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 (of valence R), H (of valence H), etc. We say that a (dependent) quantity R is proportional to a second (independent) quantity H when each component (with reference to an arbitrary vector base) of R is proportional, with different (perhaps constant) weights, to all coordinates of H . 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 H H H R R . G ; 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 2H H H G . , and postulate a second relation between the dependent and independent polyadics, the scalar 2W: H H 2H H H H H H W 2 . . . G . The existence of this scalar implies the symmetry of 2H G , i.e., the equality of the proportionality polyadic with its transpose: H 2H 2H 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 occurs int 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 N-dimensional 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.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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 (of valence R), H
(of valence H), etc. We say that a (dependent) quantity R is proportional to a second (independent) quantity H when
each component (with reference to an arbitrary vector base) of R is proportional, with different (perhaps constant)
weights, to all coordinates of H . 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
HH
HRR .G ; the valence of the polyadic
R+HG 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 2H
H H G.
, and postulate a second relation between the dependent and independent
polyadics, the scalar 2W: HH
2HH
HHH
H W2 ... G . The existence of this scalar implies the symmetry of 2HG ,
i.e., the equality of the proportionality polyadic with its transpose: H2H2H
GG . 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 2HG is to be preferred.
Some of the most interesting cases of proportionality between polyadics occurs int 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 2HG . 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 N-dimensional 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.
2
SECTION I: POLYADICS AND GEOMETRY.
I.1 - Physical Magnitudes, Polyadics and Euclidean Space.
All physical magnitudes can be represented by tensors of different orders, or by polyadics of different valences; scalars are
polyadic of zero valence, vectors are of valence one, dyadics have valence two etc..
Scalars (work, energy, temperature, entropy etc.) and vectors (force, velocity, acceleration, electric field etc.) are
well known from elementary mechanics and electromagnetics. Some dyadics are also known as stress and strain in the
theory of elasticity and in fluid mechanics; others, like dielectric permittivity, dielectric impermeability, thermal diffusivity
etc. are known but in crystal physics. Triadics are common one of the better known examples is the piezoelectric triadic.
Tetradics are even more common: the stiffness (and compliance) tetradic, the elasto-resistivity tetradic, the piezo-optical
and the electro-optical tetradic; and in theoretical geometry, the Riemann-Cristophell curvature tetradic.
Dyadics.
With two given ordered sets of vectors, say },,{ 321 eee and },,{ 321aaa , between which we can establish a bi-unique
correspondence (the ei is the correspondent of ai), we can generate dyadics and represent then by the symbolic sum
iiae , for i=1,2,3, the repeated indexes in different levels denoting sum in the range (Gibbs, 1901; Drew, 1961). The ei
are said to be the antecedent and the ai the consequent of the dyadic. If we insert between the antecedent and the
consequent a dot we obtain from the dyadic a number, called the scalar of the dyadic and denoted by s; if ewe insert a
(a inverted v) we obtain a vector, called the vector of that dyadic and denoted by V. A simple example of this
correspondence from the theory of elasticity is Cauchy's tetrahedron of tension: to each unit vector ie normal to a face "i"
corresponds one and only one stress-vector is . This correspondence generates the stress dyadic iiˆ se (for i=1,2,3). The
stress dyadic is symmetric, that is, it is equal to its transpose (obtained by interchanging antecedent and correspondent
consequent) denoted by T. We have: T
ii ˆ es , in which case V=o (o is the null vector). The converse is true, that
is, the necessary and sufficient condition that a dyadic be symmetric is that its vector vanishes. A dyadic, say A, can also
be anti-symmetric, when it is equal to the negative of its symmetric: A=-AT.
Triadics.
Using ordered and correspondent sets of vectors and dyadics we generate triadics. For piezoelectric crystals (that generate
an electric field when deformed) there is a bi-unique correspondence between each electric vector field ie in a point and
the strain dyadic i in this point (or vice-versa). This generates the piezoelectric triadic ii3e . In this form the dyadics
are the triadic antecedent and the vectors the consequent. If each one of the dyadics i could be related to other sets of
vectors, say },,{ 321 rrr and },,{ 3i2i1iaaa we could write by substitution: ij
ji3 )( era for i,j=1,2,3 where the
parentheses are necessary. Triadics can also present symmetries depending on the characteristics of the original dyadics.
The Euclidean space of a polyadic.
