Variational Theory of Balance Systems

In this work we apply the Poincare-Cartan formalism of the Classical Field Theory to study the systems of balance equations (balance systems). We introduce the partial k-jet bundles of the configurational bundle and study their basic properties: partial Cartan structure, prolongation of vector fields, etc. A constitutive relation C of a balance system is realized as a mapping between a (partial) k-jet bundle and the extended dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system corresponding to a constitutive relation C is studied. Special cases of balance systems -Lagrangian systems of order 1 with arbitrary sources and RET (Rational Extended Ther- modynamics) systems are characterized in geometrical terms. Action of auto- morphisms of the configurational bundle on the constitutive mappings C is studied and it is shown that the symmetry group Sym(C) of C acts on the sheaf of solutions Sol(C) of he balance system. Suitable version of Noether Theorem for an action of a symmetry group is presented together with the special forms for semi- Lagrangian and RET balance systems and examples of energy momentum and gauge symmetries balance laws.