VARIATIONAL THEORY OF BALANCE SYSTEMS
Abstract
In this work, we apply the Poincaré–Cartan formalism of Classical Field Theory to study the systems of balance equations (balance systems). We introduce the partial k-jet bundles 
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 
is realized as the mapping between a (partial) k-jet bundle 
and the extended dual bundle 
similar to the Legendre mapping of the Lagrangian Field Theory. The invariant (variational) form of the balance system 
corresponding to a constitutive relation 
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. The action of automorphisms of the bundle π on the constitutive mappings 
is studied and it is shown that the symmetry group 
of 
acts on the sheaf of solutions 
of balance system 
. A 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. Examples of energy momentum and gauge symmetries balance laws are provided. At the end, we introduce the secondary balance laws for a balance system and classify these laws for the Cattaneo heat propagation system.
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