Poisson brackets and generators

Our convention differentiates the first function with respect to position and the second with respect to momentum, then subtracts the opposite pairing. It gives $\{q_i,p_j\}=\delta_{ij}$ and $\{p_j,q_i\}=-\delta_{ij}$. Treat q and p as independent coordinates during these differentiations. Replacing p with a particular trajectory too early destroys that independence and can turn a nonzero bracket into a misleading derivative along one curve.

The bracket is bilinear, antisymmetric, and obeys a product rule in either argument. These properties allow a complicated polynomial bracket to be reduced to elementary coordinate brackets. It also satisfies the Jacobi identity, which makes transformations generated by brackets algebraically consistent. A useful computational habit is to check antisymmetry on the final answer: if reversing the two functions does not reverse the sign, a derivative or subtraction was probably mishandled.

Apply the chain rule to an observable f along a solution and insert Hamilton's equations. Its total rate becomes its bracket with H plus its explicit time derivative. If f has no explicit time dependence and its bracket with H vanishes, it is conserved. The converse must be phrased carefully: a quantity can have explicit time dependence whose derivative cancels a nonzero bracket. Looking only for a zero bracket would miss that kind of conserved combination.

For a free unit-mass particle, p is conserved, but $q-pt$ is conserved as well. The latter has a nonzero bracket with H and an explicit time derivative of the opposite sign. A cyclic coordinate yields another familiar example: when H is independent of q, its conjugate p has zero bracket with H. Conservation is thus a statement about a phase-space function and the chosen generator, not an intrinsic property of a symbol named momentum.

Introduce a transformation parameter ε and define the infinitesimal active change by $\delta f=\epsilon\{f,G\}$. Taking $G=p$ moves q forward while leaving p fixed, so momentum generates translations with this sign convention. Taking $G=q$ instead changes momentum by minus ε. These transformations are geometric operations on states; ε need not be physical time, and G need not have energy units. Its units must make the particular transformation dimensionally sensible.

A less obvious generator is $G=qp$. Its equations in ε stretch q and compress p by reciprocal factors. The oriented canonical area is unchanged, even though ordinary distances are changed. A Euclidean rotation also preserves area in one canonical plane, but an arbitrary area-preserving map in a higher-dimensional phase space need not preserve the full canonical bracket structure. The generator viewpoint supplies a stronger local test than visual resemblance to a rigid motion.