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Inertia tensor and cross product

Inn dimensions space

M. Hage-HassanUniversit Libanaise, Facult des Sciences Section (1)

Hadath-Beyrouth

Abstract

We demonstrated using an elementary method that the inertia tensor of a material pointand the cross product of two vectors were only possible in a three or seven dimensionalspace. The representation matrix of the cross product in the seven dimensional space andits properties were given. The relationship between the inertia tensor and the octonionsalgebra was emphasized for the first time in this work.

RsumNous montrons par une mthode lmentaire que le tenseur dinertie dun point matrielet le produit vectoriel de deux vecteurs sont possibles seulement si la dimension delespace est 3 ou 7. La reprsentation matricielle dans lespace de 7 dimensions ainsi queses proprits sont donnes. La relation entre le tenseur dinertie et lalgbre desoctonions est souligne pour la premire fois dans ce travail.

1. IntroductionThe vector cross product in the Euclidean space of 3 dimensions is largely used inphysics, but the generalization by Eckmann (1-2) to 7 dimensions is not well known by

the physicists. This generalization starts to be useful in modern physics (2-3) and asimple presentation to make these concepts available is interesting. We present theseconcepts on the basis of the inertia tensor and its generalization (4). This allows us by asimple method to obtain and to present the concepts of quaternion and octonion as well asthere representation matrix and its properties.

2. Inertia TensorThe kinetic energy of a particle of mass m=1 which moves in a system in rotation with

angular velocity )(r

is )()(2

1rrT

rrrr= .

With, rX rrr= this is written in the matrix form ))(()(

3

VX =

(1)

=

3

2

1

3

2

1

0

0

0

xy

xz

yz

x

x

x

The kinetic energy became: ))(()()(2

1)()(

2

133 VVXXT

ttt ==

tX)( is the transpose of (X).

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We write the inertia matrix: )()()( 33 VVM t=

=

=22

22

22

332313

232212

131211

)(

zryzxz

yzyrxy

xzxyxr

mmm

mmm

mmm

Mr

r

r

(2)

3. Inertia tensor and the quaternionsThe identification of two sides of the equation (2) may be written as:

.

00

00

00

2222

22332313

2322

2212

131222

11

rzyxwith

rzmyzmxzm

yzmrymxym

xzmxymrxm

r

r

r

r

=++

=+=+=+

=+=+=+

=+=+=+

We can express these systems in matrix form as IrHH t 244 )()( r

=

( ) ( )

=

2

2

2

2

33

000

000000

000

00 r

r

r

r

zyx

z

y

x

V

zyx

z

y

x

V t

r

rr

r

(3)

Iis the identity matrix.We replace the matrix by its expression in (1), we deduce the orthogonal and)( 3V

antisymmetric matrix:

= (4))( 4H

0

0

0

0

zyx

zxy

yxz

xyz

The matrix is the matrix representation of the quaternion)( 4H 321 zeyexeh +=

With (5)213132321

23

22

21

,,

1,1,1

eeeeeeeee

eee

===

===

)( 4H Is the Hurwitz matrix and are the generators of the algebra.321 , eandee

4. Inertia tensor and the octonions

If we write , withexrn

i

i

rr=

=1

}{ ier

is the base of the vector in Euclidean space nR . The

generalization of the tensor of inertia in an intuitive way (4) is written then:

=

=

2221

22

22

21

1212

12

21

22221

11211

)(

nnn

n

n

nnnn

n

n

xrxxxx

xxxrxx

xxxxxr

mmm

mmm

mmm

M

rK

MKKM

Kr

Kr

K

MKKM

K

K

(6)

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The identification of two members gives:

njietjixxm

nirxm

jiij

iii

,,1,,0

,,1,22

K

Kr

==+

==+

And the matrix system IrHH nt

n

2)()( r

= takes the form:

( ) ( )

=

2

2

2

1

1

1

1

0

0000

00 r

r

r

xx

x

x

V

xx

x

x

V

n

n

n

n

n

t

n

rKK

MKLM

Kr

Kr

K

M

K

M (7)

Hurwitz (5) showed that we can only build orthogonal and antisymmetric matrix whichlines are a linear combination of components of a vector only if n=1, 2,4 or 8.Consequently the matrix or Hurwitz matrix is orthogonal if n+1=8, it results from it

that dim (

)( nHn

R ) = 1, 3 or 7.

