How Do Poisson Brackets Function in Classical Mechanics?

[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.

Start with
\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)

Using Hamilton's equations of motion gives
\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\}

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[APhysicist]

Thanks for the overview of Poisson brackets