A covariant and extended approach to some physical problems with constrained field variables

Abstract

Many physical problems are described by means of systems S of partial differential equations, whose field variables u are restricted by relations of the type Φ_I (u) = 0. Some examples, to this regard, are the ultrarelativistic gases studied in the framework of Extended Thermodynamics, the relativistic magnetofluiddynamics and the Maxwell Equations in the relativistic form. Here a general method is proposed to deal with problems of this kind; in particular, a new system S' is proposed in the independent variables u, ψ R which are not restricted. Moreover, the solutions of S' with �Φ_I (u) = 0, ψ_R =0, are the same of the original system S. The new system S' is expressed in
the covariant form and is hyperbolic, under the assumption that the original system S satisfies these properties; Φ_I (u) = 0, ψ_R = 0 are satisfied as consequences of S' and of the intial conditions. The new variables ψ_R are only auxiliary quantities.