The matrix is the matrix representation of the octonion)( 8H =

==

7

1

i

i iiexh .

The generators of the algebra satisfy:

(8)ijji

i

eeee

ie

=

== 7,1,12 K

We can obtain this matrix by hand easily, which will be the object of paragraph 6.

5. Dimension ofn

R and cross productThe problem of the research of the dimension of space where we define the cross productis known for a long time. We know that dim=3 or 7 and we will simply find all these

results starting from number of parameters of which is)( nV 2

)1( nn. Moreover lines of

the matrix (V) are made of components of vector .exrn

i

i

rr=

=1

1- In the case where the vector is without component

02

)1(=

nnthe solution is: n=0 and n=1 therefore dim ( nR ) =1.

2- In the case where the vector have n components

nnn

=

2

)1(the solution is: n=0 and n=3 therefore dim ( nR ) =3.

3- In the case where n>3 the vector has n components that are subjected to theconstraints conditions:

,0)().( == rreere iii rrrrrr The number of the constraints is 2n then it results from it that

nnnn

22

)1(+=

the solution is: n=0 and n=7 therefore dim ( nR ) =7.

We thus checked in a simple way the theorem of Eckmann (6-7).

6. Hurwitzs Transformation and its matrixrepresentation.

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To determine the matrices (H) we must notice that these matrices are antisymmetric andorthogonal. Moreover if ur

rrr== we find the relation on sums of squares (5), that we

write

with.......),......,(

)()()(22

12

1

22

Nn

t

uuuetzzZ

uZZ

++==

=rr

r

In what follows, we will expose by two simple methods of recurrences (8) thedetermination of the matrices (V). The first one takes its starting point the transformationof Levi-Civita and the orthogonality of the matrices (H), the second is based on the lawof composition algebra of Cayley-Dickson.

6.1 Levi-Civita Transformation

For n=2 Levi-Civita introduced the conformal transformation which is anapplication of .22 RR

2122

22

11 2, uuzuuz ==

that is written (9)))(( 222

1

12

21

2

1

UHu

u

uu

uu

z

z

=

=

6.2 Hurwitzs Transformations

For the generalization of the transformations of Levi-civita we pose

)')((2u

u2

z

z22

4

3

12

21

2

1UH

uu

uu=

=

Using the orthogonality of we find)( 2H

))((2 242

32

22

12

22

1 uuuuzz ++=+

And if we put )()( 242

32

22

13 uuuuz ++=

We write

(10)))((

0

44

4

3

2

1

3412

4321

1234

2143

3

2

1

UH

u

u

u

u

uuuu

uuuu

uuuu

uuuu

z

z

z

=

=

Thus we find the transformation of known by the transformation ofKustaanheimo-steifel.

34RR

To obtain and we repeat the same process while replacing

by , we deduce then

)( 8H )( 16H

322

)'(,)( zandUH )(4

H5854

),,()'( zanduuU t K= )(8

H

we adopt the same way for .)( 16H

6.3 Hurwitzs Transformations and Cayley-Dickson algebra

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We determine the matrix using the result of our work (9) on the transformation)( 8H

of Hurwitz in the theory of the angular momentum and which consists in posing

=

34

43

12

212vv

vv

vv

vv

xy

yx (11)

With

)()(

,

,,,

443322115

874653

432211

4321

vvvvvvvvz

iuuviuuv

iuuviuuvizzyizzx

++=

+=+=

+=+=+=+=

(12)

We obtain.

