Poisson bracket

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In canonical coordinates (q^i,p_j)\, on the phase space, given a function f(p^i,q^i,t)\,, one has

\frac{d}{dt}f(p^i,q^i,t) = \sum_{i=1}^N \frac{\partial f}{\partial q^i} \frac{d q^i}{dt} + \sum_{i=1}^N \frac{\partial f}{\partial p^i} \frac{d p^i}{dt} + \frac{\partial f}{\partial t}\,.

If (q^i,p_j)\, solve the Hamilton equations of motion, then

\frac{d}{dt}f(p^i,q^i,t) = \sum_{i=1}^N \left( \frac{\partial f}{\partial q^i} \frac{\partial H}{\partial p^i} - \frac{\partial f}{\partial p^i} \frac{\partial H}{\partial q^i}\right) + \frac{\partial f}{\partial t}\,.

Then, for functions f(p^i,q^i,t)\, and g(p^i,q^i,t)\,, we define the Poisson bracket which takes the form:

\{f,g\} = \sum_{i=1}^{N} \left( \frac{\partial f}{\partial q^{i}} \frac{\partial g}{\partial p^{i}} - \frac{\partial f}{\partial p^{i}} \frac{\partial g}{\partial q^{i}} \right) ,

so that

\frac{df}{dt} = \{f, H\}+ \frac{\partial f}{\partial t}\,.

Furthermore,

\{q^i, p^j\} = \sum_{l=1}^{N} \left( \frac{\partial q^i}{\partial q^{l}} \frac{\partial p^j}{\partial p^{l}} - \frac{\partial q^i}{\partial p^{l}} \frac{\partial p^j}{\partial q^{l}} \right) = \sum_{l=1}^{N} \delta^{il} \delta^{jl} = \delta^{ij}.

Integrals of motion

The Poisson brackets immediately imply that

\frac{dH}{dt} = \frac{\partial H}{\partial t}\,.

Furthermore, if f(p^i, q^i)\, is only a function of canonical coordinates (q^i,p_j)\, and satisfies \{f, H\} = 0\,, i.e., it commutes with the Hamiltonian, then f\, is conserved, i.e., \frac{df}{dt} = 0\,.

Quantum mechanics

In Quantum mechanics, one replaces \{q^i, p^j\} = \delta^{ij}\, with the commutator [q^i, p^j] \equiv q^i p^j - p^j q^i = i \hbar \delta^{ij}\,. See Poisson brackets (Quantum mechanics).

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