With polyadics of valence H (the H-adics) and certain basic operations defined between then we create a Euclidean space
(with up to 3H dimensions). To generate a linear space these operations are the addition of polyadics (of the same valence)
and the multiplication of a polyadic by a number. These two operations are similar to their counterparts defined for
vectors. Another important operation must be defined: the multiple dot multiplication with polyadics (we will abbreviate
mdm) which is based on the dot multiplication of vectors. For example: the dot product of the dyadic kkgb by the
vector v is the vector defined by the law )( kk
.vgb.v . The double dot product of the triadic ii3e by the dyadic
is the vector defined by the law ))(( kiki3
.ge.b: . In view of the first definition this later expression can be written
as ))](([ kik
jji3
.ge.bra: . Proceeding in this manner we can define the multiple dot product (abbreviated mdp) as far
as the number of dot does not exceed the valence of the polyadic factor of smaller valence. We say that two polyadics are
equal if their mdp by a same and any polyadic are equal. After these definitions and the demonstration of some theorems it
is possible to write ijji3
era , ))(( kik
jji3
.ge.bra: and similar expressions.
3
The polyadic can be represented by an arrow in its space.
We can also calculate the double dot product of a dyadic by itself and, in general, of a H-adic H by itself; this product,
indicated in the form HH
H . , is a scalar, always positive and called the norm of that H-adic. The positive square root
of the norm of the polyadic H is its modulus and denoted by | H |. Hence, H-adics - as vectors - can be written in the
form ˆ|| HHH where H is a unit H-adic parallel to H . It may be shown that the square of the H-dot product of
two H-adics H and H is less than the product of its norms; hence, there is an angle defined by two H-adics such that its
cosine equals the H dot product between then divided by the product of their moduli; and we write:
),cos( || || HHHHHH
H .. Again we have found a similarity with vector operations. After the choice of a scale,
we can represent a polyadic in its space by an arrow whose length and direction be the magnitude and direction of the
polyadic. The angle of two polyadics, for example, is the angle apanned by their arrows.
Dimension and base in a polyadic space.
We can say that two non-vanisning H-adics are perpendicular if its H-dot product vanishes. One vector in its space is
orthogonal to at most two other vectors. What about dyadics, triadics and polyadics?. To answer this question we must first
look for the maximal number of linear independent H-adics of the generated space, that is, its dimension. We can conclude
that this number is up to 3H (3 for vectors, 9 for dyadics etc.) and say that any set of 3
H independent H-adics of a H-adic
space is a base of this space. Given the G3H H-adics of a G-space (a subspace of the H-adic space), G
H2
H1
H ..., ,, ,
they will be linearly independent if the determinant (or order G) does not vanish:
GHH
GH
2HH
GH
1HH
GH
GHH
2H
2HH
2H
1HH
2H
GHH
1H
2HH
1H
1HH
1H
H
...
.........
...
...
||
. . .
. . .
. . .
, (I.1.01).
Now, given a H-adic base any of a G-space, GH
2H
1H ..., ,, , we can determine its reciprocal, that is, the base
GH2H1H ..., ,, such that j
ijHH
iH . where the deltas are the Kronecker deltas.
Other types of products.
We can define also a multiple skew product of G-1 H-adics of a G-subspace. For example: as for vectors, the (simple)
skew product of the two H-adics H and H of a 3-subspace is a third H-adic, say H , whose direction is normal to the
directions of the factors (hence this H-adic belongs to the 3-subspace), its unit H pointing to the side on which a rotation
less than 180 from the first to the second appears positive, and whose magnitude equal to the product of their lengths
multiplied by the sine of the angle between then. We write: ˆ ),sin( || || HHHHHHHH . If
},,{ 3H
2H
1H is a base of the 3-space we can write also the pseudo-determinant:
3HH
H2HH
H1HH
H
3HH
H2HH
H1HH
H
3H2H1H
HHHH
||
...
... , (I.1.02),
a formula well known for vectors (H=1, 11H
e etc.). To extend the definition we can use this determinant as reference
and amplify it for at most H-1 H-adics since the skew product must belong to the H-space.
From the two multiple operations defined (the dot and the skew), we can define the multiple mixed product of G H-
adics H , H , ..., H of a G-space by the expression:
GHH
H2HH
H1HH
H
GHH
H2HH
H1HH
H
GHH
H2HH
H1HH
H
HHHHHHHHH
...
.........
...
... )... (
...
...
...
. , (I.1.03).
If we substitute in this expression H for 1H , H for 2H etc., we can say also that the set
1H , 2H , ..., GH form a
base if their multiple mixed product does not vanish (as for vectors).
4
The polyadic associated matrix.
