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 ×
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
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.
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