Book contents
- Frontmatter
- Contents
- Preface
- 1 The Arnold-Liouville theorem
- 2 Lagrangian fibrations
- 3 Global action-angle coordinates and torus actions
- 4 Symplectic reduction
- 5 Visible Lagrangian submanifolds
- 6 Focus-focus singularities
- 7 Examples of focus-focus systems
- 8 Almost toric manifolds
- 9 Surgery
- 10 Elliptic and cusp singularities
- Appendix A Symplectic linear algebra
- Appendix B Lie derivatives
- Appendix C Complex projective spaces
- Appendix D Cotangent bundles
- Appendix E Moser’s argument
- Appendix F Toric varieties revisited
- Appendix G Visible contact hypersurfaces and Reeb flows
- Appendix H Tropical Lagrangian submanifolds
- Appendix I Markov triples
- Appendix J Open problems
- References
- Index
1 - The Arnold-Liouville theorem
Published online by Cambridge University Press: 06 July 2023
Book contents
- Frontmatter
- Contents
- Preface
- 1 The Arnold-Liouville theorem
- 2 Lagrangian fibrations
- 3 Global action-angle coordinates and torus actions
- 4 Symplectic reduction
- 5 Visible Lagrangian submanifolds
- 6 Focus-focus singularities
- 7 Examples of focus-focus systems
- 8 Almost toric manifolds
- 9 Surgery
- 10 Elliptic and cusp singularities
- Appendix A Symplectic linear algebra
- Appendix B Lie derivatives
- Appendix C Complex projective spaces
- Appendix D Cotangent bundles
- Appendix E Moser’s argument
- Appendix F Toric varieties revisited
- Appendix G Visible contact hypersurfaces and Reeb flows
- Appendix H Tropical Lagrangian submanifolds
- Appendix I Markov triples
- Appendix J Open problems
- References
- Index
Summary
We first introduce integrable Hamiltonian systems on symplectic manifolds. We show that if a Hamiltonian system on a two–dimensional phase space has all of its orbits closed then we can modify the Hamiltonian by a diffeomorphism to ensure all the orbits have the same period. The rest of the chapter explains how to generalise this to Hamiltonian systems with more degrees of freedom, culminating in the Arnold–Liouville theorem, which underpins everything else in the book.
- Type
- Chapter
- Information
- Lectures on Lagrangian Torus Fibrations , pp. 1 - 16Publisher: Cambridge University PressPrint publication year: 2023