What is Poisson bracket? — Physics

Definition 2.10 University Physics — Year 3 · Chapter 2 — Hamiltonian Mechanics

The Poisson bracket of two functions $f(q, p, t)$ and $g(q, p, t)$ on phase space is

\{f, g\} = \sum_i\Big( \frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\Big) .

It is antisymmetric, linear in each argument, obeys the product rule $\{f, gh\} = \{f, g\}h + g\{f, h\}$ , and satisfies for the coordinates themselves the canonical relations

\{q_i, p_j\} = \delta_{ij} , \qquad \{q_i, q_j\} = \{p_i, p_j\} = 0 .

Examples

Example 2.12 (Brackets of angular momentum)

For one particle, $L_z = xp_y - yp_x$ and its cyclic companions. A direct computation from the canonical relations gives

\{L_x, L_y\} = L_z , \qquad \{L_y, L_z\} = L_x , \qquad \{L_z, L_x\} = L_y ,

and $\{L^2, L_z\} = 0$ : the components of angular momentum do not “commute” with each other, but each commutes with the total square. Remember the shape of these relations — they will return, verbatim, as commutators in quantum mechanics, where they dictate everything about atomic structure.

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Examples

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