The bases in a H-adic space can be formed with P-adics as long as PH, since we could not express a P-adic in a H-adic
base. If, say, e1, e2, e3 and e1, e
2, e
3 are reciprocal vector bases then we can generate the two following two groups of nine
dyads: e1e1, e1e
2, ... and e
1e1 etc. to compose dyadic reciprocal bases; or the two group of 27 triads: e1e
1e1, e1e
1e2, ... and
e1e1e1, e
1e
1e2 etc.) to compose triadic reciprocal bases etc.. . Taking these polyades as bases and a coupled reference system
- in which case we shall say that the H-adic is referred to a vector base - we can associate to a H-adic a (rectangular)
3H3
H-1 matrix if H is odd (a matrix whose elements are the H-adic coordinates with respect to that base); and a square
3H3
H matrix if H is even. In the latter case the trace is the H-adic scalar. This may lead to huge computational calculations
since the number of rows and column of these matrices can be large. The abstract image of a base in a space is a "star of
arrows", and a "pencil of arrows" in subspaces. If we imagine the arrow of a polyadic with its initial point coincident with
the vertex of the star of the base arrows, the coordinates of its end point are the coordinates of the polyadic, i.e., the
elements of its associated matrix (this justifies the name coordinate instead of component).
I.2 - Physical laws, Linear Transformations and Polyadic Geometry.
The operations between polyadics studied in Polyadic Calculus (the mdm in special) are appropriate to express linear and
non linear physical laws. A compact general form to express that a H-adic (the value of a function) is a function of a P-adic
(the argument of the function), that is, )( PHH , is the generalized Taylor-series:
... 2
1 PP2P
P2HPP
PH
0HH
, (I.2.01),
where H0,
H+P,
H+2P etc. are polyadics independent of the current
H and
P (perhaps functions of time, temperature etc).
We say also that H and
P are, respectively, the dependent and independent variables. If this function is linear there are
only the first two terms of the series. This means - for polyadics referred to a common base - that each one coordinate H is
a linear function of (or proportional to) all coordinates P. For most physical laws the linear description is sufficient. The
foregoing considerations means that if the end point of the arrow representing P describes a line, a plane or a sphere in the
space (of dimension 3P), then the
H ending point arrow describes a line, a plane or a sphere in its space (of dimension up
to 3H), respectively.
A new matrix operation to represent the polyadic linear laws.
With respect to specified reciprocal vector bases we can associate matrices with the four polyadics present in the linear law
(H,
H0,
H+P and
P) which can also be written in matrix form since we define a new operation between matrices (which
differs from the classic operation), called double scalar product, to translate the mdp of two polyadics. Let us consider two
arbitrary matrices, NM]A[ and N
M]B[ of the same order (with M rows and N columns), being ijA and ijB their
corresponding elements. We define as the double dot product A:B of these matrices (in arbitrary order) as the number
MN
MN12
1211
11ij
ij BA ... BABABA , (I.2.02).
Notice that the polyadic associate matrices in the linear laws are multi-ordinal, that is, the numbers of rows and columns of
one are multiples of the correspondent ones in the other: for instance, QP]B[ and [ ]A
LP
MQ (with L and M integers). The
second matrix may be resolved in LM blocks with P rows and Q columns, that is, this matrix has L rows and M columns
whose elements are matrices Aij with P rows and Q columns. We define the double dot product [ ]ALP
MQ : [B]P
Q of the multi-
ordinally linked matrices [ ]ALP
MQ and QP]B[ , in this order, as the matrix with L rows and M columns whose elements are
the double dot product of each [ ]ALP
MQ sub-matrix [ ]Aij P
Q (with i = 1, 2, ..., L e j = 1, 2, ..., M) with the matrix [B]P
Q . Thus,
QP
Q
PLMQ
PL2Q
PL1
Q
P2MQ
P22Q
P21
Q
P1MQ
P12Q
P11
B
A...AA
............
A...AA
A...AA
:
M
L
QP
Q
PLMQP
Q
PL2QP
Q
PL1
QP
Q
P2MQP
Q
P22QP
Q
P21
QP
Q
P1MQP
Q
P12QP
Q
P11
B A...B AB A
............
B A...B AB A
B A...B AB A
:::
:::
:::
, (I.2.03).
The correspondent operation - the double dot multiplication of multi-ordinally linked matrices - always exists. It is
commutative, distributive with respect to addition, and generally non associative.
5
Example 1:
2311636
0x1+1x2(-1)x1+2x25x1+3x2
0x1+3x2(-1)x1+2x24x11x212
015123014321
1
2
3x1=3
2x2=4
: .
Example 2:
Let us consider the product )R( )A( mmkji
ijk3e.eee.r ; we have: jik
ijkRA ee and the following matrix
representations
:
AAAAAAAAA
AAAAAAAAA
AAAAAAAAA3x3=9
3x1=3
333332331323322321313312311
233232231223222221213212211
133132131123122121113112111
3
3
R R R1 2 3 1
3
,
and
1
33
2
1
3x1=3
3x3=9
333323313
332322312
331321311
233223213
232222212
231221211
133123113
132122112
131121111
3
3
R
R
R
AAA
AAA
AAA
AAA
AAA
AAA
AAA
AAA
AAA
: ,
Notice that each dyadic product coordinate is a function of all vector coordinates factor.
