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.
Properties
Given two functions and 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: Anticommutativity Bilinearity Leibniz's rule Jacobi identity
Also, if a function is constant over phase space (but may depend on time), then for any .
Definition in canonical coordinates
In canonical coordinates (also known as Darboux coordinates) on the phase space, given two functions and , the Poisson bracket takes the form
The Poisson brackets of the canonical coordinates are
where is the Kronecker delta.
Hamilton's equations of motion
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,
can serve as the bracket coordinates. Poisson brackets are canonical invariants.
Dropping the coordinates,
The operator in the convective part of the derivative, , is sometimes referred to as the Liouvillian (see Liouville's theorem (Hamiltonian)).
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 is a constant of motion. This implies that if is a trajectory or solution to Hamilton's equations of motion, then
along that trajectory. Then
where, as above, the intermediate step follows by applying the equations of motion and we assume that does not explicitly depend on time. This equation is known as the Liouville equation. The content of Liouville's theorem is that the time evolution of a measure given by a distribution function is given by the above equation.
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 .)