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In this work we apply infinitesimal variational calculus to the systems of balance equations. We determine a class of the exterior n+(n+1)-forms Θ on the jet bundle of infinite order over a configurational bundle π:Yn+m→Xn similar to the class of Lepage n-forms. Systems of differential equations obtained in the way similar to one used in the Lagrangian field theory, include the Euler–Lagrange equations corresponding to a Lagrangian functions as well as arbitrary regular systems of balance equations. For a balance system with a symmetry group G we present the Noether balance laws corresponding to the generators of the Lie algebra of the group G.

Noether Theory of Symmetries and Conservation laws is a beautiful chapter in mathematical physics. Here we present the form this theory take place in the presence of non-commuting variations. Shortly speaking, in the presence of non-commutative variations, Noether laws becomes balance (not conservation) laws.

In this work we present the theory of balance equations of Continuum Thermodynamics (balance systems) in a geometrical form using the Poincare-Cartan formalism of 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. A version of the Noether Theorem for an action of a symmetry group of vertical automorphisms is presented with the usage of conventional multimomentum mapping. The "entropy principle" of Irreducible Thermodynamics (the requirement that the entropy balance should be the consequence of the balance system) is studied for 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 constitutive laws. Corresponding results may be considered as a generalization of the transition to the dual, Lagrange-Liu picture of Rational Extended Thermodynamics (RET).

In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient do- main) of a general balance system as the sum of a Lagrangian part and a complemental \pure non-Lagrangian" balance system. In the case when derivatives of the dynamical fields do not enter the constitutive relations of the balance system, the \pure non-Lagrangian" systems coincide with the systems introduced by S. Godunov (Soviet Math. Dokl. 2 (1961), 947{948) and, later, asserted as the canonical hyperbolic form of balance systems in (Muller I., Rug- geri T., Rational extended thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy, Vol. 37, Springer-Verlag, New York, 1998).

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 [Formula: see text] 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 [Formula: see text] is realized as the mapping between a (partial) k-jet bundle [Formula: see text] and the extended dual bundle [Formula: see text] similar to the Legendre mapping of the Lagrangian Field Theory. The invariant (variational) form of the balance system [Formula: see text] corresponding to a constitutive relation [Formula: see text] 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 [Formula: see text] is studied and it is shown that the symmetry group [Formula: see text] of [Formula: see text] acts on the sheaf of solutions [Formula: see text] of balance system [Formula: see text]. 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.