THE GEOMETRY HIDDEN IN THE PHYSICAL LAWS. By ELYSIO R. F. RUGGERI Abstract THE POLYADIC GEOMETRY. I.1 - Polyadics and physical magnitudes. The Polyadic Calculus (Ruggeri, 2000) defines polyadics and operations with polyadics; through this operations we can express the physical (non relativistic) laws in a unified manner. Indeed, we see the physical magnitudes mathematically represented by entities called tensors, or free polyadics of different valences. The polyadics of valence zero are the scalars; its space is 1-dimensional. The polyadics of valence one are the vectors or monadics and can be sketched in its 3-dimensional space by an arrow with a single pointer at its ending point. In general, the polyadics of valence H, shortly referred as H-adics – the dyadics (H=2), triadics (H=3), tetradics (H=4) etc – can be sketched by hyper-arrows (or, simply, arrows) in its 3 H -dimensional space with, respectively, double, triple etc pointer at its ending points; the other extreme point of these arrows are the starting points (or origin). The valence H of a polyadic defines the maximal number of ordinary and different directions inherent to it. This can be viewed by the so called 3 H-1 -nomial representation of the H-adic: the symbolic sum of 3 H-1 sets of H juxtaposed vectors (3 H-1 H-ades). To a vector is associated only one direction; to a dyadic (by the trinomial representation), as a sum of three sets of two juxtaposed vectors (three dyads); to triadics, as a sum of nine sets of three vectors (nine triads); to tetradics, twenty seven sets of four vectors (twenty seven tetrads) etc. The several physical laws dictates the relationship between the polyadics involved in a certain phenomenon. I.2 - Physics and geometry. Some simple physical laws can be more easily understood evoking single geometric concepts and figures of the 3-dimensional euclidian geometry, the like the arrows. This treasure – the geometrical picturesque view of the physical laws, left overly known and still conserved for vectors (in Mechanics, Electromagnetics etc.) - was lost in the time for polyadics of major valence perhaps owing to difficulties. We shall show here that, with the
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THE GEOMETRY HIDDEN IN THE PHYSICAL LAWS.
By ELYSIO R. F. RUGGERI
Abstract
THE POLYADIC GEOMETRY.
I.1 - Polyadics and physical magnitudes.
The Polyadic Calculus (Ruggeri, 2000) defines polyadics and operations with polyadics;
through this operations we can express the physical (non relativistic) laws in a unified
manner. Indeed, we see the physical magnitudes mathematically represented by entities
called tensors, or free polyadics of different valences. The polyadics of valence zero are the
scalars; its space is 1-dimensional. The polyadics of valence one are the vectors or
monadics and can be sketched in its 3-dimensional space by an arrow with a single pointer
at its ending point. In general, the polyadics of valence H, shortly referred as H-adics – the
dyadics (H=2), triadics (H=3), tetradics (H=4) etc – can be sketched by hyper-arrows (or,
simply, arrows) in its 3H-dimensional space with, respectively, double, triple etc pointer at
its ending points; the other extreme point of these arrows are the starting points (or origin).
The valence H of a polyadic defines the maximal number of ordinary and different
directions inherent to it. This can be viewed by the so called 3H-1
-nomial representation of
the H-adic: the symbolic sum of 3H-1
sets of H juxtaposed vectors (3H-1
H-ades). To a vector
is associated only one direction; to a dyadic (by the trinomial representation), as a sum of
three sets of two juxtaposed vectors (three dyads); to triadics, as a sum of nine sets of three
vectors (nine triads); to tetradics, twenty seven sets of four vectors (twenty seven tetrads)
etc.
The several physical laws dictates the relationship between the polyadics involved in a
certain phenomenon.
I.2 - Physics and geometry.
Some simple physical laws can be more easily understood evoking single geometric
concepts and figures of the 3-dimensional euclidian geometry, the like the arrows. This
treasure – the geometrical picturesque view of the physical laws, left overly known and still
conserved for vectors (in Mechanics, Electromagnetics etc.) - was lost in the time for
polyadics of major valence perhaps owing to difficulties. We shall show here that, with the
2
Polyadic Calculus magic touch, we can to recover and to extend this procedure for the
geometrical understanding of the more complex physical laws, in the same euclidian
geometric manner, but now inside the abstract multi-dimensional spaces.
I.3 - The polyadic space.
The polyadic space is conceived in the same way as those of vectors (arrows). The
dimension of the H-adic space, 3H, is the major number of linear independent H-adics we
can find in this space. Helped by the algebra, we can determine this number or check
weather a set really constitute a base. It is enough to apply the following theorem: the
necessary and sufficient condition for a given set of H-adics to be a base of its space is that
be non null its multiple mixed product. The H-adic space embraces all the spaces defined
by polyadics whose valence be minor than H; these ones are subspaces of the first. With the
vectors of a vector base we can construct a dyadic base of a dyadic space; with dyadic bases
we can construct a tetradic base for a tetradic space etc.
Polyadic coordinates.
