Symplectic vector field

Last updated February 26, 2019

In physics and mathematics, a symplectic vector field is one whose flow preserves a symplectic form. That is, if $(M,\omega )$ is a symplectic manifold with smooth manifold $M$ and symplectic form $\omega$ , then a vector field $X\in {\mathfrak {X}}(M)$ in the Lie algebra ${\mathfrak {X}}(M)$ is symplectic if its flow preserves the symplectic structure. In other words, the Lie derivative of the vector field must vanish:

Physics study of matter and its motion, along with related concepts such as energy and force

Physics is the natural science that studies matter and its motion and behavior through space and time and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

Mathematics includes the study of such topics as quantity, structure, space, and change.

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.

{\mathcal {L}}_{X}\omega =0

.^[1]

An alternative definition is that a vector field is symplectic if its interior product with the symplectic form is closed.^[1] (The interior product gives a map from vector fields to 1-forms, which is an isomorphism due to the nondegeneracy of a symplectic 2-form.) The equivalence of the definitions follows from the closedness of the symplectic form and Cartan's magic formula for the Lie derivative in terms of the exterior derivative.

In mathematics, an isomorphism is a homomorphism or morphism that can be reversed by an inverse morphism. Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.

In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field, along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.

On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899; it allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus.

If the interior product of a vector field with the symplectic form is an exact form (and in particular, a closed form), then it is called a Hamiltonian vector field. If the first De Rham cohomology group $H^{1}(M)$ of the manifold is trivial, all closed forms are exact, so all symplectic vector fields are Hamiltonian. That is, the obstruction to a symplectic vector field being Hamiltonian lives in $H^{1}(M)$ . In particular, symplectic vector fields on simply connected manifolds are Hamiltonian.

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.

De Rham cohomology cohomology with real coefficients computed using differential forms

In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties.

In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.

The Lie bracket of two symplectic vector fields is Hamiltonian, and thus the collection of symplectic vector fields and the collection of Hamiltonian vector fields both form Lie algebras.

In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted [X, Y].

Lie algebra A vector space with an alternating binary operation satisfying the Jacobi identity.

In mathematics, a Lie algebra is a vector space $together with a non-associative, alternating bilinear map, called the Lie bracket, satisfying the Jacobi identity.$

Related Research Articles

Symplectic group The group of matrices preserving a non-degenerate alternating quadratic form

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted $Sp(2 n, F)$ and $Sp(n)$ . 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 matrices used to 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 $Sp(n)$ is the compact real form of $Sp(2 n, C)$ . Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension $n$ .

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.

In mathematics, a symplectic vector space is a vector space V over a field F equipped with a symplectic bilinear form.

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 mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phase space, and is called a canonical transformation.

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 in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson–Lie groups are a special case. The algebra is named in honour of Siméon Denis Poisson.

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 differential geometry, the curvature form describes the curvature of a connection on a principal bundle. It can be considered as an alternative to or a generalization of the curvature tensor in Riemannian geometry.

In 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.

In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.

In mathematics, a volume form on a differentiable manifold is a top-dimensional form. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωⁿ(M) = ⋀ⁿ(T^∗M). A manifold admits a nowhere vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density.

In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T*Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system to it's momentum, thus providing a bridge between Lagrangian mechanics with Hamiltonian mechanics.

In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In mathematics, the interior product is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, should not be confused with an inner product. The interior product ι_Xω is sometimes written as X ⨼ ω.

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 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 manifold. Such vector fields find important applications in the study of Lie theory, symplectic geometry, and the study of Hamiltonian group actions.

In differential geometry, an equivariant differential form on a manifold M acted by a Lie group G is a polynomial map

References

1 2 Cannas da Silva, Ana (2001), Lectures on Symplectic Geometry, Lecture Notes in Mathematics, 1764, Springer-Verlag, p. 106, ISBN 978-3-540-42195-5 .

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[CdS-1] 1 2 Cannas da Silva, Ana (2001), Lectures on Symplectic Geometry, Lecture Notes in Mathematics, 1764, Springer-Verlag, p. 106, ISBN 978-3-540-42195-5 .