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

CRC Press
04 April 2018

Differential calculus & equations; Geometry

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: CRC Press
Country of Publication: United Kingdom
Dimensions: Height: 234mm, Width: 156mm,
Weight: 453g
ISBN: 9781138417588
ISBN 10: 1138417580
Series: Chapman & Hall/CRC Research Notes in Mathematics Series
Pages: 208
Publication Date: 04 April 2018
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

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"