The H-pointer arrows of a H-space, postulated to be free arrows, can be imagined
applied to a fixed point O of this space. This point can be taken as "origin" of the H-adics
of a base (perhaps with unit modulus), to each H-adic direction being associated a cartesian
axes; this configuration defines a cartesian system of reference for the H-adic space. With
respect to a H-adic base of a H-adic space, a H-adic has 3H coordinates, the cartesian
coordinates of the end point of its arrow when this one is applied at the origin; this arrow is
its "positional arrow". The product of a certain coordinate by its correspondent H-adic of
base - a H-ade - is a component of this H-adic. Hence, with respect to a H-adic base, a H-
adic is the sum of its H H-ade components.
With respect to bases formed with polyadics of valence R minor than the valence H of a
certain polyadic, it is always possible to ordinate the coordinates (which are polyadics of
valence H-R) and to arrange then in a definite order in a square or in a rectangular matrix.
In practice we often work with vector and dyadic bases to which we refer vector, dyadic
and tetradic magnitudes.
With respect to a vector base: 1) – a vector has 3 scalar coordinates arranged in a
column (or in a row); 2)- a dyadic has 3 vector coordinates arranged in a column; if we
substitute each one of this vectors by the row of its scalar coordinates we obtain a 33
matrix with 9 dyadic scalar coordinates; 3)- a tetradic has 3 triadic coordinates arranged in a
column; if we substitute each one of these triadics by the line of its dyadic coordinates we
obtain a 33 matrix with the 9 dyadic coordinates of this tetradic; if, now, we substitute
each dyadic by the column of its 3 vector coordinates we obtain a 93 matrix (whose
elements are the vector coordinates of this tetradic); and, finally, if we substitute each one
of these vectors by the line of its coordinates we obtain the 81 tetradic scalar coordinates
disposed in a 99 matrix.
With respect to a dyadic base: 1)- a dyadic has 9 scalar coordinates arranged in a column;
3)- a tetradic has 9 dyadic coordinates arranged in a 33 matrix and 81 scalar coordinates
arranged in a 99 matrix. Given a H-adic by its matrix scalar coordinates in a vector base,
we can determine, through that precise ordering, the coordinates of all its R-adic
coordinates of minor orders (R<H).
The similarity with vectors.
3
Starting from the definition of the multiple dot product of two H-adics, which result is a
scalar, we find the concepts of norm and modulus of a H-adic (some of its invariant) and
angle of two H-adics.
The angle of two H-adics is the ordinary angle of its arrows. The modulus of a H-adic is
a positive real number that defines its intensity. When referred to a orthogonal and unit H-
adic base – the called orthonormal bases – the norm of a polyadic is the sum of the squares
of its coordinates (and only in this case); its modulus is the positive square root of its norm
(the like the vectors). Adopting a certain scale we can draw the H-adic arrow, its length
being defined by its modulus.
I.4 – Constructing figures in the polyadic space.
I.4.1 - The main graphical property of the arrows: the plane hyper-trigonometry.
It is easy to see that we sum (or subtract) two H-arrows by the parallelogram rule.
Indeed, supposing that is acute the angle A formed by the arrows of the H-adics H and
H
and being H its sum, we can write the norm of
H (the square of its modulus) in the form
| | ( ) ( )H H H H H H 2 . , to be, | | | | | | | || |H H H H H cosA 2 2 2 2 .
This formula is exactly the so called "Carnot's formula" for the triangles and confirm the
parallelogram's law.
I.4.1 - A 3H
-angular pyramid as a natural reference system in a H-adic space.
In the 3-dimensional (3-dim) vector space, 4 is the maximal number of independent
points, that is, the position of any one fifth point in this space can be univocally determined
with respect to the triangular pyramid defined by the given four points. Taking one of
these 4 points as origin of vectors and the other three as ending points we define a vector
base in this space. In the 2-dim sub-spaces, the maximal number of independent points is 3;
in the 1-dim, it is 2. The figures in 1-dim can be constructed with segments; in 2-dim, with
triangles; in 3-dim, with triangular pyramids.
In the 3H-dim H-adic space, the even number P=3
H+1 is the maximal number of
independent points; we shall say that they define a 3H
-angular pyramid, or a (P-1)-angular
pyramid. There exist a hyper-sphere which contain all these points. Every set of 3H points
belongs to a (3H-1)-dim subspace; there exist 3
H of then and they are contained by a hyper-
circle. For brevity the pyramids in this space will be named (3H+1)-points, or simply, a P-
point. The points defining a P-point are said to be its vertex. In general, the sets of points
defined by 2k<P of the vertex of a P-point, which exist in number of CPk , are said its k-
point. Particularly, the H-adics defined by any 2-point will be sometimes called the wedges
of the P-point and the modulus of this H-adic the length of these wedges; for k>2, a k-point
any (a sub-space) will be named also a face of the P-point. Hence, there are P-1=3H wedges
and the same number of 3H-faces from a given vertex.
Taking one of the vertex of a P-point as origin and the others as ending points of H-adic
arrows, we define a H-adic base in this space; if the P-point is any, this base is not
orthogonal in general. A variable (P+1)-th point in this space will describe all the space; it
4
can be coincident with a vertex, belongs to a wedge, to a face or occupy a notable position.
This variable point is a fifth point in the vector space, a eleventh point in a dyadic space etc.
