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Willie Wong
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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.