= Momentum-space Poisson structure for a uniform magnetic field
{title2=$\pi^{ij}=e\epsilon_{ijk}B_k$}
A uniform <magnetic field> defines a constant <Poisson bivector> on momentum space by the displayed components. For the Hamiltonian $E(\mathbf p)$ and convention $\dot f=\{f,E\}$, it gives $\dot{\mathbf p}=e\nabla E\times\mathbf B$. Constancy makes every term in the coordinate <Jacobi identity> vanish. If $e\mathbf B\ne0$, the <symplectic leaves> are the planes $\mathbf B\cdot\mathbf p=c$, and energy is conserved along the flow. The whole three-dimensional momentum space is not symplectic, and an energy surface need not itself be a Poisson submanifold.
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