I.4.2 – The hyper-planes geometry and trigonometry.
We have seen that in a face (which dimension is P-1) of a P-point there exist P-1=3H
points which we will ordinate one time for all; under this condition we shall say that these
3H points define a ordered 3
H-angle (or, simply, a 3
H-angle) of which the 3
H points are the
vertex. In this face we can define a set of ordered H-adics whose starting and ending points
of its arrows are those 3H points in the chosen order such that vanish its sum; we shall say
that, under these conditions, to any 3H-angle is associated a closed 3
H-angle. The modulus
of the H-adics associated to a closed 3H-angle are said to be the sides of the 3
H-angle. The
angles of the 3H H-adic arrows will be named the 3
H-angle internal angles; the
supplementary angles of the internal angles will be said the 3H-angle exterior angles.
With these basic concepts and definitions, added to others like medians, bisector angles,
altitudes, circumscribed and inscribed circles etc. it is possible to extend all the classical
theorems about triangles to 3H-angles.
In the same way all the classical plane trigonometric formulas for triangles are to be
valid (extended) to the 3H-angles of the faces of a P-point; it could be called
I.4.3 - Constructing parallelepipeds in the polyadic space.
We have seen that there are P-1=3H wedges and 3
H (3
H-1)-faces from a given vertex. In
each one of these faces we can determine the ending point of the (H-adic) sum of its H-
adics. We add 3H=P-1 new points to the previous P. The ending point of the sum of all the
wedges is a further added point. This configuration defines a particular 2P-point which we
shall call the "hyper-parallelepiped" associated to the given P-point or to the (P-1)-angular
pyramid.
As the diagonal relative to a vertex of a parallelepiped (in the vector space) is the
modulus of the sum of the 3 vector edges co-initial in this vertex, the like the diagonal
relative to a vertex of a hyper-parallelepiped in a H-adic space is the modulus of the sum of
3H H-adic edges co-initial in this vertex; etc..
I.4.4 - Some characteristic properties of a (P-1)-angular pyramid.
Many consequences can be obtained from the parallelogram law. For example: the H-
adic position of the midpoint (also named centroid or baric point) of two any points is
determined by the one half the sum of the H-adic which define the positions of these points;
to this point we associate a weight 2. In general, the centroid of a k-point is the average of
the H-adic position of the k given points and to it we assign the weight k.
Let us numerate the vertex of a P-point from 0 to 3H in a arbitrary but fixed way. We
can divide the P-point in two sets of points having each one the same number (because P is
even); in this case the two sets are said to be opposite and they are
C P! (P
2!)P
P/2 2
in number, to be, 6 in vector space, 252 in the dyadic space etc. When the number of points
of the two sets are different they are said to be complementary. Thus, there are: CPk pairs of
sets one of then containing k points and the other the resting, for all k=1,2,3, ..., P/2.
5
Opposite and complementary sets have each one its centroid (with its correspondent
weight). For k=1 the modulus of the H-adic defined by a vertex and the centroid of the
opposite face will be called median; for k=2, the modulus of the H-adic defined by the two
centroids will be called 2-median etc. For the particular value k=P/2 the modulus of the H-
adic defined by the two centroids will be called bi-median (essentially different from the 2-
median). The total number of medians, 2-medians, ..., except bi-medians is
C C ...+C CP1
P2
P
P
2P
P
2 1
.
The centroid of a k-point, H , which weight is k, and the centroid of the resting points,
H , which weight is P-k, admits also a centroid with the total weight, P. This last centroid
is evidently the centroid of the P-point, say H . Hence we can write:
H H H Hi
P
k+(P-k)k P k
P
1 1
1
[ ( ) ] ,
if Hi is the positional H-adic of the point i. The first two members of this equality show
that over the k-median (defined by the two set) the P-point centroid is situated at the (k/P)-
th part of this k-median measured from de centroid H , or, what is the same, at the (P-
k)/P-th part of the k-median, measured from the centroid H . Hence:
The P medians of a (P-1)-angular pyramid concur in the centroid of its
vertex to the the (1/P)-th part of each one measured from the vertex;
and
The CPk k-medians of a (P-1)-angular pyramid concur in the centroid of
its vertex to the the (k/P)-th part of each one measured from the vertex.
For the bi-medians in particular, we enunciate:
The CPP/2 bi-medians of a (P-1)-angular pyramid bisect they self at the
centroid of its vertex.
For P=4 (in the 3-space) pass by the centroid 7 lines (the supports of 4 medians and 3
bi-medians); for P=10 (in the 9-space), pass 737 lines (the supports of 10 medians, 45 2-
medians, 120 3-medians, 210 4-medians and 252 bi-medians; etc..
I.4.4 – The hyper-spherical trigonometry.
We have mentioned that there ever exist a sphere containing the vertex of a P-point and
a circle containing the points of a face (a P-1 point). The center of the sphere and P-2 points
between the P-1 of a face define a maximal hyper-circle of this sphere. The (curved) hyper-
polygon formed by the P-1 arch of maximal hyper-circles in a face will be called a
spherical (P-1)-angle, or spherical 3H
-angle: a spherical triangle for vector-space (H=1), a
spherical nine-angle for dyadic space (H=2) etc..
