setting. Superquadratic Hamiltonians. Asymptotically Linear Systems. The Arnold Conjectures for Symplectic Fixed Points. Index. References.

Alberto Abbondandolo

Reviews for Morse Theory for Hamiltonian Systems

""…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