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Tautological one-form

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 mecha

Poisson–Lie group

In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold. The infinitesimal

Weinstein conjecture

In mathematics, the Weinstein conjecture refers to a general existence problem for periodic orbits of Hamiltonian or Reeb vector flows. More specifically, the conjecture claims that on a compact conta

Coadjoint representation

In mathematics, the coadjoint representation of a Lie group is the dual of the adjoint representation. If denotes the Lie algebra of , the corresponding action of on , the dual space to , is called th

Poisson algebra

In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally

Delzant's theorem

In mathematics, a Delzant polytope is a convex polytope in such for each vertex , exactly edges meet at , and these edges form a collection of vectors that form a -basis of . Delzant's theorem, introd

Moment map

In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used

Glossary of symplectic geometry

This is a glossary of properties and concepts in symplectic geometry in mathematics. The terms listed here cover the occurrences of symplectic geometry both in topology as well as in algebraic geometr

Hamiltonian vector field

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 Willia

Symplectic frame bundle

In symplectic geometry, the symplectic frame bundle of a given symplectic manifold is the canonical principal -subbundle of the tangent frame bundle consisting of linear frames which are symplectic wi

Lefschetz manifold

In mathematics, a Lefschetz manifold is a particular kind of symplectic manifold , sharing a certain cohomological property with Kähler manifolds, that of satisfying the conclusion of the Hard Lefsche

Poisson manifold

In differential geometry, a Poisson structure on a smooth manifold is a Lie bracket (called a Poisson bracket in this special case) on the algebra of smooth functions on , subject to the Leibniz rule

Symplectic spinor bundle

In differential geometry, given a metaplectic structure on a -dimensional symplectic manifold the symplectic spinor bundle is the Hilbert space bundle associated to the metaplectic structure via the m

Non-squeezing theorem

The non-squeezing theorem, also called Gromov's non-squeezing theorem, is one of the most important theorems in symplectic geometry. It was first proven in 1985 by Mikhail Gromov. The theorem states t

Symplectic basis

In linear algebra, a standard symplectic basis is a basis of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form , such that . A symplectic basis of a sym

Musical isomorphism

In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle and the cotangent bundle of a pseudo-Riemanni

Hamilton–Jacobi equation

In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such a

Moyal bracket

In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product. The Moyal bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeed

Poisson ring

In mathematics, a Poisson ring is a commutative ring on which an anticommutative and distributive binary operation satisfying the Jacobi identity and the product rule is defined. Such an operation is

Symplectic matrix

In mathematics, a symplectic matrix is a matrix with real entries that satisfies the condition where denotes the transpose of and is a fixed nonsingular, skew-symmetric matrix. This definition can be

Sasakian manifold

In differential geometry, a Sasakian manifold (named after Shigeo Sasaki) is a contact manifold equipped with a special kind of Riemannian metric , called a Sasakian metric.

Darboux's theorem

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 foundation

Mirror symmetry (string theory)

In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds lo

Symplectic vector space

In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping ω : V ×

Carathéodory–Jacobi–Lie theorem

The Carathéodory–Jacobi–Lie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem.

Poisson bracket

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 e

Symplectic category

In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds L into

Symplectic vector field

In physics and mathematics, a symplectic vector field is one whose flow preserves a symplectic form. That is, if is a symplectic manifold with smooth manifold and symplectic form , then a vector field

Symplectic representation

In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (V, ω) which preserves the symplectic form ω. H

Geometric mechanics

Geometric mechanics is a branch of mathematics applying particular geometric methods to many areas of mechanics, from mechanics of particles and rigid bodies to fluid mechanics to control theory. Geom

Geodesics as Hamiltonian flows

In mathematics, the geodesic equations are second-order non-linear differential equations, and are commonly presented in the form of Euler–Lagrange equations of motion. However, they can also be prese

Fubini–Study metric

In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CPn endowed with a Hermitian form. This metric was originally described i

Symplectization

In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.

Contact type

In mathematics, more precisely in symplectic geometry, a hypersurface of a symplectic manifold is said to be of contact type if there is 1-form such that and is a contact manifold, where is the natura

Thomas–Yau conjecture

In mathematics, and especially symplectic geometry, the Thomas–Yau conjecture asks for the existence of a stability condition, similar to those which appear in algebraic geometry, which guarantees the

Almost symplectic manifold

In differential geometry, an almost symplectic structure on a differentiable manifold is a two-form on that is everywhere non-singular. If in addition is closed then it is a symplectic form. An almost

Fukaya category

In symplectic topology, a Fukaya category of a symplectic manifold is a category whose objects are Lagrangian submanifolds of , and morphisms are Floer chain groups: . Its finer structure can be descr

Symplectic space

No description available.

Fundamental vector field

In the study of mathematics and especially differential geometry, fundamental vector fields are an instrument that describes the infinitesimal behaviour of a smooth Lie group action on a smooth manifo

Metaplectic structure

In differential geometry, a metaplectic structure is the symplectic analog of spin structure on orientable Riemannian manifolds. A metaplectic structure on a symplectic manifold allows one to define t

Gerstenhaber algebra

In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combin

Lagrangian foliation

In mathematics, a Lagrangian foliation or polarization is a foliation of a symplectic manifold, whose leaves are Lagrangian submanifolds. It is one of the steps involved in the geometric quantization

Symplectic geometry

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Sy

Poisson supermanifold

In differential geometry a Poisson supermanifold is a differential supermanifold M such that the supercommutative algebra of smooth functions over it (to clarify this: M is not a point set space and s

Langevin dynamics

In physics, Langevin dynamics is an approach to the mathematical modeling of the dynamics of molecular systems. It was originally developed by French physicist Paul Langevin. The approach is character

Lagrangian Grassmannian

In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is 1/2n(n + 1) (where the dimension of V is 2n). It may be

Perfect obstruction theory

In algebraic geometry, given a Deligne–Mumford stack X, a perfect obstruction theory for X consists of: 1.
* a perfect two-term complex in the derived category of quasi-coherent étale sheaves on X, a

Duistermaat–Heckman formula

In mathematics, the Duistermaat–Heckman formula, due to Duistermaat and Heckman, states that the pushforward of the canonical (Liouville) measure on a symplectic manifold under the moment map is a pie

Poisson superalgebra

In mathematics, a Poisson superalgebra is a Z2-graded generalization of a Poisson algebra. Specifically, a Poisson superalgebra is an (associative) superalgebra A with a Lie superbracket such that (A,

Lie bialgebra

In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the multiplication

Kähler manifold

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure.

Arnold Conjecture

No description available.

Phase-space formulation

The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position or momentum represen

Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, F) and Sp(n) for positive integer n and field F (usually C

Batalin–Vilkovisky formalism

In theoretical physics, the Batalin–Vilkovisky (BV) formalism (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the ghost structure for Lagrangian gauge theorie

Lie bialgebroid

A Lie bialgebroid is a mathematical structure in the area of non-Riemannian differential geometry. In brief a Lie bialgebroid are two compatible Lie algebroids defined on dual vector bundles. They for

Spectral invariants

In symplectic geometry, the spectral invariants are invariants defined for the group of Hamiltonian diffeomorphisms of a symplectic manifold, which is closed related to Floer theory and Hofer geometry

Symplectic manifold

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 sy

Non-autonomous mechanics

Non-autonomous mechanics describe non-relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose Lagrangians and Hamiltonian

Canonical coordinates

In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates a

Dirac bracket

The Dirac bracket is a generalization of the Poisson bracket developed by Paul Dirac to treat classical systems with second class constraints in Hamiltonian mechanics, and to thus allow them to underg

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