Variational Theory of Balance Systems

Abstract. 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 Jk p (π) of the configurational bundle π : Y → X and study their basic properties: partial Cartan structure, prolongation of vector fields, etc. A constitutive relation C of a balance system BC is realized as a mapping between a (partial) k-jet bundle Jk p (π) and the extended dual bundle Λ n+(n+1) 2/1 Y similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system BC 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 Thermodynamics) systems are characterized in geometrical terms. Action of automorphisms of the bundle π 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 SolC of balance system BC . Suitable version of Noether Theorem for an action of a symmetry group is presented together with the special forms for semiLagrangian and RET balance systems and examples of energy momentum and gauge symmetries balance laws. k-jet bundle, balance law, Poincare-Cartan form, Noether Theorem