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 ,