Abstract: Under reasonable assumptions，this paper presents a Lagrangian density of relativistic fluid from the energy-momentum tensor of the relativistic fluid，and presents the equation of motion for relativistic fluid based on the Lagrangian density and the principle of least action. In the limit of low speed and low pressure our equations of motion reduces to Navier-Stokes equations. The continuity equation of fluid is also discussed. It is found that the continuity equation of fluid in Newtonian mechanics can be established only when the influence of pressure and external force can be neglected，even at the low speed of flat space-time.

Key words: relativistic fluid, Lagrangian density, principle of least action, motion equation of fluid, continuous equation of fluid