Alternative variational formulations for first order partial differential systems

Abstract

Abstract. Two simple alternative variational principles are derived for a first order differential system with appropriate initial and boundary conditions. The problem is assumed to be well posed, and may be nonlinear, nonhomogeneous and of any type (i.e. elliptic, hyperbolic or mixed). Primitive variables are used, which allows for non-smooth solutions. Redundancy in the system is considered, and applications to fluidynamics and electrodynamics fields given.1. Introduction. Two forms of mathematical modeling of physical continua are possible: (i) a system of first order, partial differential equations for the field components and quantities; (ii) an integral representation of the field via a variational principle, where a functional of the field quantities becomes stationary at the solution of the differential system which constitutes its Euler-Lagrange equations. The two approaches are common in physics (e.g. the Newtonian and Lagrangian representations of mechanics, Hamilton's principle of least action, Fermat's principle in optics) and both have been extremely useful in many and varied areas, both in classical (e.g. mechanics, electrodynamics, relativity e.g. [1][2][3]) and quantum applications [4],