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In mathematics, a symplectic manifold is a smooth manifold M equipped with a closed, nondegenerate, 2-form ω called the symplectic form. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, especially Vector calculus and Differential topology, a closed form is a Differential form α whose differential is In Mathematics, specifically Linear algebra, a degenerate Bilinear form f(xy on a Vector space V is one such that In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic geometry is a branch of differential topology/geometry which studies Symplectic manifolds that is Differentiable manifolds equipped with a Symplectic geometry is a branch of differential topology/geometry which studies Symplectic manifolds that is Differentiable manifolds equipped with a Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds, e. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Analytical mechanics is a term used for a refined highly mathematical form of Classical mechanics, constructed from the Eighteenth century onwards as a formulation In Mathematics, especially Differential geometry, the cotangent bundle of a Smooth manifold is the Vector bundle of all the Cotangent spaces g. 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 modelled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a 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 the Hamilton–Jacobi equations. In Mathematics and Physics, a Hamiltonian vector field on a Symplectic manifold is a Vector field, defined for any energy function In Mathematics, an integral curve for a Vector field defined on a Manifold is a curve in the manifold whose tangent vector (i In Physics, the Hamilton–Jacobi equation (HJE is a reformulation of Classical mechanics and thus equivalent to other formulations such as Newton's laws of The Hamiltonian vector field defines a flow on the symplectic manifold, called a Hamiltonian flow or symplectomorphism. In Mathematics, a symplectomorphism is an Isomorphism in the category of Symplectic manifolds Formal definition Specifically By Liouville's theorem, Hamiltonian flows preserve the volume form on the phase space. In Physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian

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Definition

A symplectic form on a manifold M is a nondegenerate closed two-form ω. Explicitly, nondegeneracy of the form means that, relative to any given basis Xi of the tangent space of M at a point, the matrix

Ωij = ω(Xi,Xj)

is nonsingular (meaning that its determinant is non-zero). Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n Note that Ω, being a skew-symmetric non-singular matrix, must have an even number of rows and columns. Thus the dimension of M is necessarily an even number 2n. In intrinsic terms, ω is nondegenerate if and only if its n-th exterior power is non-zero:

\omega^{\wedge n} \not= 0.

Furthermore, ω is required to be closed, meaning that

dω = 0

where d is the exterior derivative. In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms

Linear symplectic manifold

There is a standard 'local' model, namely R2n with

ωi,n + i = 1 and ωn + i,i = − 1 for i = 1, . . . , n,
ωj,k = 0 for all j, k = 1, . . . , n with j ≠ k − n and j ≠ k + n.

This is an example of a linear symplectic space. See symplectic vector space. In Mathematics, a symplectic vector space is a Vector space V equipped with a Nondegenerate, Skew-symmetric, Bilinear form A proposition known as Darboux's theorem says that locally any symplectic manifold resembles this simple one. Darboux's theorem is a Theorem in the mathematical field of Differential geometry and more specifically Differential forms, partially generalizing In Mathematics, a phenomenon is sometimes said to occur locally if roughly speaking it occurs on sufficiently small or arbitrarily small Neighborhoods

Volume form

Directly from the definition, one can show that every symplectic manifold M is of even dimension 2n and ωn is a nowhere vanishing form, the symplectic volume form. It follows that every symplectic manifold is canonically oriented and comes with a canonical measure, the Liouville measure (often normalized to be ωn / n!). A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with

Contact manifolds

Closely related to symplectic manifolds are the odd-dimensional manifolds known as contact manifolds. In Mathematics, contact geometry is the study of a geometric structure on Smooth manifolds given by a hyperplane distribution in the Tangent bundle Any 2n+1-dimensional contact manifold (M, α) gives rise to a 2n+2-dimensional symplectic manifold (M × R, d(et α)).

Lagrangian and other submanifolds

There are several natural geometric notions of submanifold of a symplectic manifold. In Mathematics, a submanifold of a Manifold M is a Subset S which itself has the structure of a manifold and for which the Inclusion There are symplectic submanifolds (potentially of any even dimension), where the symplectic form is required to induce a symplectic form on the submanifold. On isotropic submanifolds, the symplectic form restricts to zero, i. e. each tangent space is an isotropic subspace of the ambient manifold's tangent space. Similarly, if each tangent subspace to a submanifold is coisotropic (the dual of an isotropic subspace), the submanifold is called coisotropic.

The most important case of the above is that of Lagrangian submanifolds, which are isotropic submanifolds of maximal dimension, namely half the dimension of the ambient manifold. Lagrangian submanifolds arise naturally in many physical and geometric situations. 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. In Mathematics, a symplectomorphism is an Isomorphism in the category of Symplectic manifolds Formal definition Specifically

Special cases and generalizations

A symplectic manifold endowed with a metric that is compatible with the symplectic form is an almost Kähler manifold in the sense that the tangent bundle has an almost complex structure, but this need not be integrable. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Mathematics, an almost complex manifold is a Smooth manifold equipped with smooth Linear complex structure on each Tangent space. In Mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. In Mathematics, an almost complex manifold is a Smooth manifold equipped with smooth Linear complex structure on each Tangent space. In Mathematics, certain systems of Partial differential equations are usefully formulated from the point of view of their underlying geometric and algebraic structure in terms Symplectic manifolds are special cases of a Poisson manifold. In Mathematics, a Poisson manifold is a Differential manifold M such that the algebra C &infin( M) of Smooth functions The definition of a symplectic manifold requires that the symplectic form be non-degenerate everywhere, but if this condition is violated, the manifold may still be a Poisson manifold.

A multisymplectic manifold of degree k is a manifold equipped with a closed nondegenerate k-form. See F. Cantrijn, L. A. Ibort and M. de León, J. Austral. Math. Soc. Ser. A 66 (1999), no. 3, 303-330.

See also

References

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