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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 (below symbolized by and , respectively) 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 a more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case. There are other general examples, as well: it occurs in the theory of Lie algebras, where the tensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the universal enveloping algebra article. Quantum deformations of the universal enveloping algebra lead to the notion of quantum groups.

All of these objects are named in honor of Siméon Denis Poisson. He introduced the Poisson bracket in his 1809 treatise on mechanics.^{[1]}^{[2]}

Given two functions f and g that depend on phase space and time, their Poisson bracket is another function that depends on phase space and time. The following rules hold for any three functions of phase space and time:

Also, if a function is constant over phase space (but may depend on time), then for any .

In canonical coordinates (also known as Darboux coordinates) on the phase space, given two functions and ,^{[Note 1]} the Poisson bracket takes the form

The Poisson brackets of the canonical coordinates are

Hamilton's 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 is a function on the solution's trajectory-manifold. Then from the multivariable chain rule,

Further, one may take and to be solutions to Hamilton's equations; that is,

Then

Thus, the time evolution of a function on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e., canonical transformations, area-preserving diffeomorphisms), with the time being the parameter: Hamiltonian motion is a canonical transformation generated by the Hamiltonian. That is, Poisson brackets are preserved in it, so that *any time * in the solution to Hamilton's equations,

Dropping the coordinates,

The operator in the convective part of the derivative, , is sometimes referred to as the Liouvillian (see Liouville's theorem (Hamiltonian)).

The concept of Poisson brackets can be expanded to that of matrices by defining the Poisson matrix.

Consider the following canonical transformation:

The Poisson matrix satisfies the following known properties:

where the is known as a Lagrange matrix and whose elements correspond to Lagrange brackets. The last identity can also be stated as the following:

The invariance of Poisson bracket can be expressed as: , which directly leads to the symplectic condition: .^{[3]}

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 is a constant of motion. This implies that if is a trajectory or solution to Hamilton's equations of motion, then

If the Poisson bracket of and vanishes ( ), then and are said to be **in involution**. In order for a Hamiltonian system to be completely integrable, independent constants of motion must be in mutual involution, where is the number of degrees of freedom.

Furthermore, according to **Poisson's Theorem**, if two quantities and are explicitly time independent ( ) constants of motion, so is their Poisson bracket . This does not always supply a useful result, however, since the number of possible constants of motion is limited ( for a system with degrees of freedom), and so the result may be trivial (a constant, or a function of and .)

Let be a symplectic manifold, that is, a manifold equipped with a symplectic form: a 2-form which is both **closed** (i.e., its exterior derivative vanishes) and **non-degenerate**. For example, in the treatment above, take to be and take

If is the interior product or contraction operation defined by , then non-degeneracy is equivalent to saying that for every one-form there is a unique vector field such that . Alternatively, . Then if is a smooth function on , the Hamiltonian vector field can be defined to be . It is easy to see that

The **Poisson bracket** on (*M*, *ω*) is a bilinear operation on differentiable functions, defined by ; the Poisson bracket of two functions on *M* is itself a function on *M*. The Poisson bracket is antisymmetric because:

Furthermore,

(1) |

Here *X _{g}f* denotes the vector field

If α is an arbitrary one-form on *M*, the vector field Ω_{α} generates (at least locally) a flow satisfying the boundary condition and the first-order differential equation

The will be symplectomorphisms (canonical transformations) for every *t* as a function of *x* if and only if ; when this is true, Ω_{α} is called a symplectic vector field. Recalling Cartan's identity and *d*ω = 0, it follows that . Therefore, Ω_{α} is a symplectic vector field if and only if α is a closed form. Since , it follows that every Hamiltonian vector field *X _{f}* is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations. From

This is a fundamental result in Hamiltonian mechanics, governing the time evolution of functions defined on phase space. As noted above, when {*f*,*H*} = 0, *f* is a constant of motion of the system. In addition, in canonical coordinates (with and ), Hamilton's equations for the time evolution of the system follow immediately from this formula.

It also follows from **(1)** that the Poisson bracket is a derivation; that is, it satisfies a non-commutative version of Leibniz's product rule:

(2) |

The Poisson bracket is intimately connected to the Lie bracket of the Hamiltonian vector fields. Because the Lie derivative is a derivation,

Thus if *v* and *w* are symplectic, using , Cartan's identity, and the fact that is a closed form,

It follows that , so that

(3) |

Thus, the Poisson bracket on functions corresponds to the Lie bracket of the associated Hamiltonian vector fields. We have also shown that the Lie bracket of two symplectic vector fields is a Hamiltonian vector field and hence is also symplectic. In the language of abstract algebra, the symplectic vector fields form a subalgebra of the Lie algebra of smooth vector fields on *M*, and the Hamiltonian vector fields form an ideal of this subalgebra. The symplectic vector fields are the Lie algebra of the (infinite-dimensional) Lie group of symplectomorphisms of *M*.

It is widely asserted that the Jacobi identity for the Poisson bracket,

The algebra of smooth functions on M, together with the Poisson bracket forms a Poisson algebra, because it is a Lie algebra under the Poisson bracket, which additionally satisfies Leibniz's rule **(2)**. We have shown that every symplectic manifold is a Poisson manifold, that is a manifold with a "curly-bracket" operator on smooth functions such that the smooth functions form a Poisson algebra. However, not every Poisson manifold arises in this way, because Poisson manifolds allow for degeneracy which cannot arise in the symplectic case.

Given a smooth vector field on the configuration space, let be its conjugate momentum. The conjugate momentum mapping is a Lie algebra anti-homomorphism from the Lie bracket to the Poisson bracket:

This important result is worth a short proof. Write a vector field at point in the configuration space as

The above holds for all , giving the desired result.

Poisson brackets deform to Moyal brackets upon quantization, that is, they generalize to a different Lie algebra, the Moyal algebra, or, equivalently in Hilbert space, quantum commutators. The Wigner-İnönü group contraction of these (the classical limit, ħ → 0) yields the above Lie algebra.

To state this more explicitly and precisely, the universal enveloping algebra of the Heisenberg algebra is the Weyl algebra (modulo the relation that the center be the unit). The Moyal product is then a special case of the star product on the algebra of symbols. An explicit definition of the algebra of symbols, and the star product is given in the article on the universal enveloping algebra.

**^**means is a function of the independent variables: momentum, ; position, ; and time,

**^**S. D. Poisson (1809)**^**C. M. Marle (2009)**^**Giacaglia, Giorgio E. O. (1972).*Perturbation methods in non-linear systems*. Applied mathematical sciences. New York Heidelberg: Springer. pp. 8–9. ISBN 978-3-540-90054-2.

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