Symplectic geometry

"Complex" comes from the Latin com-plexus, meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek sym-plektikos (συμπλεκτικός); in both cases the stem comes from the Indo-European root *pleḱ- The name reflects the deep connections between complex and symplectic structures.A symplectic geometry is defined on a smooth even-dimensional space that is a differentiable manifold.To specify the trajectory of the object, one requires both the position q and the momentum p, which form a point (p,q) in the Euclidean planeAnother difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are certain topological restrictions.Mikhail Gromov, however, made the important observation that symplectic manifolds do admit an abundance of compatible almost complex structures, so that they satisfy all the axioms for a Kähler manifold except the requirement that the transition maps be holomorphic.
Phase portrait of the Van der Pol oscillator , a one-dimensional system. Phase space was the original object of study in symplectic geometry.
Phase portraitVan der Pol oscillatorPhase spacedifferential geometrydifferential topologysymplectic manifoldsdifferentiable manifoldsclosednondegenerate2-formHamiltonian formulationclassical mechanicsHermann WeylcalqueDarboux's theoremsymplectic vector spacedifferentiable manifoldmetric tensorRiemannian geometrypositionmomentumEuclidean planearea formconservative dynamical systemsmetric tensorscurvatureorientablede Rham cohomologyn-sphere2-spheregeodesicspseudoholomorphic curvesKähler manifoldWilliam ThurstonRobert Gompffinitely presented groupfundamental groupcomplex structureMikhail Gromovalmost complex structurestransition mapsholomorphicGromov–Witten invariantsAndreas FloerFloer homologyContact geometryGeometric mechanicsMoment mapPoisson geometrySymplectic integrationQuanta MagazineCiteSeerXAbraham, RalphMarsden, Jerrold E.Russian Mathematical SurveysMcDuff, DusaSalamon, D.de Gosson, Maurice A.Weinstein, AlanBulletin of the American Mathematical SocietyWeyl, HermannPrinceton University PressEncyclopedia of MathematicsEMS Press