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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).

Differential geometry. Global analysis --- Differential geometry. --- Manifolds (Mathematics). --- Complex manifolds. --- Physics. --- Differential Geometry. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Mathematical Methods in Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Analytic spaces --- Manifolds (Mathematics) --- Geometry, Differential --- Topology --- Differential geometry