Multisymplectic Theory of Balance Systems, I

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 extended 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. 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.