The sides of a spherical 3H-angle can be calculated by the analogous methods of
spherical trigonometry; the set of these new methods and formulas will be called hiper-
spherical trigonometry; hence, the object of the hiper-spherical trigonometry is to calculate
the spherical 3H-angles.
6
2 – A GENERAL LINEAR PHYSICAL LAW.
2.1- Concepts and definitions.
At any point O of a field, the magnitudes are represented by polyadics, say R of valence R,
H of valence H etc. The relationship between these polyadics – defining a physical law –
are translated by the operations defined in the Polyadic Calculus. When this relationship is
linear it can be written as a multiple dot multiplication
R R+H
H H G . ,
the polyadic R+H
G being not dependent on R or
H. This means that each
R coordinate is
proportional to all the coordinates of H, each one of these later entering with different
weight; these weights are defined by R+H
G, perhaps as functions of time, temperature,
position etc. (but never as functions of the H or
R coordinates). The polyadics
R and
H
are the dependent and independent variables, respectively. The more simple examples of
this general law are the linear constitutive laws in Continuum Mechanics.
The aim theme of this paper will be those linear laws for which R=H, that is, for
proportional magnitudes of the same order; hence, H 2H
H H G . , (2.01).
Thus, for H=1,we shall be dealing with vectors linked by a dyadic; for H=2, with dyadics
linked by a tetradic etc..
The equation (2.01) express many general laws in Physics. In Classical Mechanics we
can cite, for H=1: the third Newton's law f=ma with m=mI (I being the unit dyadic); the
law of the dynamic of rigid body, j=J.w, where j is the angular moment vector, J is the
inertia dyadic and w the angular velocity vector. In Elasticity, for H=1, we have the
Cauchy's law, t . n , where t is the vector stress on a element of area with unit normal
vector n ; for H=2 we have the generalized Hooke's law, 4 G : , where (epsilon) is
the (adimensional) strain dyadic and 4G the Green's tetradic (also named Hooke's tetradic).
A lot of laws in Crystallography and many other branches of Physics could be cited.
In each particular law the three polyadics can assume a special form, being for example
(simple or multiple) symmetric, anti-symmetric etc; or to be of some special nature like a
tonic, a rotation, simple shearer, a complex shearer, a strictily triangular etc.
7
2.2- The physical law is a mapping.
For practical purposes, the physical space has up to three dimensions. A sheet can be seen
as a 2-dimensional space and a cable as a one dimensional space. This realistic approach
brings us to conceive the polyadics of valence Q existing in euclidian spaces of dimension
G3Q.
The variable and independent H-adic H, applied in the fixed point O of its 3
H-space, is
the positional arrow of a "object point" E of this space and the proportionality 2H-adic 2H
G
can be seen as a linear operator that transforms E into the end point S of the positional H
(inside this same space); S is the "image point" of E. If, with respect to the point O, the
multiple mixed product (H1
H2...
H3H ) of the positional H-adics of the points E1, E2,
... E3H is non null, this H-adics constitute a base of this space; the 3
H+1 points O, E1, E2, ...
E3H are said to be independent. The 3
H hyper-planes defined by the sets of 3
H points
defines a hyper-polyhedra in the space.
If, with respect to a fixed O in the field, by any process or law of correspondence, we
know the image S1, S2, ..., S3H of the 3
H object points E1, E2, ... E
3H of this space, which,
with O, define a independent set, the 2H-adic stays univocally determined and can be used
as a pre-factor in multiple H-dot multiplication by H-adics. We write:
H 2H H H G . , for =1,2, ...,3
H and 2H H H G
(2.02),
since we state that that in this last equality is established a sum in the index and that the
H-adics H are the reciprocals of the positional
H. In particular, the positional
H. can
constitute a mutually orthogonal set, that is, they can be orthogonal two by two; in this case
the H-adic base are said to be orthogonal. If, besides a H-adic base to be orthogonal, their
H-adics are taken unitary (what is always possible), the base is said to be orthonormal.
If { , , }H H HG ..., 1 2 is an orthonormal base we can write
Hi
Hj ij . (i,j=1,2, ..., G),
where the ij are the deltas of Kronecker; and we can conclude that if a base in orthonormal
it is coincident with its reciprocal.
One notable particular case is that for H=2 in which the base dyadics are dyads formed with
vectors of a orthonormal vector base { }i j k , that is,
, , , , 1 9 ii ij ik ji kk ...., 2 3 4 ,
being, evidently,
||| || ( ... )
...
...
... ... ...
1 22 1G
i i:i i i i:ij i i:kk
ij:i i ij:ij ij:kk
kk:ii kk:ij kk:kk
.
8
2.3- The 2H
G determination.
To complete the structure of the constitutive equation (2.01) we need first to determine the
proportionality polyadic since it is a characteristic of the media through which the variable
magnitudes are linked, be this media full or empty with matter. The second of expressions
(2.02) furnishes naturally the necessary and more general way (perhaps not ever simple
when it is possible) to get that determination. Laboratory devices and accurate
measurements fulfil the necessary conditions to construct the polyadics H and
H (23
H
in number), and hence the polyadic 2H
G, being enough to chose a convenient vector base
with respect to which the measures could be easily made.
