and Pseudoeuclidean Spaces 
.
3. Lagrangean Conventional Euclidean Spaces
and Pseudoeuclidean Spaces .
Let N denote the number of material particles making a classical closed
system. Time back, Lagrange has noted that kinematics and dynamics of,
say, two material particles A (xA, yA, zA)
and B (xB, yB, zB) having masses
of rest 
and 
may be formally interpreted in a conventional Euclidean space 
in a way as if only one material point having mass 
is moving in that space. Here the only idea is that six coordinates zs
(s=1,2,...,6) of this "united" particle should be defined as follows:
(3.1)
If these coordinates are set off in the directions of six orthogonal
unit vectors 
then at
any moment of absolute time t the position of the "united" particle
and its kinetic energy T(t) will be described by expressions
. (3.2)
On the grounds of six-dimensional conventional Euclidean space 
having basis unit vectors 
and the coordinates (3.1) of a single material particle M00
it is easy to construct a seven-dimensional pseudoeuclidean vector space 
exactly same way how the space 
has been constructed on the basis of a space 
by adding the coordinate t and the unit vector 
.
Now we must only to change all 
into such 
, when 
and 
, and then to add 
and coordinate t. The same idea may be generalised to get a pseudoeuclidean
space 
where the only one
material particle is regarded instead of regarding N points in space 
.
In the space 
one can
construct different groups of orthogonal transformations of seven coordinates 
and of co-transformations of unit vectors 
.
In fact, all those groups of transformations will be subgroups of the most
general group of ortogonal transformations in space 
of the type 
. This group
(D-group) can be represented by matrixis of type 
,
where 
generalizes the
former 
, and 
has a block-structure 
,
where 
-describe usual
rotation in subspace 
.
Our objective, though, is to construct such a group (subgroup) of transformations,
which will have not less than two independent numerical or vector invariants.
The author thinks it to be his great achievement that he managed to find
(in D-group) so called g- group of orthogonal
transformations in 
, which
has an additional invariant dAB with the following generalisation
of this method upon the more general case of space 
having 2N-3 additional invariants (in the case of N=2 there
is only one additional invariant dAB and this invariant
is absent in the case of N=1, i.e. in the space 
).
The first (traditional) invariant in the space 
is existing in the whole D-group and follows from the property of
the vector 
, which fix
the "united" mass M00 at each moment t
:
. (3.3)
By this, in 
-subgroup
seven coordinates 
of (3.1)
type are transformed into coordinates 
and the unit vectors 
are
co-transformed into the basis 
using orthogonal transformations belonging to the abovesaid g
- group, which is represented by above mentioned matrixes 
.
These transformations from g -group also consist
of pseudorotations in space 
(special block-matrix [V]7,7 ) and of conventional rotations
in subspace 
(special ortogonal
block-matrix [U]6,6 ) leaving the first unit vector 
untouched.
It is shown that from six coordinates 
of the type (3.1) one may proceed to six new (more convenient) coordinates
as follows: 
(this is the centre of mass, but in “our” space 
)
and yet three coordinates xAB = xA - xB,
yAB = yA - yB, zAB
= zA - zB. These six new coordinates are
set off along new unit vectors 
and 
, which are linear
combinations of former unit vectors 
in 
:
(3.4)
Above mentioned vectors 
and accordingly the expression 
of
the form (3.3) may be presented in the new form:
(3.5)
(3.6)
In case of transformations, belonging to g
-group, four unit vectors 
are transformed in the subspace 
(from the whole space 
)
into new unit vectors 
,
using above mentioned matrix 
and 
from the
Pointcare group in the space 
.
But the three vectors 
undergoes
into 
by only usual transformation
of conventional rotation, using an orthogonal matrix 
from the same Poincare group (pseudoratation matrix 
in -subgroup do not change
both coordinates 
and basis
vectors 
). It means that
subspace 
(having the unit
vectors 
or 
)
is an orthogonal supplement to the subspace 
,
i.e. 
.
So we will have (in case of transformation belonging to -subgroup)
an additional vectorial invariant 
which may be regarded as a function of the invariant argument ?
, which represents the length of an arc drawn by the vector
of the form (3.5) in the whole space 
.
It is a natural consequence that a scalar quantity
will also be invariant. The quantity dAB is not the
length of the vector 
being
just a numerical invariant in space 
,
but it is a "great luck" that this invariant has happened to resemble the
well-known expression from the space 
geometry. The vector 
from
subspace 
consequently
is an invariant too. But of course, the “part” 
of vector 
is not invariant.
All this gives us a possibility to construct an invariant metric tensor
gss(dAB) as a function from invariant dAB
. This diagonal tensor gss(dAB) may serve
as a tool for arranging a pseudoriemannean space 
using a pseudoeuclidean space 
with its invariant vectors 
and 
of the form (3.4).
If a potential function U describing mutual interaction of two material
particles A and B, has as its argument an invariant function
dAB(? ), for example, 
, then as a quadric form 
characterising the metric of the space V7(?A,
?B) we can accept the
expression
(3.7)
where 
are the differentials
of the vectors 
of the
form (3.5). Vectors 
are
realized in the space R7(?
) which is tangent to space V7(?A,
?B) in every moment ?
.
After this we can construct a neorelativistic functional of action which in case of metric (3.7) is defined as an extremum condition (in the pseudoriemannean space V7(?A, ?B) this is the condition of a maximum) for the functional of action S having the form:
(3.8)
All elements 
in expression
(3.8) do not change after any transformation of coordinates (and co-transformation
of the basis), when the transformation belongs to -subgroup
(abovementioned block-matrexes 
and 
).
There are at least three reasons to take the guadric quantic 
just the form (3.7):
1) when noninvariant coordinate dt of the vector 
tends to differential dt of the classical absolute time t,
then from (3.7) one can obtain an aproximate equality 
,
where L is the Lagrange function, and thus the condition of maximum
for functional S will give the same result as the minimum condition
for classical action;
2) when 
and expression
(3.7) will be close to that of Einstein’s GTR;
3) two coefficients before vectors 
in (3.6) have a product which not depends on 
,
and only by this condition the very important future formulae (4.3) and
(4.5) for H and 
can be obtained.
From the geometrical point of view the condition of the first full variation
being equal to zero, 
,
corresponds to the definition of a geodestic line connecting the starting
point A0 with a current point 
in pseudo-Riemannean space 
.
From the mechanical point of view seven equalities t = ?0(?
), x0 = ?1(?
), y0 = ?2(?
), z0 = ?3(?
), xAB = ?4(?
), yAB = ?5(?
), zAB = ?6(?
), which realize the maximum value of S , are representing equations
of motion of the two regarded particles A and B, when thier
interaction is governed by a potential function U(dAB)
under the given initial conditions.
The expression (3.8) is generalised upon the case of N interacting
material particles. Just this is what invariantly "immerses" mechanics
into geometry, when (in case 
)
the transition to a new neorelativistic frame of reference of type 
changes nothing in the expression (3.8) and, consequently, does not change anything in those differential equations which are the sequence of extremum conditions for the functional of action S.