Consider a classical, non-relativistic field of dust particles in outer space that are so diluted they do not exert any measurable pressure on one another. There is no gravitational, or other, potential density, i.e., V(xi)=0. The density of particles is ρ(xi), which for our purposes we can consider constant in time. The displacement of the field (movement of each dust particle at each point) from its initial position is designated by the field value ϕr(xi), r=1,2,3, here, as there is a component of displacement, measured in length units, in each of the three spatial directions. ϕr and xi are both measures of length, but the xi are fixed locations in space, whereas the ϕr are displacements of the particles, in three spatial directions, relative to their initial positions.
What is the kinetic energy density in terms of the field displacement ϕr (actually, it is in terms of the time derivatives of ϕr and ϕ˙r)? What is the Lagrangian density for the field? Use (2-13) to find the differential equation of motion for the displacement ϕr. You should get ρϕ¨r=0. Is this just Newton's second law for a continuous medium with no internal or external force?