Example 3:
The product 3 : , with 3 ijk
i j kA e e e and B
rs
r se e , is the vector v e e A B Vijk
jk i
i
i. This expression can be
written in matrix form as
V
V
V
A A AA A AA A AA A AA A AA A AA A AA A AA A A
B B B
B B B
B B B
1
2
3
111 112 113
121 122 123
131 132 133
211 212 213
221 222 223
231 232 233
311 312 313
321 322 323
331 332 333
3x3
1x3
11 12 13
21 22 23
31 32 33
3
1
3
3
: ,
or
3
1321 VVV
A A A A A A A A AA A A A A A A A AA A A A A A A A A
111 112 113 211 212 213 311 312 313
121 122 123 221 222 223 321 322 323
131 132 133 231 232 233 331 332 333
3=1x3
9=3x3
:
B B B
B B B
B B B
11 12 13
21 22 23
31 32 33
3
3
,
but these forms are not unique. Again notice that each vector coordinate is a function of all coordinates of the dyadic
factor.
Example 4:
For the same polyadics of example 3 we have, on the other hand: 3 3 ijk
ks i j
s A B . e e e . The corresponding matrix
notation is
3
9
3
111 112 113
121 122 123
131 132 133
211 212 213
221 222 223
231 232 233
311 312 313
321 322 323
331 332 333
9
3
11 12 13
21 22 23
31 32 33 3
3
A A A
A A A
A A A
A A A
A A A
A A A
A A A
A A A
A A A
B B B
B B B
B B B
. .
6
We notice that in this case the individual coordinates of the triadic product is not a function of all coordinates of the
dyadic factor, but only of some ones.
Polyadic Geometry.
The mdm of polyadics can be interpreted geometrically; particularly a linear mdm can be regarded as a linear
transformation (LT) between polyadics (of different spaces) achieved by a polyadic operator whose valence is the sum of
the valences of the input and output polyadics. Linear Transformations from one vector space (valence 1 and dim3) to
another vector space, operated by a dyadic (valence 2), are well known and sometimes mentioned as a "Vectorial
Geometry". In this LT the operator may perform translation, rotation - i.e., rigid transformations - and deformation
(implying changes in distances and angles in the neighborhood of a point). Such linear transformations occur frequently in
the theory of classical mechanics and electromagnetics. The more complex cases - the LTs between a vector space and a
dyadic space (performed by a triadic), or between two dyadic spaces (dim9) performed by a tetradic - are rarely
mentioned. Such transformations occur in the theory of elasticity and electromagnetism, and in Crystallography.
The polyadic operator may perform also translation, rotation and deformation. For example: a rotation dyadic can
rotate vectors by simple dot multiplication, a rotation tetradic can rotate dyadics by double dot multiplication. In the same
way a tetradic may stretch or shrink the dyadic defined by two points (in a dyadic space) and diminish or enlarge the angle
between two dyadics. There exist also a polyadic that performs the identity transformation: this polyadic is called the unit
polyadic of the space (or subspace) and has always even valence; it is denoted by 2H and its associated matrix is the GG
unit matrix (for G3H).
It follows that there is a multi-dimensional purely Euclidean geometry hidden in the physical laws, which can be as
useful as the common two- and three dimensional ones. It could be called a "Polyadic Geometry" and has still to be
explored.
From this point of view, the triangle defined by three points, e.g., is a universal entity whether its sides are vectors,
dyadics or arbitrary H-adics, each defined in the corresponding space with the corresponding dimension. The so called
"cosine law for triangles" holds universally, whether the squares of the sides of the triangle - the norm of the H-adic - are
determined in a space of 3, 9 or any other dimension.
This approach to the physical laws is now unified; it extends or complements the isolated cases mentioned above.
But what are the consequences of theses geometrical concepts for the physical laws they represent?
I.3 - Linear Transformations, Experimental Measurements and Statistical Polyadics.
This geometrical interpretation of physical laws suggests us a single way to determine the LT polyadic operator. A
fundamental theorem: If in a G-space G independent P-adics Pi. (i=1,2, ..., G) is associated bi-uniquely with G H-adics
Hi, then the linear transformation polyadic operator
H+P (or the proportionality polyadic) is determined as
H+P=
Hi
P
i
for i=1,2, ..., G. From the physical point of view, the physical law connecting two physical magnitudes can be determined
by measuring under specified physical conditions (of time, temperature etc.) G pairs of the correspondent magnitudes
under the geometrical condition that one set of one member of the pairs, say P1,
P2, ...,
PG, is composed of independent
polyadics, i.e., (P1
P2...
