Symplectic manifold

Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds.For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system.[1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differentialis the interior product): so that, on repeating this argument for different smooth functionsspan the tangent space at each point the argument is applied at, we see that the requirement for the vanishing Lie derivative along flows ofSince in odd dimensions, skew-symmetric matrices are always singular, the requirement thatCotangent bundles are the natural phase spaces of classical mechanics.The point of distinguishing upper and lower indexes is driven by the case of the manifold having a metric tensor, as is the case for Riemannian manifolds.Upper and lower indexes transform contra and covariantly under a change of coordinate frames.The phrase "fibrewise coordinates with respect to the cotangent vectors" is meant to convey that the momentaThe soldering is an expression of the idea that velocity and momentum are colinear, in that both move in the same direction, and differ by a scale factor.A large class of examples come from complex algebraic geometry.There are several natural geometric notions of submanifold of a symplectic manifold: One major example is that the graph of a symplectomorphism in the product symplectic manifold (M × M, ω × −ω) is Lagrangian.Their intersections display rigidity properties not possessed by smooth manifolds; the Arnold conjecture gives the sum of the submanifold's Betti numbers as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the Euler characteristic in the smooth case.with the canonical symplectic form There is a standard Lagrangian submanifold given byThe cotangent bundle of a manifold is locally modeled on a space similar to the first example.A less trivial example of a Lagrangian submanifold is the zero section of the cotangent bundle of a manifold.This example can be repeated for any manifold defined by the vanishing locus of smooth functionsSimplify the result by making use of the canonical symplectic form onAs local charts on a symplectic manifold take on the canonical form, this example suggests that Lagrangian submanifolds are relatively unconstrained.The classification of symplectic manifolds is done via Floer homology—this is an application of Morse theory to the action functional for maps between Lagrangian submanifolds.Another useful class of Lagrangian submanifolds occur in Morse theory.is called special if in addition to the above Lagrangian condition the restrictionThe following examples are known as special Lagrangian submanifolds, The SYZ conjecture deals with the study of special Lagrangian submanifolds in mirror symmetry; see (Hitchin 1999).The Thomas–Yau conjecture predicts that the existence of a special Lagrangian submanifolds on Calabi–Yau manifolds in Hamiltonian isotopy classes of Lagrangians is equivalent to stability with respect to a stability condition on the Fukaya category of the manifold.Since M is even-dimensional we can take local coordinates (p1,...,pn, q1,...,qn), and by Darboux's theorem the symplectic form ω can be, at least locally, written as ω = ∑ dpk ∧ dqk, where d denotes the exterior derivative and ∧ denotes the exterior product.The critical value set of π ∘ i is called a caustic.Two Lagrangian maps (π1 ∘ i1) : L1 ↪ K1 ↠ B1 and (π2 ∘ i2) : L2 ↪ K2 ↠ B2 are called Lagrangian equivalent if there exist diffeomorphisms σ, τ and ν such that both sides of the diagram given on the right commute, and τ preserves the symplectic form.