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: 
where is any operator of quantum mechanics, is its expectation value, and
is the commutator of and .
- ↑ Smith, Henrik (1991). Introduction to Quantum Mechanics. World Scientific Pub Co Inc. pp. 108–109. ISBN 978-9810204754.