POLYADIC FORMULATION OF LINEAR PHYSICAL LAWS. by
Elysio R. F. Ruggeri Ouro Preto, MG, Brazil Klaus Helbig Hannover, Ge
ABSTRACT Physical fields are represented by tensors or polyadics of different valence (rank, order): R Σ (of valence R), H ε (of valence H), etc. We say that a (dependent) quantity R Σ is proportional to a second
ε when each component (with reference to an arbitrary vector base) of R Σ is proportional, with different (perhaps constant) weights, to all coordinates of H ε . The proportionality between two physical quantities expresses a linear physical law. In polyadic calculus this proportionality is formulated as a "multiple dot multiplication" written in the form R Σ = R +H G H. H ε ; the valence of the polyadic R+H G is the sum of the valences of the other two polyadics. The coordinates of R+H G define the weights with which the coordinates of the independent polyadic enter in the constitution of each coordinate of the dependent polyadic. For R=H the proportionality exists between two fields of the same valence. We concentrate on this type of proportionality, expressed by H Σ = 2H G H. H ε , and postulate a second relation between the (independent) quantity
H
dependent and independent polyadics, the scalar 2W: of this scalar implies the symmetry of r H
2H
H
ε
H H Σ .
= 2W = H ε
H 2H G H. H ε .
. The existence
G , i.e., the equality of the proportionality polyadic with its
transpose: G = G . Many physical laws are expressed as symmetric relationships of this type with H=1 or H=2. Symmetric polyadic relationships can be expressed with reference to an arbitrary external vector base, but often the expression with respect to the orthonormal polyadic base is more convenient. In particular, the orthonormal polyadic base defined by the eigenH-adics of 2H G is to be preferred. Some of the most interesting cases of proportionality between polyadics occur in the theory of elasticity. For H=2, 2W is the energy density stored at each point of a stressed body; for H=1, 2W is the normal stress. For any H, the particular concepts of normal and tangential stress are extended to "radial" and "tangential" values of 2H G . Stationary radial and tangential values at a point of a field allow the generalization of classical theorems known in the theory of stresses, as Cauchy's and Lamè's quadrics and the representation in Mohr's plane. From Mohr's circle one can derive a general criterion of proportionality, closely related to the failure criterion in the theory of materials. When one uses dyadic bases to study the natural laws with H=2, it is necessary to introduce a new 9-dimensional space that is closely linked to the core of the problem. This space allows us to use intuitively some concepts of nine-dimensional Euclidean geometry. The main concepts of this geometry were established within the Polyadic Calculus (Ruggeri 1999), but are outside the scope of this contribution. It is not difficult to generalize the properties to arbitrary H and to establish the Ndimensional analytic geometry associated with the physical laws. They follow immediately if one regards the linear law as linear transformations (a mapping) of the "space defined by one polyadic" into the space defined by another polyadic through a "polyadic operator" (the proportionality polyadic). Some aspects of the geometry hidden in these laws suggest interesting experiments to define the polyadic operator and a statistical polyadic to define "probable" values. The main objective of this paper is to show that all linear physical laws in continuum physics (particularly for H=1, i.e., vector quantities linked by dyadics, and for H=2, i.e., for dyadics linked by tetradics) can be treated mathematically by a unified method. This method is algebraic as well as geometrical. It is based on a synthesis of Polyadic Calculus and multidimensional Euclidean (analytic) geometry. 2H
2H