PG)0 does not vanish. For example: to determine the tetradic which connects the stress dyadic
with the strain dyadic in linear elasticity, we must chose and measure six independent strains dyadics (instead of nine in
view of the symmetry of the strain dyadic) and the corresponding stress dyadics (which are not necessarily independent).
This approach may require appropriate laboratory devices and accurate measurements. Moreover, the
measurements must be collected within the "media" in a "state of proportionality". These measurements will be performed
with respect to a convenient chosen vector base, say e1, e2, e3 (and its reciprocal e1, e
2, e
3 if the base is not orthonormal).
With this base, the associate matrix and the ensuing calculations many other physical problems can be solved.
Though one such determination of G independent pairs of polyadics is theoretically sufficient to define the true
associate polyadic coordinate matrix, there may be practical difficulties. The matrix obtained by two such experiments
may be not equal, mainly due to observational errors. To deal with these, the classical theory of "probability and error
statistics" has to be extended to polyadics. Such a theory has to be based on the representation of polyadics by their
invariants.
I.4 - The Physical Phenomenon is Equivalent to a System of Linear Polyadic Equations.
Physical phenomena occur in definite regions of the physical space. These regions are seen as a field of the various
"quantities" participating in the phenomenon. This quantities (scalar, vector, dyadic etc.) are continuous functions of
position (even in the limit) and time (often non zero only after a definite initial time) with continuous first derivatives.
In a given point P and time t of a field, one quantity of a set (which we select as dependent) can be proportional to
one or more of the magnitudes of different orders of a second set (selected as independent), each one of these latter varying
with P and t.
Hence we can conclude, from the mathematical point of view, that:
7
1) the physical phenomenon is equivalent to a system of linear polyadic equations involving (arbitrarily selected)
dependent and independent magnitudes;
2) this system must be compatible, that is, to a given set of values of the independent variables there always
correspond one and only one set of values for the dependent variables;
3) as this system must be true in space and time, there must be defined its values in the beginning of the time
measurements (with correspondent position) and at the boundaries of the region (with the correspondent time).
It should be noticed that when one of the variables undergoes a differential operation (for example, when it derives
from a potential) the system pass to be a system of linear or non linear differential equations (according as the derivatives
appears as simple derivatives, or as second derivatives, as third etc. or, also, as products of different derivatives) but
always with degree one (the power of the derivatives is always one).
I.5 - Eigenvalues and Eigenpolyadics.
In vector geometry we look for "special bases" with respect to which we can simplify the geometrical studies; in physics,
besides this geometrical simplification, we may be interested in the facilitation of experimental measurements. This is
always possible when the LTs are to be performed between polyadics of the same valence, say H and
H, in which case
the polyadic operator has even valence, 2H. We ask: when HHH
2H X . for some scalar X, i.e., is there any H-adic
H which is transformed into a H-adic parallel to itself?. Or, what is the same, when HHH
2H2H )X( . ?. The
existence of this equality for some H implies that the 2H-adic between the parentheses must be incomplete, that is, its
associated mixed matrix must be degenerate (its determinant must vanish).
If we put 2H H
i
H i for i=1,2, ..., G with respect to some H-adic reciprocal bases {H*} and {
H
*} of the G-
space, we define the 2H adjoint, and denote it by G
~2H , by the expression
fatores 1G
mH
jH
iHmHjHiHG
~2H
... ...
1)!(G
1
for (i, j, ... ,m = 1, 2, ..., G), (I.5.01).
where we are using the ready defined multiple skew multiplication of H-adics. The G~
2H associated matrix is the adjoint
of the 2H associated matrix. This adjoint and its leading (diagonal) minors express the condition for 2H2H X to be
incomplete:
0 )1(X )1(X )1(
... X X X X
G2HGG
~
E 2HG2)
~1(G
E 2HG
3G3~
E 2H2G2
~
E 2H1G
E2HG
, (I.5.02),
where 2H
G is the
2H determinant; the coefficient of the linear term is the sum of the leading minor of degree one of this
determinant, that is, the scalar of G~
2H ; the coefficient of the quadratic term is the sum of the leading minors of degree
two of this determinant, that is, the scalar of )~
1-(G2H ; etc.. This equation is the "2H-adic characteristic equation". The
solution of this problem brings us to the determination of the 3H (invariant) eigenvalues and eigenH-adics of the (statistical
measured) 2H-adic operator. Then it is possible to demonstrate the Cayley-Hamilton theorem (for future usefulness) for
polyadics, that is: every 2H-adic satisfy its own characteristic equation.
SECTION II: THE ESSENTIAL CONDITION FOR A GEOMETRICAL APPROACH TO
PHYSICAL LAWS.
II.1 - A particular situation, largely useful in Physics.
Let it H and H the H-adics (polyadics of valence H) representing two H-order proportional and continuous
variable quantities defined in the current point O of a G-space (G<3H), one of then, say H , taken as independent
variable. We have:
HH
HHHHH | | | | ,ˆ|| . , and H
H H .
