  Various formulations of smooth-particle hydrodynamics (SPH) have been
proposed, intended to resolve certain difficulties in the treatment of fluid
mixing instabilities. Most have involved changes to the algorithm which either
introduce artificial correction terms or violate arguably the greatest
advantage of SPH over other methods: manifest conservation of energy, entropy,
momentum, and angular momentum. Here, we show how a class of alternative SPH
equations of motion (EOM) can be derived self-consistently from a discrete
particle Lagrangian (guaranteeing manifest conservation) in a manner which
tremendously improves treatment of instabilities and contact discontinuities.
Saitoh & Makino recently noted that the volume element used to discretize the
EOM does not need to explicitly invoke the mass density (as in the 'standard'
approach); we show how this insight, and the resulting degree of freedom, can
be incorporated into the rigorous Lagrangian formulation that retains ideal
conservation properties and includes the 'Grad-h' terms that account for
variable smoothing lengths. We derive a general EOM for any choice of volume
element (particle 'weights') and method of determining smoothing lengths. We
then specify this to a 'pressure-entropy formulation' which resolves problems
in the traditional treatment of fluid interfaces. Implementing this in a new
version of the GADGET code, we show it leads to good performance in mixing
experiments (e.g. Kelvin-Helmholtz & blob tests). And conservation is
maintained even in strong shock/blastwave tests, where formulations without
manifest conservation produce large errors. This also improves the treatment of
sub-sonic turbulence, and lessens the need for large kernel particle numbers.
The code changes are trivial and entail no additional numerical expense. This
provides a general framework for self-consistent derivation of different
'flavors' of SPH.