We can choose 3H states inside the same phenomenon (like to put a body in charge with
different efforts), and a point any or a set of points in the field (the body), under which the
variables to be measured, H and
H, may assume simple forms (stresses and strains),
presenting the smallest number of coordinates. Practice, apprenticeship and lucky can help
in this choose. Since there is correspondence between each pair (H,
H), since the G
measured H-adics of the independent variable (H, say strains) are independent – and this
will be confirmed if non vanish the multiple mixed product of the 3H measured H-adics (see
section 1) – we really have got a 2H
G measure.
2.4- A decisive and simplifier assumption.
If, in all the points of the field, the variable scalar 2W H H H H H H . . exists
univocally determined and can be measured, then:
( ) H H H 2H 2H H H H : . .G G
0 , to be, 2H 2H HG G
,
and only in this condition. In resume:
2W H H H H H H . . 2H 2H HG G
, (2.03).
Be Sij ...l and Ei'j' ...l' (for =1,2, ...,3H) the measurable
H and
H coordinates,
respectively, with respect to an orthogonal and unit vector base { , , }e e e1 2 3 , in which case
the two sets of H indexis i,j, ...,l, and i',j',...l' assume the values 1,2,3. We can write, in
cartesian coordinates, 2H
ij ... l i' j' ... l' i j l i' j' l'S E ... ... G e e e e e e
and, hence, associate to 2H
G, in this vector base, the 3Hx3
H symmetric matrix [Gij ...l i'j' ...l']
product of the 3Hx3
H symmetric matrices [Sij ...l] and [Ei'j' ...l']. The column of order in
[Sij ...l] is formed with the H coordinates; the coordinates of
H form the [Ei'j' ...l'] row of
order . As we see, to determine the (1+3H)3H/2 elements of [Gij ...l i'j' ...l'] we need to make
23H3
H measures, to be, up to 18 for H=1 (vector magnitudes), up to 162 for H=2 (dyadic
magnitudes) etc. Troubles in experimental measures will be ever present.
Now, for H=2Q2 (or Q1), be Si,j, ... l and Ei',j',... l' the 2Q and
2Q coordinates,
respectively, with respect to an orthogonal and unitary dyadic base { , , } 1 2 9 ..., for the
Q index i,j, ...,l and the Q others i',j', ...,l' running now from 1 to 9. We can write, in
cartesian coordinates,
9
4Qij ... l i' j' ... l' i j l i' j' l'S E ... ... G .
We can bring this problem in the same way we have brought for vector bases. Perhaps it
could present a bit less laboratory work to accomplish but we still can not to construct a
dyadic base in a way to determine polyadic coordinates in these bases; for while they are
useful only for handling and researching properties as we shall see.
Under this fault we will admit now on that we posses a 2H
G associated 3Hx3
H symmetric
matrix referred to some conveniently chosen orthonormal vector base, which we will
denote by [2H
G]. The 2H
G determinant, det2H
G, equals the determinant of the matrix [2H
G],
sometimes denoted by det[2H
G], is one of the 2H
G invariant. If this determinant is non null, 2H
G is named complete and indicates that it can be inverted; otherwise, 2H
G is named
incomplete and its inversion is not defined.
2.5- A convenient change of variables.
To simplify the mathematical handling we will introduce the new variables
HH
H
| |,
H
H
HH
00
H H 2W=| | 2W and 2w=
2W
| |
2W
|| ||
| |, , , (2.04),
in which case, besides the unchanged law (01), we have:
H H H H H
H H 2 2G G. . , (2.05),
2w H H H H
H H H
H 2H
H H . . . .G , (2.06),
2W 0H
H H H
H H . . , (2.07),
H H H . 1 , (2.08).
The magnitudes H and 2W0 will be named "specific magnitudes", or "magnitudes by
unit of H intensity (modulus)" since | H | represents a quantity of the (independent
variable) magnitude H . Recalling isomorphic concepts with vectors we shall name 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 will be named also the
radial (specific) value of 2HG relative to the direction H .
For two different directions H and H we will write (in accordance with
polyadic algebra) 2 w H H 2H
H H H
H 2H
H H . . . .G G , or
2 w H H H H
H H . . , (2.06').
Hence we have demonstrate the following theorem (of Betty):
The 2H
G projection relative to a direction H , H , projected on a
second direction H , is equal to the
2HG projection relative to this
direction H , H , projected on the first direction H .
We will name the scalar 2w''' the 2H
G tangential value relative to the directions H and
H .
10
It is interesting this change of variables because the independent variable can be
considered now with unit fixed norm or modulus. This permits a significant geometrical
interpretation: when H varies with fixed origin O, assuming all positions about the point,
its end point, being always a unit distance from O, describes the (hyper)spherical surface
(08), centered at the point, with unit radius.
3 - THE QUADRICS ASSOCIATED TO THE PHYSICAL LAW.
Suppose that 2H
G is complete in the 3H-space. From (05), considering (08), we can
deduce: H
H H
H H . .G2 2 1 , (3.01).