1 (II.1.01)1,
1 - When a polyadic is expressed in an arbitrary vector base by its "coordinates" (covariant, contravariant, etc.), its norm (always a positive number) is
equal to the sum of the product of the coordinates of one type with the corresponding coordinates of opposite type; the square root of its norm is its
8
where ˆ and || | | ,| | HHH denotes the norm, the modulus and the unitary (a H-adic of norm 1 parallel to H ) of the H-
adic H .
The proportionality of the magnitudes – the linear physical law - can be expressed as the linear polyadic equation
HH
H2H .G , (II.1.021),
where the dependent variable H , besides to have variable direction, has also variable norm (hence, a variable modulus);
and 2H G - the proportionality polyadic, independent of the point O and the current H and H (a constant or, perhaps,
a function of time, temperature or other parameters) - must be a complete 2H-adic into the G-space if it is necessary to
express H as a function of H . Hence, admitting that the 2H G determinant in this G-space does not vanish, we can
invert (II.1.021) and write
0)det( 2H G and H H
H H 2G
. H 2H
H H G .
1 , (II.1.022).
The pair of inverse laws (II.1.021) and (II.1.022) exist in the G-space if 2H G has non vanishing eigenvalues (in this
space). Else the law exist in a space of dimension one unit less (a subspace if G3H) for each vanishing 2H G eigenvalue;
and in this subspace the 2H G will be seen complete.
The polyadic G2H characterizes the medium completely for the phenomenon governed by the law (II.1.01). Thus
we can say that these polyadics are the parameters of the medium with respect to the phenomenon under consideration;
with our laboratory devices we can determine its cartesian coordinates by specifying a convenient vector base. Before hand
we must observe that the polyadic coordinates for the specified phenomenon are different for different observers (because
each one chooses his own vector base).
A Postulate, Specific Magnitudes and 2HG Symmetry.
In view of physical usefulness we might admit the following
Postulate:
There exists a continuous variable function of scalar value 2W with several continuous derivatives,
defining a new physical magnitude by the law
W2 HH
2HH
HHH
HHH
H .... G , (II.1.031).
To simplify the mathematical handling we will introduce the new variables
HH
HH
00
H H 2W=| | 2W and 2w=
2W
| |
2W
|| ||
| |, , , (II.1.04),
so that - besides the unchanged law (II.1.01) - we have,
H H H H H
H H H 2 2G G. .
, (II.1.02),
2w H H H H
H H H
H 2H
H H . . . .G , (II.1.03).
ˆ ˆ W2 HH
HHH
H0 .. , (II.1.031).
We could name magnitudes H and 2W0 "specific magnitudes", or "magnitudes by unit of H intensity
(modulus)" since | H | represents a quantity of the magnitude H .
From (II.1.02) and (II.1.03) we deduce 0= ˆ )( ˆ : ˆ HH
H2H2HH
HH .. GG
, which is possible if and only if
2H 2H H 2H G G
, that is,
2H 2H H G G
, (II.1.05).
Hence, the acceptance of the postulate (II.1.03) carries the 2HG single symmetry2 for any H.
The unit H-adic H (in a G-space) has G coordinates when resolved in a H-adic base of this space but only G-1 are
independent because its norm is equal to one. Taking the point O of this euclidean G-space as origin of polyadics, H can
modulus. If the base is orthonormal the norm is equal to the sum of the square of its coordinates.
2 This concept is valid only for polyadics of even valence (as dyadics, tetradics etc.).
9
be seen as the H-adic position of a point on a hiperspherical surface (or, simply, a spherical surface) of unit radius
centered at O; it defines a hiperdirection in this space (or, simply, a direction). In elasticity, for H=1 and G=3 for
example, H is a vector p representing the stress vector on a plane with normal unit vector H n ; and 2w is the normal
stress, , on this plane. Still in elasticity3, for H=2 and G=6, H is the stress dyadic on a subspace of dimension six4 at
the point O (of the 6-space spanned by six independent H ) with normal unit dyadic H ; and 2W0 represents twice the
specific density energy (the stored strain energy by unit of volume at this point), although the value 2W is more commonly
used.
In view of the isomorphism with vector spaces we call H ( H ) the H-adic (specific) projection of 2HG in the
direction H . Similarly, 2W (2w) is the scalar (specific) projection of H ( H ) in the direction H ; it is also called the
radial (specific) value of 2HG relative to the direction H .
For two different directions H and H we write (in accordance with polyadic algebra)
ˆ ˆ ˆ ˆ w2 HH
2HH
HHH
2HH
H.... GG , (II.1.061),
or
HH
HHH
H ˆ ˆ w2 .. (II.1.06).
