Momentum-space Poisson structure for a uniform magnetic field (${\pi }^{ij}=e{ϵ}_{ijk}{B}_k$)

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 $e\mathbf{B}\ne 0$, 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.