7.4: Poisson’s Theorem
( \newcommand{\kernel}{\mathrm{null}\,}\)
If f and g are two constants of the motion (i.e., they both have zero Poisson brackets with the Hamiltonian), then the Poisson bracket [f,g] is also a constant of the motion. Of course, it could be trivial, like [p,q]=1 or it could be a function of the original variables. But sometimes it’s a new constant of motion. If f,g are time-independent, the proof follows immediately from Jacobi’s identity. A proof for time dependent functions is given in Landau—it’s not difficult.