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In the year 2001, Spanish mathematicians C.Murial and J.Romero, introduced the new procedure of prolongations of vector fields from the space Y, of a configuration bundle π : Y → R to the jet spaces J k (π) (λ-prolongations), new class of “symmetries” (λ-symmetries) for the ordinary differential equations and the new method of the symmetry reduction of the ordinary differential equations ([102]) with the λ-symmetries corresponding to the λ-prolongations.

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.

The 4-dim intrinsic (material) Riemannian metric G of the material 4-D space-time continuum P is utilized as the characteristic of the aging processes developing in the material. Manifested through variation of basic material characteristics such as density, moduli of elasticity, yield stress, strength, and toughness., the aging process is modeled as the evolution of the metric G (most importantly of its time component G00) of the material space-time P embedded into 4-D Newtonian space-time with a Euclidean metric. The evolutional equation for the metric G is derived by the classical variational approach. Construction of the Lagrangian for an aging elastic medium and the derivation of a system of coupled elastostatic and aging equations constitute the central part of the work. The external and internal balance laws associated with symmetries of material and physical space-time geometries are briefly reviewed from a new viewpoint.

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.

We analyze mathematical underpinnings of Davini's theory of defective crystals (1) when the defectiveness of a kinematic state may be material point dependent. We show how the underlying space can be identified with a suitably chosen homogeneous space and how the uniformly defective structure is just a special case. 2000 Mathematics Subject Classification: 53Z05, 74B99