Similarly, from (2.06),
Q 2w
2w|H
H 2H
H H . .G
|1 , (3.02),
where H is the H-adic parallel to the unitary H-adic
H whose modulus is the inverse of
|2w| square root, that is,
H H
2w
1 , (3.03).
Hence, while the H end point describes the spherical surface, the H-adics
H and
H
end points, P and Y – whose distance to O are the modulus of H and
H - describe the
(hiper)surfaces (3.02) and (3.03), respectively. These are quadric surfaces centered at the
point O. The first – representing the 's variations – is the Lamé's ellipsoid. The second –
representing the 2w variations and named Cauchy's quadric or indicator quadric – could
be an ellipsoid or an hyperboloid (of one or two sheets) depending on the 2H
G scalars
invariant; by these reasons 2H
G will be named, in correspondence, elliptic and hyperbolic.
The indicator exists always independent on 2H
G to be complete.
Imagined outlined the quadrics (3.02) and (3.03) centered at the point O, |H| and 2w,
related to a given H , can be easy determined. Indeed, to calculate 2w it is enough to
determine the point Y where H intercepts the indicator Q, in which case, according to
(3.03), |2w|=1/(OY)2. To calculate |
H| it is enough to fix its direction in space and to write
|H|=|OP.|
Representing by Q+ and Q- the indicator (3.03) correspondent to signals (+) and (–),
respectively, we can conclude: 1) - If Q+ is a real ellipsoid, Q- will not be sketched because
it is a imaginary ellipsoid. In this case 2w>0, whatever be H and all the
2HG eigenvalues
are positive. The angle between H and
H is acute always; 2) - If Q- is a real ellipsoid, Q+
is imaginary and 2w<0; the angle between H and
H is obtuse always; 3) - If Q+ is an
hyperboloid, Q- is its conjugate hyperboloid; both are separated in the space by the
common assyntoptic cone, with the equation
C =0H
H 2H
H H . .
G .
The point Y, intersection of H with Q, could be a proper point, situated on Q+ or on Q-
, or a improper point (if H should be parallel to any generator of the cone). In the first
11
case, the angle between H and
H should be always acute and 2w>0; or obtuse, with
2w<0; in the second case, this angle is 90 and 2w=0.
The hipercurves intersection of the hipercone C and the hypersphere define over this
hipersphere the regions with respect to which correspond 2w>0 and 2w<0; their normal
projected upon the coordinate planes will be ellipses or arch of hyperbolas.
4 – THE PROPORTIONALITY POLYADIC CHARACTERISTIC ELEMENTS.
4.1- Concepts.
The real and symmetric proportionality polyadic 2H
G defines the maximal dimension of the
spaces to be considered: Gmax=32H
=9H. This animates us to look for particular bases
concerning facilities.
It can be proved that a polyadic 2H
G, any, can transform some non null H-adic H in a
H-adic parallel to H, that is,
2H.G X
H H H , (4.01),
where X is a scalar. This means that there exist solution for the equation
( )2 2H H.G O X
H H H , (a), with X and the H–adic H as unknowns; this equation is
named the 2H
G polyadic characteristic equation. The necessary and sufficient condition for
the existence of H is that the 2H-adic between parentheses – the
2HG characteristic
polyadic - be incomplete. Geometrically this means that if we represent this 2H-adic in a H-
adic base { , , }H H H ..., H 1 2 3, say in the form
2 2H HG X F X) H H
( ,
then its 3H H-adic coordinates F X) F X) H H
( , ( etc belongs to a same hyper-plane
to which the H-adic H might be orthogonal. Algebraically this the same to say that the
2 2H HGX determinant – named the 2H
G characteristic determinant - might vanish.
With the 2H
G associated matrix (that one determined by experiments, see section 2) we can
solve the 3H degree algebraic equation – named the
2HG algebraic characteristic equation -
defined by this determinant, to each of its 3H roots – named the
2HG eigenvalues and
represented by G1, G2, ..., G3H - correspond a certain H-adic solution. This 3
H H-adics are
the 2H
G eigenH-adics and will be represented by H H H3
..., H 1 2, , .
The set of the 2H
G eigenvalues and correspondent 2H
G eigenH-adics are the 2H
G
characteristic elements, sometimes called the 2H
G eigensystems.
4.2- The 2H
G characteristic elements are all real.
Let us prove that for 2H
G single symmetric, 2H 2H HG G
,
the eigenvalues must to be all real.
There must exist a possibly complex H-adic H satisfying (4.01) with X complex scalar,
hence complex H,
H being its conjugate. Hence we write, evidently:
12
H H 2H H H
H X . . .G , (b). Taking complex conjugate from (4.01) it comes:
H H 2H H H H
H 2H X . .G G
since : ( 2H H 2H H H H
H 2H H
, ). .
and 2H
G is real and symmetric. Hence, H H 2H H H
H X . . .G , (c). Now,
comparing (b) and (c) we infer that (X X 0H H H ) . . But for every non null
H,
H H H 0 . ; hence X X . This implies that the eigenvalues are all real.
If the 2H
G eigenvalues – which now on will be denoted by G1, G2, ..., G3H - are all real,
the correspondent eigenH-adics – taken unitary and denoted by H H2
H3
, ..., H , 1 - are
also real and
To each pair of different 2H
G eigenvalues correspond a pair of orthogonal unitary
eigenH-adics.
