In differential geometry, a subject of mathematics, a **symplectic manifold** is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , 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.

- Motivation
- Definition
- Examples
- Symplectic vector spaces
- Cotangent bundles
- Kähler manifolds
- Almost-complex manifolds
- Lagrangian and other submanifolds
- Examples 2
- Special Lagrangian submanifolds
- Lagrangian fibration
- Lagrangian mapping
- Special cases and generalizations
- See also
- Notes
- References
- External links

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 differential *dH* of a Hamiltonian function *H*.^{ [2] } So we require a linear map *TM* → *T*^{∗}*M* from the tangent manifold *TM* to the cotangent manifold *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 *V _{H}* such that

so that, on repeating this argument for different smooth functions such that the corresponding 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 corresponding to arbitrary smooth is equivalent to the requirement that *ω* should be closed.

A **symplectic form** on a smooth manifold is a closed non-degenerate differential 2-form .^{ [3] }^{ [4] } Here, non-degenerate means that for every point , the skew-symmetric pairing on the tangent space defined by is non-degenerate. That is to say, if there exists an such that for all , then . Since in odd dimensions, skew-symmetric matrices are always singular, the requirement that be nondegenerate implies that has an even dimension.^{ [3] }^{ [4] } The closed condition means that the exterior derivative of vanishes. A **symplectic manifold** is a pair where is a smooth manifold and is a symplectic form. Assigning a symplectic form to is referred to as giving a **symplectic structure**.

Let be a basis for We define our symplectic form *ω* on this basis as follows:

In this case the symplectic form reduces to a simple quadratic form. If *I _{n}* denotes the

Let be a smooth manifold of dimension . Then the total space of the cotangent bundle has a natural symplectic form, called the Poincaré two-form or the canonical symplectic form

Here are any local coordinates on and are fibrewise coordinates with respect to the cotangent vectors . Cotangent 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 momenta are "soldered" to the velocities . The 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 Kähler manifold is a symplectic manifold equipped with a compatible integrable complex structure. They form a particular class of complex manifolds. A large class of examples come from complex algebraic geometry. Any smooth complex projective variety has a symplectic form which is the restriction of the Fubini—Study form on the projective space .

Riemannian manifolds with an -compatible almost complex structure are termed almost-complex manifolds. They generalize Kähler manifolds, in that they need not be integrable. That is, they do not necessarily arise from a complex structure on the manifold.

There are several natural geometric notions of submanifold of a symplectic manifold :

**Symplectic submanifolds**of (potentially of any even dimension) are submanifolds such that is a symplectic form on .

**Isotropic submanifolds**are submanifolds where 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 co-isotropic (the dual of an isotropic subspace), the submanifold is called**co-isotropic**.

**Lagrangian submanifolds**of a symplectic manifold are submanifolds where the restriction of the symplectic form to is vanishing, i.e. and . Lagrangian submanifolds are the maximal isotropic submanifolds. In physics, Lagrangian submanifolds are frequently called branes.

The most important case of the isotropic submanifolds is that of **Lagrangian submanifolds**. A Lagrangian submanifold is, by definition, an isotropic submanifold of maximal dimension, namely half the dimension of the ambient 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.

Let have global coordinates labelled Then, we can equip with the canonical symplectic form

There is a standard Lagrangian submanifold given by . The form vanishes on because given any pair of tangent vectors we have that To elucidate, consider the case . Then, and Notice that when we expand this out

both terms we have a factor, which is 0, by definition.

The cotangent bundle of a manifold is locally modeled on a space similar to the first example. It can be shown that we can glue these affine symplectic forms hence this bundle forms a symplectic manifold. A less trivial example of a Lagrangian submanifold is the zero section of the cotangent bundle of a manifold. For example, let

Then, we can present as

where we are treating the symbols as coordinates of We can consider the subset where the coordinates and , giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions and their differentials .

Consider the canonical space with coordinates . A parametric submanifold of is one that is parameterized by coordinates such that

This manifold is a Lagrangain submanifold if the Lagrange bracket vanishes for all That is, it is Lagrangian if

for all This can be seen by expanding

in the condition for a Lagrangian submanifold . This is that the symplectic form must vanish on the tangent manifold ; that is, it must vanish for all tangent vectors:

for all . Simplify the result by making use of the canonical symplectic form on :

and all others vanishing.

As 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. In physics, the action describes the time evolution of a physical system; here, it can be taken as the description of the dynamics of branes.

