In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. In a more general setting, the Poisson bracket is used to define a Poisson algebra, of which the Poisson manifolds are a special case. These are all named in honour of Siméon-Denis Poisson. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... A Poisson algebra is an associative algebra together with a Lie bracket, satisfying Leibniz law. ... A Poisson manifold is a differential manifold M such that the algebra of smooth functions over it, is equipped with a bilinear map called the Poisson bracket turning it into a Poisson algebra. ... Siméon Poisson. ...

Definition

Let M be symplectic manifold, that is, a manifold on which there exists a symplectic form: a 2-form ω which is both closed (dω = 0) and non-degenerate, in the following sense: when viewed as a map , ω is invertible to obtain . Here d is the exterior derivative operation intrinsic to the manifold structure of M, and i_ξθ is the inner derivative or contraction operation, which is equivalent to θ(ξ) on 1-forms θ. In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, the inner derivative is a grade −1 derivation on the exterior algebra of differential forms on a differential manifold. ... In multilinear algebra, a tensor contraction is a sum of products of scalar components of one or more tensors caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression. ...

Using the axioms of the exterior calculus, one can derive: This article may be too technical for most readers to understand. ...

i_[v,w]ω = d(i_vi_wω) + i_vd(i_wω) − i_wd(i_vω) − i_wi_vdω

Here [v,w] denotes the Lie bracket on smooth vector fields, whose properties essentially define the manifold structure of M. In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented...

If v is such that d(i_vω) = 0, we may call it ω-coclosed (or just coclosed). Similarly, if i_vω = df for some function f, we may call v ω-coexact (or just coexact). Given that dω = 0, the expression above implies that the Lie bracket of two coclosed vector fields is always a coexact vector field, because when v and w are both coclosed, the only nonzero term in the expression is d(i_vi_wω). And because the exterior derivative obeys , all coexact vector fields are coclosed; so the Lie bracket is closed both on the space of coclosed vector fields and on the subspace within it consisting of the coexact vector fields. In the language of abstract algebra, the coclosed vector fields form a subalgebra of the Lie algebra of smooth vector fields on M, and the coexact vector fields form an algebraic ideal of this subalgebra. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to A. Since the axioms of algebraic structures in universal algebra are described by equational laws, the... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...

Given the existence of the inverse map , every smooth real-valued function f on M may be associated with a coexact vector field . (Two functions are associated with the same vector field if and only if their difference is in the kernel of d, i. e., constant on each connected component of M.) We therefore define the Poisson bracket on (M,ω), a bilinear operation on differentiable functions, under which the (smooth) functions form an algebra. It is given by: In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... Partial plot of a function f. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ...

The skew-symmetry of the Poisson bracket is ensured by the axioms of the exterior calculus and the condition dω = 0. Because the map is pointwise linear and skew-symmetric in this sense, some authors associate it with a bivector, which is not an object often encountered in the exterior calculus. In this form it is called the Poisson bivector or the Poisson structure on the symplectic manifold, and the Poisson bracket written simply . This article may be too technical for most readers to understand. ... THE TRIVECTOR: the sum of all coolness encapsulated in three rocking houses surrounding victoria park, located in Kingston Ontario Canada. ... What is the poisson curve? In mathematics, a Poisson manifold is a differential manifold M such that the algebra of smooth functions over it, is equipped with a bilinear map called the Poisson bracket, turning it into a Poisson algebra. ... In mathematics, a Poisson manifold is a differential manifold M such that the algebra of smooth functions over it, is equipped with a bilinear map called the Poisson bracket, turning it into a Poisson algebra. ...

The Poisson bracket on smooth functions corresponds to the Lie bracket on coexact vector fields and inherits its properties. It therefore satisfies the Jacobi identity: In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. ...

{f,{g,h}} + {g,{h,f}} + {h,{f,g}} = 0

The Poisson bracket {f,_} with respect to a particular scalar field f corresponds to the Lie derivative with respect to . Consequently, it is a derivation; that is, it satisfies Leibniz' law: In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ... In mathematics, the product rule of calculus, which is also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...

{f,gh} = {f,g}h + g{f,h}

It is a fundamental property of manifolds that the commutator of the Lie derivative operations with respect to two vector fields is equivalent to the Lie derivative with respect to some vector field, namely, their Lie bracket. The parallel role of the Poisson bracket is apparent from a rearrangement of the Jacobi identity: In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ...

