Morse Theory for Hamiltonian Systems.
"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 t...
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| Format: | eBook |
| Language: | English |
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Boca Raton, FL :
CRC Press,
2001.
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| Edition: | First edition. |
| Series: | Chapman & Hall/CRC Research Notes in Mathematics Series
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| Online Access: | Connect to the full text of this electronic book |
Table of Contents:
- Cover; Half Title; Title Page; Copyright Page; Table of Contents; Preface; 1: The Maslov index; 1.1 The symplectic group; 1.2 The Maslov index in dimension 2; 1.2.1 The topology of Sp(2); 1.2.2 The rotation function on Sp(2); 1.2.3 The Maslov index for non-degenerate paths in Sp(2); 1.3 The Maslov index in dimension 2N; 1.3.1 The Krein signature on Sp(2N); 1.3.2 Normal forms of semi-simple symplectic matrices; 1.3.3 The topology of Sp(2N); 1.3.4 The rotation function on Sp(2N); 1.3.5 Decomposition of Sp(2N); 1.3.6 The Maslov index of non-degenerate paths in Sp(2N).
- 1.3.7 The Maslov index of degenerate paths in Sp(2N)1.4 The Maslov index of a linear Hamiltonian system; 1.4.1 Iteration formulas; 1.5 The Maslov index of an autonomous system; 1.6 Some bibliography and further remarks; 2: The relative Morse index; 2.1 Commensurable spaces and relative dimension; 2.1.1 Further properties; 2.2 Fredholm pairs of subspaces; 2.3 Relative Morse index of quadratic forms; 2.3.1 Further properties; 2.4 Relative Morse index of critical points; 2.5 Finite dimensional reductions; 2.6 Some bibliography and further remarks; 3: Functional setting.
- 3.1 Fractional Sobolev spaces3.2 Linear Hamiltonian systems; 3.2.1 Comparison between the relative Morse index and the Maslov index; 3.3 Nonlinear Hamiltonian systems; 3.4 Linear Lagrangian systems; 3.4.1 Comparison between the Morse index and the Maslov index; 3.5 Nonlinear Lagrangian systems; 3.6 Some bibliography and further remarks; 4: Superquadratic Hamiltonians; 4.1 Abstract critical point theory; 4.1.1 The (PS) and (PS)* condition; 4.1.2 A review of Morse theory; 4.1.3 Linking theorems and Morse index estimates; 4.1.4 Strongly indefinite functionals; 4.2 Superquadratic Hamiltonians.
- 4,2.1 Minimality of the period4.3 A Birkhoff-Lewis type theorem; 4.4 Some bibliography and further remarks; 5: Asymptotically linear systems; 5.1 Non-resonant systems; 5.1.1 Abstract asymptotically quadratic functionals; 5.1.2 Morse relations for non-resonant systems; 5.1.3 Solutions of higher period; 5.2 Morse relations for autonomous systems; 5.2.1 Growth of the number of periodic solutions of autonomous systems; 5.3 Systems with resonance at infinity; 5.3.1 Abstract asymptotically quadratic functionals, degenerate at infinity; 5.3.2 Resonant systems.
- 5.4 Some bibliography and further remarks6: The Arnold conjectures for symplectic fixed points; 6.1 The Arnold conjectures; 6.1.1 Hamiltonian diffeomorphisms of the two-torus; 6.1.2 Non-degenerate fixed points on the torus; 6.1.3 Degenerate fixed points on the torus; 6.2 The Arnold conjectures on the projective space; 6.3 Periodic points on the torus; 6.4 Some bibliography and further remarks; Bibliography; Index.