Hence the proposition:
The 2H
G projection H relative to a direction H projected on a second direction H , is
equal to the 2H
G projection H relative to this direction projected on the first direction H .
We call the scalar 2w''' the tangential (specific) value of 2H
G relative to the directions H and
H . It is interesting to
note that in the theory of elasticity equality (II.1.06) translates into Betti's law. For H=1 - in which case H is a force
vector, H is a unit displacement vector and the tangential value a work - we state:
"the work done by a certain force f1 (or a system of forces f1, f1', ...) in virtue of the application of the force f2 (or a
system of forces f2, f2', ...) is equal to the work that should be produced by this latter, in virtue of the application of
the first".
For H=2 - in which case H is a stress dyadic and H a unit strain dyadic - we state:
"for a linear elastic body the work done by a first state of stress in the strain of a second state of stress is equal to
the work done by the second state of stress in the strain of the first state of stress".
The problem consist in the study of the simultaneous laws (II.1.021) and (II.1.031) when the independent and
continuous variable H assumes all possible finite values of a certain defined domain that will be not discussed here from
physical point of view.
II.2 – The 2H
G Characteristic Elements or Eigensystems.
Orthogonal and unit H-adic bases.
There is a well-known theorem:
In a G-dimensional H-adic space there exists H-adic orthogonal bases.
If } ..., ,,{ GH
2H
1H constitute an orthogonal base, then }ˆ..., ,ˆ,ˆ{ G
H2
H1
H - the set of the unit dyadics of the
former - also constitute a base whether the metric matrix of this set is the GxG unit matrix, or the principal of this matrix
(which is equal to 1) is of degree G. Hence, 1 is the norm of this base. The unit and orthogonal H-adic bases are called
orthonormal bases; for these bases we can write
ijjH
1H ˆ ˆ : (i,j=1,2, ..., G), (II.2.01),
where the ij are the Kronecker deltas. One notable particular case is that in which the base dyadics (H=2) of a 9-space are
3 We shall show further down that the space of stress surrounding the point O is six dimensional.
4 In a 6-space, a (non null) dyadic can be orthogonal to at most five other dyadics.
10
dyads formed with vectors of a orthonormal vector base { }i j k , that is, ijkjkkjjii ˆˆˆ ...., ,ˆˆˆ ,ˆˆˆ ,ˆˆˆ ,ˆˆˆ94321 ,
where evidently the norm of this base is equal to one.
The following theorem is also known:
To each pair of different eigenvalues corresponds orthogonal unit eigenH-adics.
If all G eigenvalues of 2H
G are different we have G distinct mutually orthogonal eigenH-adics that can be assumed
to have unit norms: G21ˆ ..., ,ˆ,ˆ ; this means,
ijjiG21ˆ ˆ H ... HH : (i,j=1,2, ..., G), (II.2.021).
The metric matrix associated to this set is ]ˆˆ[ ji : , that is, the GxG unit matrix whose determinant (the norm of the base)
is 1. Hence, the set constitute an orthonormal base in the entire space. So we can represent 2H
G in the form:
iiiH2 ˆˆG G (sum for i=1,2, ..., G), (II.2.022).
This diagonal representation is preferable because of the properties of the eigenH-adics (Kelvin, 1856; Mehrabadi and
Cowin, 1994; Helbig, 1994).
Let us suppose now that 2H
G has a double eigenvalue, say G1G GG . Then (II.2.022) is valid for 1=1,2, ..., G-1,
i.e., there exist G-1 mutually orthogonal eigenvectors in a (G-1)-space of the G-space. It can be proved that the cross
product of this G-1 eigendyadics, 1G21ˆ ... ˆˆ , is still an eigendyadic of the tetradic.
In general, if a 2H
G has S simple eigenvalues, hence S different eigenH-adics, the cross product of this S eigenH-
adics is still a 2H
G eigenH-adic; the cross product of these S+1 eigenH-adics is also a 2H
G eigenH-adic; and so on until we
can complete the set of G eigenH-adics.
II.3 – The stationary proportionality polyadic specific radial value (2w).
The extreme of w at the point O is a linked extreme because H might satisfy (II.1.01). If a direction exist at O that makes
w an extreme then dw=0 in this direction. By differentiating (II.1.03) we get: 0ˆd ˆ 2dw2 HH
2HH
H .. G . From
(II.1.01), we deduce H
H H d .
0 ; hence, we conclude that H and H 2H H H G .
are orthogonal to d H , that is,
orthogonal to the same plane (hyperplane) tangent to the spherical surface H
H H .
1 . This means that the H-adics H
and H 2H H H G .
must be parallels.
