# What is a Poisson bracket

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

In the Hamiltonian formulation of classical mechanics, equations of motion can be expressed very conveniently using Poisson brackets. They are also useful for expressing constraints on changed canonical variables.

They are also related to commutators of operators in quantum mechanics.

Equations

For canonical variables (q,p), the Poisson bracket is defined for functions f and g as
$\{f,g\} = \sum_a \left( \frac{\partial f}{\partial q_a}\frac{\partial g}{\partial p_a} - \frac{\partial f}{\partial p_a}\frac{\partial g}{\partial q_a} \right)$

The equation of motion for quantity f is
$\dot f = \frac{\partial f}{\partial t} + \{f,H\}$

A change of variables from canonical variables (q,p) to canonical variables (Q,P) has these constraints:
$\{Q_i,P_j\} = \delta_{ij} ,\{Q_i,Q_j\} = \{P_i,P_j\} = 0$

Extended explanation

Proof of equation of motion.

$\frac{df}{dt} = \frac{\partial f}{\partial t} + \sum_a \left( \frac{\partial f}{\partial q_a} \frac{dq_a}{dt} + \frac{\partial f}{\partial p_a} \frac{dp_a}{dt} \right)$
$\frac{df}{dt} = \frac{\partial f}{\partial t} + \sum_a \left( \frac{\partial f}{\partial q_a} \frac{\partial H}{\partial p_a} - \frac{\partial f}{\partial p_a} \frac{\partial H}{\partial q_a} \right) = \frac{\partial f}{\partial t} + \{f,H\}$