Another useful class of Lagrangian submanifolds occur in Morse theory. Given a Morse function and for a small enough one can construct a Lagrangian submanifold given by the vanishing locus . For a generic Morse function we have a Lagrangian intersection given by .

In the case of Kahler manifolds (or Calabi–Yau manifolds) we can make a choice on as a holomorphic n-form, where is the real part and imaginary. A Lagrangian submanifold is called **special** if in addition to the above Lagrangian condition the restriction to is vanishing. In other words, the real part restricted on leads the volume form on . The following examples are known as special Lagrangian submanifolds,

- complex Lagrangian submanifolds of hyperKahler manifolds,
- fixed points of a real structure of Calabi–Yau manifolds.

The SYZ conjecture has been proved for special Lagrangian submanifolds but in general, it is open, and brings a lot of impacts to the study of mirror symmetry. see ( Hitchin 1999 )

A **Lagrangian fibration** of a symplectic manifold *M* is a fibration where all of the fibres are Lagrangian submanifolds. Since *M* is even-dimensional we can take local coordinates (*p*_{1},…,*p*_{n}, *q*^{1},…,*q*^{n}), and by Darboux's theorem the symplectic form *ω* can be, at least locally, written as *ω* = ∑ d*p*_{k}∧ d*q*^{k}, where d denotes the exterior derivative and ∧ denotes the exterior product. This form is called the Poincaré two-form or the canonical two-form. Using this set-up we can locally think of *M* as being the cotangent bundle and the Lagrangian fibration as the trivial fibration This is the canonical picture.

Let *L* be a Lagrangian submanifold of a symplectic manifold (*K*,ω) given by an immersion *i* : *L* ↪ *K* (*i* is called a **Lagrangian immersion**). Let *π* : *K* ↠ *B* give a Lagrangian fibration of *K*. The composite (*π* ∘ *i*) : *L* ↪ *K* ↠ *B* is a **Lagrangian mapping**. The critical value set of *π* ∘ *i* is called a caustic.

Two Lagrangian maps (*π*_{1} ∘ *i*_{1}) : *L*_{1} ↪ *K*_{1} ↠ *B*_{1} and (*π*_{2} ∘ *i*_{2}) : *L*_{2} ↪ *K*_{2} ↠ *B*_{2} 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.^{ [4] } Symbolically:

where *τ*^{∗}*ω*_{2} denotes the pull back of *ω*_{2} by *τ*.

- A symplectic manifold is
**exact**if the symplectic form is exact. For example, the cotangent bundle of a smooth manifold is an exact symplectic manifold. The canonical symplectic form is exact.

- 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.

- Symplectic manifolds are special cases of a Poisson manifold. 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.^{ [5] }

- A
**polysymplectic manifold**is a Legendre bundle provided with a polysymplectic tangent-valued -form; it is utilized in Hamiltonian field theory.^{ [6] }

- Almost symplectic manifold
- Contact manifold − an odd-dimensional counterpart of the symplectic manifold.
- Fedosov manifold
- Poisson bracket – Operation in Hamiltonian mechanics
- Symplectic group – Mathematical group
- Symplectic matrix
- Symplectic topology
- Symplectic vector space
- Symplectomorphism
- Tautological one-form
- Wirtinger inequality (2-forms)
- Covariant Hamiltonian field theory

- ↑ Webster, Ben. "What is a symplectic manifold, really?".
- ↑ Cohn, Henry. "Why symplectic geometry is the natural setting for classical mechanics".
- 1 2 de Gosson, Maurice (2006).
*Symplectic Geometry and Quantum Mechanics*. Basel: Birkhäuser Verlag. p. 10. ISBN 3-7643-7574-4. - 1 2 3 Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M. (1985).
*The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1*. Birkhäuser. ISBN 0-8176-3187-9. - ↑ Cantrijn, F.; Ibort, L. A.; de León, M. (1999). "On the Geometry of Multisymplectic Manifolds".
*J. Austral. Math. Soc*. Ser. A.**66**(3): 303–330. doi: 10.1017/S1446788700036636 . - ↑ Giachetta, G.; Mangiarotti, L.; Sardanashvily, G. (1999). "Covariant Hamiltonian equations for field theory".
*Journal of Physics*.**A32**: 6629–6642. arXiv: hep-th/9904062 . doi:10.1088/0305-4470/32/38/302.

In vector calculus and differential geometry, the **generalized Stokes theorem**, also called the **Stokes–Cartan theorem**, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. It is a generalization of Isaac Newton's fundamental theorem of calculus that relates two-dimensional line integrals to three-dimensional surface integrals.

