  Carter and Lichnerowicz have established that barotropic fluid flows are
conformally geodesic and obey Hamilton's principle. This variational approach
can accommodate neutral, or charged and poorly conducting, fluids. We show
that, unlike what has been previously thought, this approach can also
accommodate perfectly conducting magnetofluids, via the Bekenstein-Oron
description of ideal magnetohydrodynamics. When Noether symmetries associated
with Killing vectors or tensors are present in geodesic flows, they lead to
constants of motion polynomial in the momenta. We generalize these concepts to
hydrodynamic flows. Moreover, the Hamiltonian descriptions of ideal
magnetohydrodynamics allow one to cast the evolution equations into a
hyperbolic form useful for evolving rotating or binary compact objects with
magnetic fields in numerical general relativity. Conserved circulation laws,
such as those of Kelvin, Alfv\'en and Bekenstein-Oron, emerge simply as special
cases of the Poincar\'e-Cartan integral invariant of Hamiltonian systems. We
use this approach to obtain an extension of Kelvin's theorem to baroclinic
(non-isentropic) fluids, based on a temperature-dependent time parameter. We
further extend this result to perfectly or poorly conducting baroclinic
magnetoflows. Finally, in the barotropic case, such magnetoflows are shown to
also be geodesic, albeit in a Finsler (rather than Riemann) space.
