Morse Theory for Hamiltonian Systems

ISBN-13: 9781584882022 / Angielski / Miękka / 2001 / 208 str.

Alberto Abbondandolo

Opis

This Research Note in Mathematics introduces a new Morse (index) theory that is easier to use, less technical and more flexible than existing theories for strongly indefinite functionals. It also contains an extensive survey of material that is now classical but has never appeared in book form. This text provides an accessible, self-contained treatment of the subject matter with full proofs of the major results, and complete with applications of the theory to Hamiltonian systems and other areas of current research, including wave equations, Chern-Simon functionals and Lorentzian geometry.

Recenzje

"…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

Spis treści

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

O autorze

Alberto Abbondandolo