  Isotropic fluids in two spatial dimensions can break parity symmetry and
sustain transverse stresses which do not lead to dissipation. Corresponding
transport coefficients include odd viscosity, odd torque, and odd pressure. We
consider an isotropic Galilean invariant fluid dynamics in the adiabatic regime
with momentum and particle density conservation. We find conditions on
transport coefficients that correspond to dissipationless and separately to
Hamiltonian fluid dynamics. The restriction on the transport coefficients will
help identify what kind of hydrodynamics can be obtained by coarse-graining a
microscopic Hamiltonian system. Interestingly, not all parity-breaking
transport coefficients lead to energy conservation and, generally, the fluid
dynamics is energy conserving but not Hamiltonian. We show how this dynamics
can be realized by imposing a nonholonomic constraint on the Hamiltonian
system.
