Something that Newtonian and relativistic fluids have in common is that there are preferred frames for measuring changes – those that are attached to the fluid elements. In the parlance of hydrodynamics, one refers to Lagrangian and Eulerian frames, or observers. A Newtonian Eulerian observer is one who sits at a fixed point in space, and watches fluid elements pass by, all the while taking measurements of their densities, velocities, etc. at the given location. In contrast, a Lagrangian observer rides along with a particular fluid element and records changes of that element as it moves through space and time. A relativistic Lagrangian observer is the same, but the relativistic Eulerian observer is more complicated to define. Smarr and York  define such an observer as one who would follow along a worldline that remains everywhere orthogonal to the family of spacelike hypersurfaces.
The existence of a preferred frame for a one fluid system can be used to great advantage. In the next Section 6.2 we will use an “off-the-shelf” analysis that exploits a preferred frame to derive the standard perfect fluid equations. Later, we will use Eulerian and Lagrangian variations to build an action principle for the single and multiple fluid systems. These same variations can also be used as the foundation for a linearized perturbation analysis of neutron stars . As we will see, the use of Lagrangian variations is absolutely essential for establishing instabilities in rotating fluids [44, 45]. Finally, we point out that multiple fluid systems can have as many notions of Lagrangian observers as there are fluids in the system.
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