  Classical relativistic field theory is applied to perfect and
magneto-hydrodynamic flows. The fields for Hamilton's principle are shown to be
the Lagrangian coordinates of the fluid elements, which are potentials for the
matter current 4-vector and the electromagnetic field 2-form. The energy
momentum tensor and equations of motion are derived from the fields. In this
way the theory of continua is shown to have the same form as other field
theories, such as electromagnetism and general relativity.
  Waves are treated as an example of the power of field theoretic methods. The
average or background flow and the waves are considered as two interacting
components of the system. The wave-background interaction involves the transfer
of energy and momentum between the waves and the average flow, but the total
energy and momentum are conserved. The average Lagrangian for the total
wave-background system is found by expanding the Lagrangian about the
background flow and averaging over the phase. The total energy-momentum tensor
is constructed, and the conservation of energy and momentum are discussed.
Varying the wave amplitude gives the dispersion and polarization relations for
the waves, and varying the phase gives the rays and conservation of wave quanta
(or wave action). The wave quanta move with the group velocity along the
bi-characteristic rays.