Indeed, if G1 and G2 are different, we can write from (a) :
21 1
H H H
1H G G . , and 2
2H
H H
2H
2 G G . , (d)
or, since 2H
G is symmetric, H
H H H
1H H
H H G = 1
21 1
2. .G G
, (e).
Hence, by double dot pre-multiplication of the second of equations (d) by H 1 , post-
multiplication in (e) by H 2 and consequent subtraction member by member, we have:
(G G 1 2H
H H ) 1 2 0. .
As G1G2 we get H H H H
H H 1 2 1 2 0. . , that is, H 1 is perpendicular to H 2 .
If all the 3H
2HG eigenvalues are to be different, we have 3
H different unitary eigenH-
adics orthogonal two by two, that is,
G G ... G 1 2 3H
i H H
j ijH . (i,j=1,2, ..., 3H), (4.02).
The metric matrix associated to this set is the 3H3
H unit matrix whose determinant equals
one. Hence, the set constitute a orthonormal base in the space, being coincident with its
reciprocal. So, we can write: 2H H H
G G (=1,2, ..., 3
H), (4.03),
where is established a sum in . The form (4.03) to represent 2H
G is named the tonic form.
The H-adic base { , }H H2
H3
, ..., H 1 is the "principal base" of the field.
4.3- Vanishing and multiples eigenvalues.
Let us suppose that 2H
G has, for instance, a triple eigenvalue, say G G G3 3 3H H H
2 1
.
Using multiple cross product, put
H3 -1
H H H3 -3
H3 -2H H H ..., , 1 2 and H
3H H H
3 3
H
3 2
H
3 -1H H H H ..., , 1 2 .
13
The H-adics H3 -1H and H
3H belongs to the 3H-1 and 3
H spaces, respectively, are unitary
and perpendicular. Besides, both are perpendicular to all the eigenH-adics; hence they
constitute with then a orthonormal base of the hole space. Then we can write, resolving 2H
G
in this base:
2Hi
HiH
iH
3 -1
H
3 -1H
3
H
3G Y Z H H H HG for i=1,2, ..., 3
H-2,
becoming obvious that H3 -1H and H
3H are two 2H
G eigenH-adics to which correspond
the eigenvalues Y and Z. Hence Y=Z= G3H2
because, by hypothesis, 2H
G has only 3H-2
different eigenvalues and G1G2 ... G3H2
. What we have deduced for a triple
eigenvalue we can also deduce for a R-ple eigenvalue since R3H. Hence:
If, in a 3H-space, a
2HG has a R-ple eigenvalue, the multiple cross product of all the 3
H-R+1
different 2H
G eigenH-adics, a H-adic of the (3H-R+2)-space, is also a
2HG eigenH-adic with
respect to that R-ple eigenvalue; the multiple cross product of this new eigenH-adic and the
precedings is also a 2H
G eigenH-adic with respect to that R-ple eigenvalue and belongs to
the (3H-R+3)-space, and so on, each multiple cross product belonging to a space one
dimension higher than the anterior.
This property can be applied to all groups of equal eigenvalues (in particular to that
group for which the eigenvalues are single). When all the eigenvalues are equal, say to G, it
is a spherical 2H-adic, G 2H
I, to whom every H-adic of the space is an eigenH-adic.
If R of the 3H eigenvalues vanish we say that the polyadic exist in G=3
H-R subspace, the
correspondent eigenH-adic being determined like proved before.
4.4- The Cauchy and Lamé's quadric reduced equation.
The quadrics associated to the 2H
G-adic at the point O can be represented in a more simple
form – in the reduced form – if the field is referred to its principal H-adic base. In this case,
(3.02) and (3.03) are written in the respective forms
(
)H
H H
G
.
2 1 and ( )H H H G .
2 1 ,
in whose first members is established a sum on . Denoting by S and Y the H and
H
coordinates we can write:
( )S
G
2 1 and (|)
Y
|G
1/12 , (4.04),
the signal of each piece in the first member of the second equation being the G signal. In
the reduced form it is more easy to classify the indicator quadric.
Let us consider that the polyadic 2H
G is variable with some variable other than H and
H. If, in a particular stage,
2HG has some eigenvalue tendind to zero, say G1, the first piece
in the second of equations (4.03) tends also to zero. This means that the quadric is
orthogonal projected (into quadrics) on the space defined by H H3
..., H , 2 . By (3.03) we
14
see that the distance OP tends to infinite for directions parallel to H 1 . Hence the indicator
quadric (in the 3H-space) tends to a hyper-cylindrical surface whose generators are parallel
to that direction. If vanish more than one eigenvalues we can deduce similar results; there
are occurrence of new degeneration. It could not vanish all the eigenvalues without 2H
G to
be the null 2H-adic.
If some eigenvalue is double, triple etc, we say that the quadrics are of revolution with
respect to two, three etc axis.
5 – THE STATIONARY PROPORTIONALITY POLYADIC RADIAL VALUE.
The 2w stationary value at the point O is a linked extreme because H might satisfy (2.08).
