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 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 formula_1 and formula_2, 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 formula_3 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 f and g that depend on phase space and time, their Poisson bracket formula_4 is another function that depends on phase space and time. The following rules hold for any three functions formula_5 of phase space and time:
Also, if a function formula_10 is constant over phase space (but may depend on time), then formula_11 for any formula_12.
Definition in canonical coordinates.
In canonical coordinates (also known as Darboux coordinates) formula_13 on the phase space, given two functions formula_14 and formula_15, the Poisson bracket takes the form
The Poisson brackets of the canonical coordinates are
where "formula_18" 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 formula_19 is a function on the solution’s trajectory-manifold. Then from the multivariable chain rule,
Further, one may take formula_21 and formula_22 to be solutions to Hamilton's equations; that is,
Then 
Thus, the time evolution of a function formula_12 on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e., canonical transformations, area-preserving diffeomorphisms), with the time formula_26 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 formula_26" 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, formula_30, 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 formula_31 is a constant of motion. This implies that if formula_32 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 supposed that formula_12 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 (or "distribution function" on the phase space) is given by the above.
If the Poisson bracket of formula_12 and formula_37 vanishes (formula_38), then formula_12 and formula_37 are said to be in involution. In order for a Hamiltonian system to be completely integrable, formula_41 independent constants of motion must be in mutual involution, where formula_41 is the number of degrees of freedom.
Furthermore, according to Poisson's Theorem, if two quantities formula_43 and formula_44 are explicitly time independent (formula_45) constants of motion, so is their Poisson bracket formula_46. This does not always supply a useful result, however, since the number of possible constants of motion is limited (formula_47 for a system with formula_41 degrees of freedom), and so the result may be trivial (a constant, or a function of formula_43 and formula_44.)
The Poisson bracket in coordinate-free language.
Let formula_51 be a symplectic manifold, that is, a manifold equipped with a symplectic form: a 2-form formula_52 which is both closed (i.e., its exterior derivative formula_53 vanishes) and non-degenerate. For example, in the treatment above, take formula_51 to be formula_55 and take
If formula_57 is the interior product or contraction operation defined by formula_58, then non-degeneracy is equivalent to saying that for every one-form formula_59 there is a unique vector field formula_60 such that formula_61. Alternatively, formula_62. Then if formula_63 is a smooth function on formula_51, the Hamiltonian vector field formula_65 can be defined to be formula_66. It is easy to see that
The Poisson bracket formula_68 on ("M", ω) is a bilinear operation on differentiable functions, defined by formula_69; the Poisson bracket of two functions on "M" is itself a function on "M". The Poisson bracket is antisymmetric because:
Furthermore,
\omega)(X_g) = df(X_g) \\
 &= X_gf = \mathcal{L}_{X_g} f
Here "Xgf" denotes the vector field "Xg" applied to the function "f" as a directional derivative, and formula_71 denotes the (entirely equivalent) Lie derivative of the function "f".
If α is an arbitrary one-form on "M", the vector field Ωα generates (at least locally) a flow formula_72 satisfying the boundary condition formula_73 and the first-order differential equation
The formula_72 will be symplectomorphisms (canonical transformations) for every "t" as a function of "x" if and only if formula_76; when this is true, Ωα is called a symplectic vector field. Recalling Cartan's identity formula_77 and "d"ω = 0, it follows that formula_78. Therefore, Ωα is a symplectic vector field if and only if α is a closed form. Since formula_79, it follows that every Hamiltonian vector field "Xf" is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations. From above, under the Hamiltonian flow "XH",
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 formula_81 and formula_82), Hamilton's equations for the time evolution of the system follow immediately from this formula.
It also follows from that the Poisson bracket is a derivation; that is, it satisfies a non-commutative version of Leibniz's product rule:
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 formula_84, Cartan's identity, and the fact that formula_85 is a closed form,
It follows that formula_87, so that
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,
follows from the corresponding identity for the Lie bracket of vector fields, but this is true only up to a locally constant function. However, to prove the Jacobi identity for the Poisson bracket, it is sufficient to show that:
where the operator formula_90 on smooth functions on "M" is defined by formula_91 and the bracket on the right-hand side is the commutator of operators, formula_92. By , the operator formula_90 is equal to the operator "Xg". The proof of the Jacobi identity follows from because the Lie bracket of vector fields is just their commutator as differential operators.
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 . 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.
A result on conjugate momenta.
Given a smooth vector field formula_94 on the configuration space, let formula_95 be its conjugate momentum. The conjugate momentum mapping is a Lie algebra anti-homomorphism from the Poisson bracket to the Lie bracket:
This important result is worth a short proof. Write a vector field formula_94 at point formula_98 in the configuration space as
where the formula_100 is the local coordinate frame. The conjugate momentum to formula_94 has the expression
where the formula_2 are the momentum functions conjugate to the coordinates. One then has, for a point formula_104 in the phase space,
The above holds for all formula_106, giving the desired result.
Quantization.
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, ) 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.