Multisymplectic Theory of Balance Systems and the 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 C of a balance system BC is realized as a mapping between a (partial) 1-jet bundle of the configurational bundle π : Y → X and the dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system BC is presented in three different forms and the space of admissible variations is defined and studied. 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 the constitutive relation C acts on the space of solutions of balance system BC . Suitable version of Noether Theorem for an action of a symmetry group is presented with the usage of conventional multimomentum mapping. The ”entropy principle” of Irreducible Thermodynamics requirement that the entropy balance should be the consequence of the balance system is studied for a general balance systems. The structure of corresponding (secondary) balance laws of a balance system BC is studied for the cases of different (partial) 1-jet bundles as domains of a constitutive laws. Corresponding results may be considered as a generalization of the transition to the dual, Lagrange-Liu picture of the Rational Extended Thermodynamics (RET). Finally, the geometrical (bundle) picture of the RET 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.