By (II.1.031) we write5: 2w=| | cos( , )H H H , whence we deduce that the 2w extreme value is |H| if the H-adics
H and
H are parallels (a maximum corresponding to the null angle and a minimum to 180). The parallel condition may
be expressed in the form X H H H H H 2 G .
, where X and H are a scalar and a H-adic to be determined, which, as
we know, are the 2H
G eigenvalues and correspondent eigenH-adics. Hence:
The 2H
G radial value, 2w, given by (II.1.03), is stationary at the point O of the G- space for directions H drawn by O and parallels to the
2HG eigenH-adics.
The G 2H
G eigenvalues Gu are all real (because it is symmetric) and we will suppose they are single and non null;
representing its corresponding (real) unit eigenH-adics by H
u, we write:
2H
u
H
u
H
uG G (sum for u=1,2, ..., G), (II.3.01),
since
2H
H H
1
H G G.
1 1 ,
2H
H H
2
H G G.
2 2 , ..., (II.3.011),
and
H
H 2H
H H 1 2
0. .
G =H
H 2H
H H 1 3. .
G .....= H
H 2H
H H = ... 2 3. .
G , (II.3.012).
5 - For multiple dot multiplication of polyadics essentially same concepts hold as for scalar multiplication of vectors.
11
The principal polyadic and principal directions.
The polyadic 2H
G, written in the form (II.3.01), is said to be sad represented in its principal form in the point O; its
eigenH-adics are its principal directions and constitute the principal (orthonormal) H-adic base in the point O. Referred
to this principal H-adic base, the 2H
G associated matrix is a (GxG) diagonal matrix, its diagonal elements being the 2H
G
eigenvalues; hence, (II.1.061) and (II.3.012) permits us to conclude:
The 2H
G tangential values relative to any two different principal directions at a point are always nil.
Substitution of (II.3.01) into (II.1.03) gives:
2w=( ) G
( ) G ( ) G ...,
H H H
u2
u
H H H
12
1H
H H
22
2
.
. .
(u=1,2, ..., G) (II.3.04),
whence we conclude:
Each eigenvalue of 2H
G is a stationary value of 2w in the point O of the G-space, which occur for the
corresponding 2H
G eigenH-adic direction.
If we denote by Eu and Su the projections (coordinates) of H and H on the eigenH-adic of base H
u, that is, if
we put
H
H H
u u E
. and H
H H
u u S . , (II.3.05),
the law (II.1.02) is then equivalent to the system
S G ES G E
S G E
1 1 1
2 2 2
G G G
...
, (II.3.06).
We conclude:
When, in the vicinity of a point, the G-space is referred to the eigenH-adic orthogonal base of the
symmetric polyadic 2H
G, the ratio of the H-order magnitudes with the same subscript is equal to the
corresponding 2H-adic eigenvalue.
II.4 - The Projection Norm and Octahedral Directions.
For an arbitrary direction H in the vicinity of the point O of the G-space we can write, with respect to the principal base
}ˆ ..., ,ˆ,ˆ{ GH
2H
1H :
H H
H H
u
H
u ( )
. (u=1,2, ..., G), (II.4.01),
being
1)ˆ ˆ(
G
1
2u
HH
H . , (II.4.011),
because H
H H .
1 . The numbers H
H H
u
. are the G principal director cosines of the direction. In general they are
all different, but for a particular direction they can be all equal. For a given and ordered set of G squares, whose sum is
equal to one, there are 2G directions (that is, all the arrangements with repetition of the signs + and – taken G by G with the
modulus of the director cosines, GG2 2(AR) ) whose director cosines have the same modulus.
We shall call octahedral directions, or octahedral H-adics of 2H
G, the unit H-adics equally inclined to its
principal directions. Denoting a octahedral direction by H
oct we can write from (II.4.01),
H
oct
H
oct
H H
u u ( )
. (u=1,2, ..., G), (II.4.012),
and from (II.4.011), since the cosines (cos oct) are all equal:
12
for (u=1,2, ..., G) G
G
G
1 cosˆ ˆ
octuHH
octH . , (II.4.013).
So, for G=2, 45oct , for G=3, '4454oct , for G=4, 60oct , for G=9, "44'3170oct
etc..
For any octahedral direction to which w=woct corresponds we have
G
1
uu2
uHH
octH
octHH
2HH
octH
oct GG
1G)ˆ ˆ(ˆ ˆ =2w ... G , (II.4.014),
that is:
In any octahedral direction, the 2H
G radial value 2woct is equal to the invariant average of the
eigenvalues.
For an arbitrary direction any in the space we can write the norm of the correspondent 2H
G projection as
ˆ ˆ =ˆ ˆ | || | HH
22HH
HHH
2HH
2HH
HH..... GGG
, with 22H2HH
2H =
GGG . , (II.4.02).
The polyadic 2H2 G is, by definition, the double dot power of the polyadic