Variational Theory of Balance Systems

TL;DR

This paper develops a geometric framework for balance systems in classical field theory using the Poincare-Cartan formalism, introducing partial jet bundles and analyzing symmetries and invariants of such systems.

Contribution

It introduces partial k-jet bundles and a geometric approach to balance systems, extending the classical variational formalism to include constitutive relations and symmetries.

Findings

Characterization of Lagrangian and RET systems in geometric terms

Identification of symmetry groups acting on solutions

Formulation of a Noether theorem for 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 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.