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.

All of these objects are named in honor of Siméon Denis Poisson.

[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.

Hamilton's equations of motion have an equivalent expression in terms of the Poisson bracket.

on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e., canonical transformations, area-preserving diffeomorphisms), with the time

is the symplectic matrix under the same conventions used to order the set of coordinates.

[3] An integrable system will have constants of motion in addition to the energy.

Such constants of motion will commute with the Hamiltonian under the Poisson bracket.

is a trajectory or solution to Hamilton's equations of motion, then

where, as above, the intermediate step follows by applying the equations of motion and we assume that

The content of Liouville's theorem is that the time evolution of a measure given by a distribution function

In order for a Hamiltonian system to be completely integrable,

This does not always supply a useful result, however, since the number of possible constants of motion is limited (

degrees of freedom), and so the result may be trivial (a constant, or a function of

on (M, ω) is a bilinear operation on differentiable functions, defined by

Furthermore, 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

; when this is true, Ωα is called a symplectic vector field.

Therefore, Ωα is a symplectic vector field if and only if α is a closed form.

, it follows that every Hamiltonian vector field Xf is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations.

This is a fundamental result in Hamiltonian mechanics, governing the time evolution of functions defined on phase space.

), Hamilton's equations for the time evolution of the system follow immediately from this formula.

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:

The proof of the Jacobi identity follows from (3) because, up to the factor of -1, the Lie bracket of vector fields is just their commutator as differential operators.

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.