  A neutron star in a compact binary is expected to be well-approximated by a
barotropic flow during the inspiral phase. During the merger phase, where tidal
disruption and shock-heating occur, a baroclinic description is needed instead.
In the barotropic case, a Hamiltonian formulation potentially offers unique
benefits for numerical relativity simulations of the inspiral phase, including
highly accurate conservation of circulation and superconvergence of the fluid
variables, and is actively being explored. In this work, we investigate the
viability of a Hamiltonian formulation in the baroclinic case. At odds with the
barotropic case, this formulation is non-conservative, yet it can be treated
well with approximate Riemann solver algorithms since the non-conservative
terms vanish across genuinely nonlinear fields. Nonetheless, using numerical
1-dimensional shock tube tests we find that the weak solutions of the
Hamiltonian system differ from the standard ones obtained by enforcing
conservation of rest mass density, momentum density, and energy density across
discontinuities. We also show that barotropic Hamiltonian formulations can
admit shockwaves at fluid-vacuum interfaces, which may be related to the
unstable behavior of stellar surfaces observed in past numerical tests. In
light of the unphysical weak solutions, we expect that in future
implementations of the Hamiltonian formulation of hydrodynamics in numerical
relativity it will be necessary to use an explicitly barotropic formulation
during the inspiral phase, and then switch to a robust baroclinic formulation
prior to merger.
