Inspired by “fracton hydrodynamic” universality classes of dynamics with unusual conservation laws, we present a dynamical universality class that arises out of local area-preserving dynamics in the noncommutative plane. On this symplectic manifold, the area-preserving spatial symmetry group $\mathrm{SL}(2,\mathbb{R})⋊{\mathbb{R}}^2$ is a symmetry group compatible with nontrivial many-body dynamics. The conservation laws associated with this symmetry group correspond to the dipole and quadrupole moments of the particles. We study the unusual dynamics of a crystal lattice subject to such symmetries, and argue that the hydrodynamic description of lattice dynamics breaks down due to relevant nonlinearities. Numerical simulations of classical Hamiltonian dynamical systems with this symmetry are largely consistent with a tree-level effective field theory estimate for the endpoint of this instability.

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• Fractons
• Nonequilibrium lattice models
• Statistical hydrodynamics