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7.4: Poisson’s Theorem

Last updated

Dec 30, 2020
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- 7.3: The Jacobi Identity
- 7.5: Example- Angular Momentum Components

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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 $\begin{equation} [f, g] \end{equation}$ is also a constant of the motion. Of course, it could be trivial, like $\begin{equation} [p, q]=1 \end{equation}$ 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.

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