Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. 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. Any real-valued differentiable function, H, on a symplectic manifold can serve as an energy function or Hamiltonian. Associated to any Hamiltonian is a Hamiltonian vector field; the integral curves of the Hamiltonian vector field are solutions to Hamilton's equations. The Hamiltonian vector field defines a flow on the symplectic manifold, called a Hamiltonian flow or symplectomorphism. By Liouville's theorem, Hamiltonian flows preserve the volume form on the phase space. Symplectic manifolds arise from classical mechanics, in particular, they are a generalization of the phase space of a closed system. 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 differential dH of a Hamiltonian function H. So we require a linear map TM → T∗M, or equivalently, an element of T∗M ⊗ T∗M. Letting ω denote a section of T∗M ⊗ T∗M, the requirement that ω be non-degenerate ensures that for every differential dH there is a unique corresponding vector field VH such that dH = ω(VH, · ). Since one desires the Hamiltonian to be constant along flow lines, one should have dH(VH) = ω(VH, VH) = 0, which implies that ω is alternating and hence a 2-form. Finally, one makes the requirement that ω should not change under flow lines, i.e. that the Lie derivative of ω along VH vanishes. Applying Cartan's formula, this amounts to (here ι X {displaystyle iota _{X}} is the interior product): so that, on repeating this argument for different smooth functions H {displaystyle H} such that the corresponding V H {displaystyle V_{H}} span the tangent space at each point the argument is applied at, we see that the requirement for the vanishing Lie derivative along flows of V H {displaystyle V_{H}} corresponding to arbitrary smooth H {displaystyle H} is equivalent to the requirement that ω should be closed. A symplectic form on a manifold M is a closed non-degenerate differential 2-form ω. Here, non-degenerate means that for all p ∈ M, if there exists an X ∈ TpM such that ω(X,Y) = 0 for all Y ∈ TpM, then X = 0. The skew-symmetric condition (inherent in the definition of differential 2-form) means that for all p ∈ M we have ω(X,Y) = −ω(Y,X) for all X,Y ∈ TpM. In odd dimensions, antisymmetric matrices are not invertible. Since ω is a differential two-form, the skew-symmetric condition implies that M has even dimension. The closed condition means that the exterior derivative of ω vanishes, dω = 0. A symplectic manifold consists of a pair (M,ω), of a manifold M and a symplectic form ω. Assigning a symplectic form ω to a manifold M is referred to as giving M a symplectic structure. There is a standard linear model, namely a symplectic vector space R 2 n . {displaystyle mathbb {R} ^{2n}.} Let { v 1 , … , v 2 n } {displaystyle {v_{1},ldots ,v_{2n}}} be a basis for R 2 n . {displaystyle mathbb {R} ^{2n}.} We define our symplectic form ω on this basis as follows: In this case the symplectic form reduces to a simple quadratic form. If In denotes the n × n identity matrix then the matrix, Ω, of this quadratic form is given by the 2n × 2n block matrix: There are several natural geometric notions of submanifold of a symplectic manifold.