),(2),(22

4

2

3

2

2

2

15

324143423121

+=

=++=+

z

izzizz

After identification and a rather simple arrangement we obtain:

(13)

=

8

7

6

5

4

3

2

1

56781234

65872143

78563412

8765421

12345678

21436587

34127856

43218765

5

4

3

2

1

0

0

0

u

u

u

u

u

u

u

u

uuuuuuuu

uuuuuuuu

uuuuuuuu

uuuuuuuu

uuuuuuuu

uuuuuuuu

uuuuuuuu

uuuuuuuu

z

z

z

z

z

Finally if we write we deduce the matrix :i

n

i

iexr rr=

=

1

)( 7V

(14)

=

0

0

0

0

0

0

0

)(

563412

572143

671234

321765

412756

143657

234567

7

xxxxxx

xxxxxx

xxxxxx

xxxxxx

xxxxxx

xxxxxx

xxxxxx

V

To determine the matrices ,,5,4),(2

K=nH n we suppose that elements x, y,

and are defined on H, O and more generally they may be elements of thealgebra of Cayley-Dickson (9). Then we adopt the same method of recurrence as above.

321 ,, vvv 4v

7. Properties of the matrix (V)

In the case where n=7 we find for the matrix analogue properties of the matrix

as follows:

)( 7V

)( 3V

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)()()(

)()()(

723

7

323

3

VrV

VrVr

r

=

= (15)

If the vector is unitary the following expression is valid for n=3 and n=7.(16)2))(cos1()(sin1)]([ nnn VViViExp =

Consequently we find a striking analogy between the two cases:(17)

)()(,)(

)()(,)(

73

72

8

33

32

4

VVIH

VVIH

==

==

8. Hurwitz transformation and spinor theoryThere is a close link between the Hurwitz transformations and spinor theory.

In this regard, we put in quadratic form in terms ofiz ( )vandv t)(

8.1 Transformation .28 RR =iz ))(()( vv i

t

With and)()(21vvv

t = )( i denotes the Pauli matrices (11).

, ,

=

01

10)( 1

=

0

0)( 2

i

i

=

10

01)( 3

8.2 Transformation .58 RR Put , then by explicit calculation we find)()( 4321 vvvvv

t =

)()(

)()(),()(

)()(),()(

04

14

23

22

51

vvz

vvizvviz

vvizvvz

t

tt

tt

=

==

==

It is clear that -matrices are the famous Dirac representation.

=

=

=

00,

00,

00 50

I

I

I

I

i

ii

Finally we can change the Euclidean by a pseudo-Euclidean space (11) which doesnt

affect our treatment. Moreover the study of Hurwitz transformations will be the subject ofanother paper.

7. References[1] B.Eckmann,Stetige Lsungen linearer Gleichungssysteme, Comm. Math. Helv.

15,318-339 (1943).

[5] Z. K. Silagadze,Multi-dimensional vector product, arXiv :math.RA/0204357v1[3] D. B. Fairlie and T. Ueno,Higher-dimensional generalizations of the Euler topquation, hep-th/9710079.

[4] J. Bass, Cours de Mathmatiques Tome 1 (Masson Editeur 1961, Paris)[5] K. Conrad, The Hurwitz theorem on sums of squares, Internet.[6] A. Elduque,Vector cross Products, Internet 2004.[7] R. L. Brown and A. Gray, Vector cross products, Comm. Math. Helv. 42, 222-236

(1967).

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[8] M. Hage Hassan; Rapport de Recherche Universit Libanaise (1991)[9] M. Hage Hassan and M. Kibler, Non-bijective Quadratic transformation and the

Theory of angular momentum, in Selected topics in statistical physics Eds: A.A.Logunovand al. World Scientific: Singapore (1990).

[10] P.Kutaanheimo and E. Steifel; J. Reine Angew. Math. 218, 204 (1965)

[11] M. Kibler, On Quadratic and Non-quadratic Forms: application, Symmetries inSciences Eds. B. Gruber and M. Ramek (Plenum Press, New York, 1977))