If there exist a direction by O that makes 2w stationary, in this direction will be d(2w)=0.
Differentiating (2.06) it comes (in accordance with polyadic analysis):
2 2 0dw w
d d
H H H H
H 2H
H H
. . .G .
Being H
H H d .
0 , we conclude that H and H 2H H H G .
are orthogonal to
d H , that is, orthogonal to the same hyper-plane tangent to the spherical surface H
H H .
1 . This means that these two H-adics must be parallels.
From the two first members of (2.06) it comes1:
2w=| | cos( , )H H H , (5.01),
from where we deduce that the 2w stationary value is |H| if the H-adics
H and
H are
parallels (the maximum corresponding to the null angle and the minimum to 180). The
parallel condition may be expressed in the form X H H H H H 2 G .
, (1.03).
Remembering the section 4 we conclude:
The 2H
G radial value, 2w, given by (2.06), is stationary at the point O
of the 3H- space for directions H lined by O and parallels to the
2HG
eigenH-adics.
From the general expression we deduce also immediately:
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 , (5.02),
that is:
The tangencial value of 2H
G relative to any two different principal
directions at a point are always null.
If we represent by Eu the projections (coordinates) of H on the principal base, and
remembering that Su are those of H , that is, if we put
1 - It is valid for multiple dot multiplication of polyadics the same concepts valid for scalar multiplication of
vectors.
15
H H H
u u E . and H H H
u u S . , (5.03),
then the law (01) is equivalent to the system
S G E
S G E
S G E
1 1 1
2 2 2
G G G
..., (5.04).
We conclude:
When, in the vicinity of a point O of a field, the space is referred to the
principal base of this point, the ratio of the same name proportional
magnitude coordinates is equal to the correspondent 2H-adic
proportionality eigenvalue.
Substitution of (1.04) into (03,Intr.) gives:
2w=( ) G
( ) G ( ) G ...,
H H H
u2
u
H H H
12
1H
H H
22
2
.
. .
(u=1,2, ..., G) (1.05),
from where we conclude:
Each one 2H
G eigenvalue is a stationary value of its radial value, 2w,
in the point O of the G-space, which occur for the correspondent 2H
G
eigenH-adic direction.
6 - THE PROJECTION NORM AND OCTAHEDRAL DIRECTIONS.
For an arbitrary direction H considered by the point O of the G-space we can write: H H
H H
uH
u ( ) . , for running from 1 to 3H, with
( )H
H H
u
2G
.
1
1 , (6.01),
because H
H H .
1 . The numbers H
H H
u
. are the 3
H 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 3H squares, whose sum is equal to one, there are 23H
directions (that is, all the arrangements with repetition of the signs + and – taken 3H by 3
H
with the modulus of the director cosines, (AR)23 3H H
2 ) whose director cosines have the
same modulus.
We shall call octahedral directions, or octahedral H-adics of a field in a point – and
will denote then by H( )oct
- the unitary H-adics equally inclined to the principal
directions of the point. For a general octahedral direction we can write H
oct
H
oct
H H
u u ( )
.; and from (6.01), since the cosines (cos oct) are all equal:
16
for (u=1,2, ..., 3H) H
oct H H
u octH
H
H cos
3
3
3 .
1, (6.02).
So, in 2-space, oct
45 , in 3-space oct54 44 ' , in 4-space
oct60 etc..
For any octahedral direction, to which correspond w=woct, we have
2w = GG
GoctH
oct H 2H
H H
octH
oct H H
u u u1
G ( ) . . .G 2 1
, (6.03),
that is:
In any octahedral direction, the 2H
G radial value, 2woct, is equal to its
eigenvalues average invariant.
For a direction any of the space we can write the norm of the correspondent 2H
G
projection as
|| || H H H 2H
H 4
H H H
H 2H
H H = . . . . .G G G2 , being 2H
H 2H 2H = G G G.
2 , (6.04).
But from (4.03) we write also 2H
u
H
u
H
u=(G G
) 2 2 ; from where we deduce
2H E2
u
3
(G
H
G
) 2
1
, (6.05).
Hence: || || ( ) ( )H H H H
u u2 G . 2 , or, writing in full:
|| || ( ) ( ) ( ) ( ) ( ) ( ) H H H
1 1H
H H
2 2H
H H
3 3 G G ... GH H. . .
2 2 2 2 2 2 , (6.06).
The Gu are invariant, that is, they don't depend on the H . Hence, ||H|| varies with the
square of the 3H director cosines ( )H
H H
u
.
2 whose sum, in conformity with (6.01), is
equal to one. Thus, when these factors are all equal, that is, for all octahedral directions,
representing by H oct the
2HG projection correspondent to any one of the octahedral
directions, we have:
|| || ( )H
oct H u
3
H
H E2
3 G
3
H
1 12
1
2 G , (6.07),
in which case ||H||max is a invariant. We conclude:
The norms of the 2H
G octahedral projections at a point are all equal to
the 2H
G eigenvalue squares average (or equal to the 3H-th part of the
2HG dot square scalar).
17
7- SOME RELATIONSHIP BETWEEN EIGENVALUES.
Let us calculate now the difference between the 2H