Symplectic Geometry of Integrable Hamiltonian Systems [Paperback]

Among all the Hamiltonian systems, the integrable ones have special geometric properties; in particular, their solutions are very regular and quasi-periodic. This book serves as an introduction to symplectic and contact geometry for graduate students, exploring the underlying geometry of integrable Hamiltonian systems. Includes exercises designed to complement the expositiont, and up-to-date references.

Among all the Hamiltonian systems, the integrable ones have special geometric properties; in particular, their solutions are very regular and quasi-periodic. The quasi-periodicity of the solutions of an integrable system is a result of the fact that the system is invariant under a (semi-global) torus action. It is thus natural to investigate the symplectic manifolds that can be endowed with a (global) torus action. This leads to symplectic toric manifolds (Part B of this book). Physics makes a surprising come-back in Part A: to describe Mirror Symmetry, one looks for a special kind of Lagrangian submanifolds and integrable systems, the special Lagrangians. Furthermore, integrable Hamiltonian systems on punctured cotangent bundles are a starting point for the study of contact toric manifolds (Part C of this book).

A Lagrangian Submanifolds.- I Lagrangian and special Lagrangian immersions in C.- I.1 Symplectic form on C, symplectic vector spaces.- Ll.a Symplectic vector spaces.- I.l.b Symplectic bases.- I.l.c The symplectic form as a differential form.- I.l.d The symplectic group.- I.l.e Orthogonality, isotropy.- 1.2 Lagrangian subspaces.- I.2.a Definition of Lagrangian subspaces.- I.2.b The symplectic reduction.- 1.3 The Lagrangian Grassmannian.- I.3.a The Grassmannian At as a homogeneous space.- I.3.b The manifold An.- I.3.c The tautological vector bundle.- I.3.d The tangent bundle to A.- I.3.e The case of oriented Lagrangian subspaces.- I.3.f The determinant and the Maslov class.- I.4 Lagrangian slÓ4