Universal Field Equations with Covariant Solutions D.B. Fairlie

Metric independent $\sigma$ models are constructed. These are field theories which generalise the membrane idea to situations where the target space has fewer dimensions than the base manifold. Instead of reparametrisation invariance of the independent variables, one has invariance of solutions of the field equations under arbitrary functional redefinitions of the field quantities. Among the many interesting properties of these new models is the existence of a hierarchical structure which is illustrated by the following result. Given an arbitrary Lagrangian, dependent only upon first derivatives of the field, and homogeneous of weight one, an iterative procedure for calculating a sequence of equations of motion is discovered, which ends with a universal, possibly integrable equation, which is independent of the starting Lagrangian. A generalisation to more than one field is given.