{f,{g,h}} − {g,{f,h}} = {{f,g},h}

If the Poisson bracket of f and g vanishes ({f,g} = 0), then f and g are said to be in mutual involution, and the operations of taking the Poisson bracket with respect to f and with respect to g commute.

Canonical coordinates

In canonical coordinates (qⁱ,p_j) on the phase space, the Poisson bracket takes the form In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ... Phase space of a dynamical system with focal stability. ...

Equations of motion

The Hamilton-Jacobi equations of motion have an equivalent expression in terms of the Poisson bracket. This may be most directly demonstrated in an explicit coordinate frame. Suppose that f(p,q,t) is a function on the manifold. Then one has In physics and mathematics, the Hamilton-Jacobi equation (HJE) is a reformulation of classical mechanics and, thus, equivalent to other formulations such as Newtons laws of motion, Lagrangian mechanics and Hamiltonian mechanics. ...

Then, by taking p = p(t) and q = q(t) to be solutions to the Hamilton-Jacobi equations and , one may write

Thus, the time evolution of a function f on a symplectic manifold can be given as a one-parameter family of symplectomorphisms, with the time t being the parameter. Dropping the coordinates, one has In mathematics, flow refers to the group action of a one-parameter group on a set. ... In mathematics, a symplectomorphism (or Hamiltonian flow) is an isomorphism in the category of symplectic manifolds. ...

The operator is known as the Liouvillian. In mathematics, the Liouvillian is another expression for the Liouville differential operator given by the mathematical expression: See also In Statistical Mechanics, see Liouville equation. ...

Constants of motion

An integrable dynamical system will have constants of motion in addition to the energy. Such constants of motion will commute with the Hamiltonian under the Poisson bracket. Suppose some function f(p,q) is a constant of motion. This implies that if p(t),q(t) is a trajectory or solution to the Hamilton-Jacobi equations of motion, then one has that 0 = df / dt along that trajectory. Then one has In mathematics and physics, an integrable system refers to a system of partial differential equations that may be integrated to obtain the solutions to the equations. ... In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. ... In physics, Hamiltonian has distinct but closely related meanings. ... In physics and mathematics, the Hamilton-Jacobi equation (HJE) is a reformulation of classical mechanics and, thus, equivalent to other formulations such as Newtons laws of motion, Lagrangian mechanics and Hamiltonian mechanics. ...

where, as above, the intermediate step follows by applying the equations of motion. This equation is known as the Liouville equation. The content of Liouville's theorem is that the time evolution of a measure (or "distribution function" on the phase space) is given by the above. The Liouville equation is the most important equation of Statistical Mechanics. ... Liouvilles theorem has various meanings: In complex analysis, see Liouvilles theorem (complex analysis). ... In mathematics, a measure is a function that assigns a number, e. ... In physics, a particles distribution function is a function of seven variables, , which gives the number of particles per unit volume in phase space. ...

In order for a Hamiltonian system to be completely integrable, all of the constants of motion must be in mutual involution. In mathematics and physics, an integrable system refers to a system of partial differential equations that may be integrated to obtain the solutions to the equations. ...

Lie algebra

The Poisson brackets are anticommutative. Note also that they satisfy the Jacobi identity. This makes the space of smooth functions on a symplectic manifold an infinite-dimensional Lie algebra with the Poisson bracket acting as the Lie bracket. The corresponding Lie group is the group of symplectomorphisms of the symplectic manifold (also known as canonical transformations). A mathematical operator (typically a binary operator, represented by *) is anticommutative iff it is true that x * y = −(y * x) for all x and y on the operators valid domain (e. ... In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...

Given a differentiable vector field X on the tangent bundle, let P_X be its conjugate momentum. The conjugate momentum mapping is a Lie algebra anti-homomorphism from the Poisson bracket to the Lie bracket: Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

This important result is worth a short proof. Write a vector field X at point q in the configuration space as In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ...

where the is the local coordinate frame. The conjugate momentum to X has the expression

where the p_i are the momentum functions conjugate to the coordinates. One then has, for a point (q,p) in the phase space, Phase space of a dynamical system with focal stability. ...

The above holds for all (q,p), giving the desired result.

See also

• Lagrange bracket
• Moyal product
• Peierls bracket
• superPoisson algebra
• superPoisson bracket

In mathematics, the Moyal product is an example for an associative, non-commutative product on the functions of a Poisson manifold. ... In theoretical physics, the Peierls bracket is an equivalent description of the Poisson bracket. ... A Poisson superalgebra A is a Z2-graded algebra with two products, . and [,] (which both share the same grading) such that . ...

References