Poisson Bracket

Let and be any functions of a set of variables . Then the expression

(1)

is called a Poisson bracket (Poisson 1809; Whittaker 1944, p. 299). Plummer (1960, p. 136) uses the alternate notation .

The Poisson brackets are anticommutative,

(2)

(Plummer 1960, p. 136).

Let be independent functions of the variables . Then the Poisson bracket is connected with the Lagrange bracket by

(3)

where is the Kronecker delta. But this is precisely the condition that the determinants formed from them are reciprocal (Whittaker 1944, p. 300; Plummer 1960, p. 137).

If and are physically measurable quantities (observables) such as position, momentum, angular momentum, or energy, then they are represented as non-commuting quantum mechanical operators in accordance with Heisenberg's formulation of quantum mechanics. In this case,

(4)

where is the commutator and is the Poisson bracket. Thus, for example, for a single particle moving in one dimension with position and momentum ,

(5)

where is -bar.