  Hydrodynamic equations for a one-component plasma are derived as a
generalization of the Euler equations to include the effects of the long-range
Coulomb interaction. By using a variational principle, these equations
self-consistently unify thermodynamics, dispersion laws, nonlinear motion, and
conservation laws. In the moderate and strong coupling limits, it is argued
that these equations work down to the length scale of the average interparticle
spacing. Closure and self-consistency of these nonlocal equations are achieved
via the implementation of a variational principle. Hydrodynamic equations are
evaluated in both the Eulerian frame, where the dependent variables depend on
the position in the laboratory, and the Lagrangian frame, where they depend on
the position in some reference state, such as the initial position. Each frame
has its advantages and our final theory combines elements of both. The
properties of longitudinal and transverse dispersion laws are calculated for
the hydrodynamic equations. A simple step function approximation for the pair
distribution function is used that allows for simple calculations that reveal
the structure of the equations of motion. The obtained dispersion laws are
compared to molecular dynamics simulations and the theory of quasilocalized
charge approximation. The action, which gives excellent agreement for both
longitudinal and transverse dispersion laws for a wide range of coupling
strengths, is elucidated. Agreement with numerical experiments shows that such
a hydrodynamic approach can be used to accurately describe a one-component
plasma at very small length scales comparable to the average interparticle
spacing. The validity of this approach suggests considering nonlinear flows and
other systems with long-range interactions in the future.
