What are the applications (physical and mathematical) of classical field theory beyond electrodynamics and gravity?
By such applications, I mean that either the field theory viewpoint adds some genuinely new insight into the underlying physics or that it gives rise to interesting mathematical problems. So I'm not thinking about:
-field-theoretical description of something that is very well understood with other tools (for example, describing classical electrodynamics in language of fibre bundles, differential forms etc. is very nice and elegant, but doesn't add much to physics)
-quantum field theory (in QFT you always write down the classical lagrangian and then turn the fields into operators, but there is not much actual classical field theory here)
Of course, you can always write some lagrangian like phi^34 + 14*phi^8 + …, and study the resulting PDE (existence and uniqueness of solutions etc.), but I guess that lacks real motivation.
Best Answer
How about the study of minimal surfaces (physical applications in soap films etc.)? In fact one might argue the Lagrangian formulation of minimal surfaces (the problem of Plateau) is one of the oldest "classical field theory" problems, and led to the revival of calculus of variations in the early twentieth century (see esp. the works of Morrey).
Slightly related is the general study of continuum mechanics and (non-linear) elasticity. Which is kind of like fluid mechanics except for deformations of solids.
Another well-known application of the general frame work is the study of harmonic maps and wave maps (also known as non-linear sigma model in physics). The study of such systems led to developments of the techniques of compensated compactness and multilinear product estimates in partial differential equations (see, e.g. works of Helein, Klainerman, Tao, Krieger, and many others). The regularity properties of the harmonic maps are still under active study (Li and Tian, Nguyen, Weinstein, and others). And in physics, the sigma models find application from particle physics (as a model for equivariant Yang-Mills equation) to general relativity (stationary solutions in Einstein-vacuum or Einstein-Maxwell theories).
The sigma models are also generalized by Tony Skyrme in his namesake quasilinear model (both hyperbolic and elliptic), which is not yet well understood. This model has found applications from nucleon physics to condensed matter, and now to topological material science. The study of the stationary problem (and its generalization in the Fadeev-Skyrme model) led to interesting developments in topology and geometry (since the model admits topological solitons), see for example the work of Kapitansky.