In mathematics, the name **symplectic group** can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2*n*, **F**) and Sp(*n*) for positive integer *n* and field **F**. The latter is called the **compact symplectic group**. Many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2*n*, **C**) is denoted *C _{n}*, and Sp(

**Hamiltonian mechanics** emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) *momenta*. Both theories provide interpretations of classical mechanics and describe the same physical phenomena.

In mathematics, especially differential geometry, the **cotangent bundle** of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories.

In mathematics and classical mechanics, the **Poisson bracket** is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called *canonical transformations*, which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself as one of the new canonical momentum coordinates.

In the calculus of variations and classical mechanics, the **Euler-Lagrange equations** is a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.

In mathematics, **contact geometry** is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem.

In mathematics, an **almost complex manifold** is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in symplectic geometry.

In differential geometry, a **Poisson structure** on a smooth manifold is a Lie bracket on the algebra of smooth functions on , subject to the Leibniz rule

In physics and mathematics, **supermanifolds** are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.

**Darboux's theorem** is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry. The theorem is named after Jean Gaston Darboux who established it as the solution of the Pfaff problem.

In mathematics and physics, a **Hamiltonian vector field** on a symplectic manifold is a vector field defined for any **energy function** or **Hamiltonian**. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics.

In mathematics, the **tautological one-form** is a special 1-form defined on the cotangent bundle of a manifold . In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between Lagrangian mechanics with Hamiltonian mechanics.

In differential geometry, a field of mathematics, a **normal bundle** is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding.

In differential geometry, a discipline within mathematics, a **distribution** on a manifold is an assignment of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle .

In mathematics, specifically in symplectic geometry, the **momentum map** is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including **symplectic** (**Marsden–Weinstein**) **quotients**, discussed below, and symplectic cuts and sums.

In mathematics, a **holomorphic vector bundle** is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : *E* → *X* is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A **holomorphic line bundle** is a rank one holomorphic vector bundle.

In the field of mathematics known as differential geometry, a **generalized complex structure** is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized complex structures were introduced by Nigel Hitchin in 2002 and further developed by his students Marco Gualtieri and Gil Cavalcanti.

In physics, particularly in quantum field theory, the **Weyl equation** is a relativistic wave equation for describing massless spin-1/2 particles called **Weyl fermions**. The equation is named after Hermann Weyl. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the Dirac and the Majorana fermions.

In geometry of normed spaces, the **Holmes–Thompson volume** is a notion of volume that allows to compare sets contained in different normed spaces. It was introduced by Raymond D. Holmes and Anthony Charles Thompson.

- McDuff, Dusa; Salamon, D. (1998).
*Introduction to Symplectic Topology*. Oxford Mathematical Monographs. ISBN 0-19-850451-9. - Auroux, Denis. "Seminar on Mirror Symmetry".
- Meinrenken, Eckhard. "Symplectic Geometry" (PDF).
- Abraham, Ralph; Marsden, Jerrold E. (1978).
*Foundations of Mechanics*. London: Benjamin-Cummings. See Section 3.2. ISBN 0-8053-0102-X. - de Gosson, Maurice A. (2006).
*Symplectic Geometry and Quantum Mechanics*. Basel: Birkhäuser Verlag. ISBN 3-7643-7574-4. - Alan Weinstein (1971). "Symplectic manifolds and their lagrangian submanifolds".
*Advances in Mathematics*.**6**(3): 329–46. doi: 10.1016/0001-8708(71)90020-X . - Arnold, V. I. (1990). "Ch.1, Symplectic geometry".
*Singularities of Caustics and Wave Fronts*. Mathematics and Its Applications.**62**. Dordrecht: Springer Netherlands. doi:10.1007/978-94-011-3330-2. ISBN 978-1-4020-0333-2. OCLC 22509804.

- "How to find Lagrangian Submanifolds".
*Stack Exchange*. December 17, 2014. - Lumist, Ü. (2001) [1994], "Symplectic Structure",
*Encyclopedia of Mathematics*, EMS Press - Sardanashvily, G. (2009). "Fibre bundles, jet manifolds and Lagrangian theory".
*Lectures for theoreticians*. arXiv: 0908.1886 . - McDuff, D. (November 1998). "Symplectic Structures—A New Approach to Geometry" (PDF).
*Notices of the AMS*. - Hitchin, Nigel (1999). "Lectures on Special Lagrangian Submanifolds". arXiv: math/9907034 .

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