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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.

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@Mikhal: Please read carefully FAQ. Such philosophical Qs are community wiki. – Wadim Zudilin Jul 7 '10 at 14:30

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.

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Well, I'm not really sure about whether you wish to refer just to relativistic classical field theories or you are interested on non-relativistic ones as well.

Either way, you have:

  • Classical thermodynamics, where you study the internal energy, entropy, temperature, pressure and volume fields of a classical sytem. This theory serves to:

Understand and apply at wish the mechanisms of heat conduction (freezers, air-conditioners, determining the hour of death of a corpse...).

Make rough cosmological assesments (heat death of the universe and the like).

Understand, measure and determine the restrictions on life at a thermodynamical level (need for feeding, maximal safe temperature, maximal attainable speed, construction of thermomethers).

Understand how to construct, manage and study combustion-powered engines (Carnot efficiency, automotive industry).

  • Fluid dynamics, where you study the velocity, temperature, density and pressure fields of a (liquid or gaseous) fluid. This theory serves to:

Design fluid-efficient machines, whether they are flying machines - planes and spaceships -, running machines - Formula 1 cars - or navigating machines - ships and submarines -.

Understand and implement interesting flows through pipes, nozzles and turbines (we need them, for example, to conduct or transport liquids, gasses and colloids and to know how to expel them properly).

Get insight on some transport phenomena that occur in Biology (since water is an important component of most living creatures and of their environments).

Predict the weather on a given (not too big) zone, in a given time (not too far in the future).

Get knowledge on the important ocean currents and on how to predict tsunamis.

Have the correct tool for (fluid) acoustical engineering, since sound is no more than disturbances of the average pressure of the fluid.

  • Magnetohydrodynamics, which studies the dynamics of electrically conducting fluids (like plasmas, liquid metals or salt water). Its equations are the mixture of Navier-Stokes' and Maxwell's. This theory serves to:

Model the core of the Earth as a liquid-metal dynamo (generation of its global magnetic field, Seismology).

Accurately describe the internal dynamics of stars and another space objects made mostly from plasma (predict the number, size and movement of the Sunspots).

Know how to cool high temperature systems by means of liquid metals.

Research a totally new generation of engines.

  • Electrohydrodynamics, which studies the dynamics of ionised particles on the sine of a fluid, subject to electric fields. This theory serves to:

In general, to convert electrical energy into kinetic energy (and vice versa).

Know how to cool systems by means of ionized liquids.

Construct liquid electrical generators.

Build propulsion devices without moving parts (EHD thrusters).

Add some more pseudoscience to our already-too-magical world: Designing air ionizers and claiming that they have all kind of health benefits.

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OK, I basically meant 'relativistic field theory' (every smooth function on R^n could be treated as a 'field'). – Michal Kotowski Jul 7 '10 at 18:23

Statistical field theory? This plays an essential role in the statistical mechanical analysis of continuous phase transitions. See, e.g., the books by Itzykson and Drouffe.

Continuum mechanics or fluid mechanics would also apply.

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Statistical and quantum field theories generally correspond to each other via Wick rotation. – Steve Huntsman Jul 7 '10 at 14:28
@Steve: Of course, from a purely technical point of view I agree. But (part of) the question was about applications to physics, and applications to statistical physics, continuum mechanics are examples of applications of this formalism to other problems than those mentioned in the question... – Yvan Velenik Jul 7 '10 at 17:02
I wholeheartedly agree that statistical field theory is a distinct discipline that shares many features with classical field theory. It's certainly not like statistical physics is just taking Laplace transforms and quantum physics is just taking Fourier transforms and a factor of $i$ turns one into the other...I also think that the rational mechanics approach to continuum and fluid mechanics is quite interesting. – Steve Huntsman Jul 7 '10 at 17:14

The applications of classical gauge theory to mathematics are well known. Instantons play a central role here. Two good references are Ward and Wells Jr's Twistor Geometry and Field Theory and Rubakov's Classical Theory of Gauge Fields.

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I should mention that instantons are fundamentally classical, though they are also of course prominent in the quantum theory through their contribution to the path integral. – Steve Huntsman Jul 7 '10 at 14:56

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