The Poisson bracket of any two functions,
and
, is:
In two dimensions, the multivariable chain rule, is
. Using implied summation notation (for the index j), we apply this to Hamilton's equations:
As an aside, we note a connection to Quantum Mechanics: Ehrenfest theorem involves the operators and expectation values of quantum mechanics. It states: [1]
where
is any operator of quantum mechanics,
is its expectation value, and
is the commutator of
and
.