Poisson tensor ($\pi$)

A Poisson tensor is a smooth bivector field defining $\{f,g\}=\pi (df,dg)$ whose bracket obeys the Jacobi identity. In coordinates, ${\pi }^{i\ell }{\partial }_{\ell }{\pi }^{jk}+{\pi }^{j\ell }{\partial }_{\ell }{\pi }^{ki}+{\pi }^{k\ell }{\partial }_{\ell }{\pi }^{ij}=0$. Its Hamiltonian distribution integrates into symplectic leaves, even though its rank can vary across the manifold. The tensor specifies a bracket without requiring a globally nondegenerate symplectic form.