Poisson bracket

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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 qi and pi, 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 H = H(q, p; t) 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 honour of Siméon Denis Poisson.

Properties[edit]

For any functions of phase space and time:

Anticommutativity
Distributivity
Product rule
Jacobi identity

Also, if a function is time-dependent but constant over phase space, then for any .

Definition in canonical coordinates[edit]

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

where δij is the Kronecker delta.

Hamilton's equations of motion[edit]

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 manifold. Then from the multivariable chain rule,

Further, one may take p = p(t) and q = q(t) to be solutions to Hamilton's equations; that is,

Then

Thus, the time evolution of a function f on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e., canonical transformations, area-preserving diffeomorphisms), with the time t 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 t in the solution to Hamilton's equations, q(t) = exp(−t{H, •}) q(0), p(t) = exp(−t{H, •}) p(0), can serve as the bracket coordinates. Poisson brackets are canonical invariants.

Dropping the coordinates,

The operator in the convective part of the derivative, i = −{H, •} , is sometimes referred to as the Liouvillian (see Liouville's theorem (Hamiltonian)).

Constants of motion[edit]

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's equations of motion, then

along that trajectory. Then

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.

If the Poisson bracket of f and g vanishes ({f,g} = 0), then f and g are said to be in involution. In order for a Hamiltonian system to be completely integrable, all of the constants of motion must be in mutual involution.

Furthermore, according to the Poisson's Theorem, if two quantities and are 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 n degrees of freedom), and so the result may be trivial (a constant, or a function of and .)

The Poisson bracket in coordinate-free language[edit]

Let M be symplectic manifold, that is, a manifold equipped with a symplectic form: a 2-form ω which is both closed (i.e., its exterior derivative dω = 0) and non-degenerate. For example, in the treatment above, take M 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 α there is a unique vector field such that . Alternatively, . Then if H is a smooth function on M, the Hamiltonian vector field XH 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 Xgf denotes the vector field Xg applied to the function f as a directional derivative, and 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 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, Ωα is called a symplectic vector field. Recalling Cartan's identity and dω = 0, it follows that . Therefore Ωα is a symplectic vector field if and only if α is a closed form. Since , it follows that every Hamiltonian vector field Xf is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations. From (1) 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 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:

, and

 

 

 

 

(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,

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 on smooth functions on M is defined by and the bracket on the right-hand side is the commutator of operators, . By (1), the operator is equal to the operator Xg. The proof of the Jacobi identity follows from (3) 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 (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.

A result on conjugate momenta[edit]

Given a smooth vector field X on the configuration space, let PX 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 X at point q in the configuration space as

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

where the pi are the momentum functions conjugate to the coordinates. One then has, for a point (q,p) in the phase space,

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

Quantization[edit]

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.

See also[edit]

References[edit]

External links[edit]

Notes[edit]

  1. ^ means f is a function of the 2N + 1 independent variables: momentum, p1…N; position, q1…N; and time, t