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.