Multisymplectic Theory of Balance Systems and Entropy Principle

In this paper we are presenting the theory of balance equations of the Continuum Thermodynamics (balance systems) in a geometrical form using Poincare-Cartan formalism of the Multisymplectic Field Theory. A constitutive relation $\mathcal{C}$ of a balance system $B_{C}$ is realized as a mapping between a (partial) 1-jet bundle of the configurational bundle $\pi:Y\to X$ and the dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system $B_{C}$ is presented in three different forms and the space of admissible variations is defined and studied. Action of automorphisms of the bundle $\pi$ on the constitutive mappings $C$ is studied and it is shown that the symmetry group $Sym(C)$ of the constitutive relation $C$ acts on the space of solutions of the balance system $B_{C}$. Suitable version of Noether Theorem for an action of a symmetry group is presented with the usage of conventional multimomentum mapping. Finally, the geometrical (bundle) picture of the Rational Extended Thermodynamics in terms of Lagrange-Liu fields is developed and the entropy principle is shown to be equivalent to the holonomicy of the current component of the constitutive section.