Partial Differential Equations of Physics

The physical world is traditionally organized into various systems: electromagnetism, perfect fluids, Klein-Gordon fields, elastic media, gravitation, etc. Our descriptions of these individual systems have certain features in common: Use of fields on a fixed space-time manifold M , a geometrical interpretation of the fields in terms of M , partial differential equations on these fields, an initial-value formulation for these equations. Yet beyond these common features there are numerous differences of detail: Some systems of equations are linear, and some are not; some have constraints, and some do not; some arise from Lagrangians, and some do not; some are firstorder, and some higher-order. Systems also differ in other respects, e.g., as to what fields they need as background, what interactions they permit (or require). It almost seems as though, in the end, every physical system has its own special character. It might be useful to have a systematic treatment of the fields and equations that arise in the description of physical systems. Thus, there would be a general definition of a “field”, and a general form for a system of partial differential equations on such fields. The treatment would consist of a framework sufficiently broad to encompass the systems found in nature, but no broader. One would, for example, treat the initial-value formulation once and for all within this broad framework, with the formulations for individual physical systems emerging as special cases. In a similar way, one would treat—within a quite general context—constraints, the geometrical character of physical fields, how some systems require other fields as a background,