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Inertia and cross In n-dimensions space Mehdi Hage-Hassan

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Mehdi Hage-Hassan. Inertia tensor and cross product In n-dimensions space. 2006. ￿hal-00022701￿

HAL Id: hal-00022701 https://hal.archives-ouvertes.fr/hal-00022701 Preprint submitted on 12 Apr 2006

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Inertia tensor and cross product In n-dimensions space

M. Hage-Hassan Université Libanaise, Faculté des Sciences Section (1) Hadath-Beyrouth

Abstract

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

Résumé Nous montrons par une méthode élémentaire que le tenseur d’inertie d’un point matériel et le produit vectoriel de deux vecteurs sont possibles seulement si la dimension de l’espace est 3 ou 7. La représentation matricielle dans l’espace de 7 dimensions ainsi que ses propriétés sont données. La relation entre le tenseur d’inertie et l’algèbre des octonions est soulignée pour la première fois dans ce travail.

1. Introduction The vector cross product in the of 3 dimensions is largely used in , 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 a simple presentation to make these concepts available is interesting. We present these concepts on the of the inertia tensor and its generalization (4). This allows us by a simple method to obtain and to present the concepts of and as well as there representation matrix and its properties.

2. Inertia Tensor The kinetic energy of a particle of mass m=1 which moves in a system in with 1 angular ωr)( is r r ωω r ×⋅×= rrT r)()( . 2 r r r With, ω ×= rX this is written in the matrix form = VX 3 )(()( ω) ⎛ x ⎞ ⎛ 0 − yz ⎞⎛ω ⎞ ⎜ 1 ⎟ ⎜ ⎟⎜ 1 ⎟ ⎜ x2 ⎟ = ⎜− 0 xz ⎟⎜ω2 ⎟ (1) ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ x3 ⎠ ⎝ − xy 0 ⎠⎝ω3 ⎠ 1 1 The kinetic energy became: t XXT )()( =⋅= ω tt VV ω))(()()( 2 2 33 (X )t is the of (X). t We write the inertia matrix: = VVM 33 )()()( ⎛ mmm ⎞ ⎛ r 22 −−− xzxyxr ⎞ ⎜ 11 12 13 ⎟ ⎜ ⎟ r 22 M )( = ⎜ 12 22 mmm 23 ⎟ = ⎜ − −− yzyrxy ⎟ (2) ⎜ ⎟ ⎜ r 22 ⎟ ⎝ 13 23 mmm 33 ⎠ ⎝ −− − zryzxz ⎠

3. Inertia tensor and the The identification of two sides of the equation (2) may be written as: r 22 11 =+ 12 =+ 0 13 xzmxymrxm =+ 0 0 =+=+ r 22 yzmrymxym =+ 0 12 22 23 r 22 13 =+ 0 23 0 33 =+=+ rzmyzmxzm with =++ rzyx r 2222 . t r 2 We can express these systems in matrix form as 44 )()( = IrHH r 2 ⎛ − x⎞⎛ x ⎞ ⎛r 000 ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ()V t − y ()V y ⎜ r 2 000 ⎟ ⎜ 3 ⎟⎜ 3 ⎟ = (3) ⎜ 2 ⎟ ⎜ − z ⎟⎜ z ⎟ r 000 ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ r 2 ⎟ ⎝ zyx 0 ⎠⎝ −−− zyx 0⎠ ⎝ 000 r ⎠ I is the identity matrix.

We replace the matrix (V3 ) by its expression in (1), we deduce the orthogonal and antisymmetric matrix: ⎛ 0 − xyz ⎞ ⎜ ⎟ ⎜− 0 yxz ⎟ H )( = (4) 4 ⎜ − 0 zxy ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ −−− zyx 0⎠

The matrix H 4 )( is the matrix representation of the quaternion = +− zeyexeh 321 2 2 eee 2 −=−=−= 1,1,1 With 1 2 3 (5) ,, === eeeeeeeee 213132321

H 4 )( Is the Hurwitz matrix and , 21 eandee 3 are the generators of the algebra.

4. Inertia tensor and the octonions n rr r n If we write = ∑ iexr , with {ei}is the base of the vector in Euclidean space R . The i=1 generalization of the tensor of inertia in an intuitive way (4) is written then: ⎛ mmm ⎞ ⎛ r 2 2 −− xxxr − xx ⎞ ⎜ 11 12 K 1n ⎟ ⎜ 1 21 K 1 n ⎟ mmm r 2 2 ⎜ 21 22 K 2n ⎟ ⎜ − 21 − xrxx 2 K − 2 xx n ⎟ M )( = ⎜ ⎟ = ⎜ ⎟ (6) ⎜ MKKM ⎟ ⎜ MKKM ⎟ ⎜ mmm ⎟ ⎜ r 2 2 ⎟ ⎝ 21 nn K nn ⎠ ⎝ 1 n −− 2 n K − xrxxxx n ⎠

The identification of two members gives: 2 r 2 ==+ ,,1, nirxm iii K jiij =≠=+ K,,1,,0 njietjixxm t r 2 And the matrix system n n )()( = IrHH takes the form: ⎛ r 2 ⎞ ⎛ − x1 ⎞⎛ x1 ⎞ r K 00 ⎜ ⎟⎜ ⎟ ⎜ ⎟ t 0 r 2 0 ⎜ ()Vn M ⎟⎜ ()Vn M ⎟ ⎜ K ⎟ = ⎜ ⎟ (7) ⎜ − x ⎟⎜ x ⎟ ⎜ n ⎟⎜ n ⎟ ⎜ MKLM ⎟ xx 0 −− xx 0 ⎜ r 2 ⎟ ⎝ 1 K n ⎠⎝ 1 K n ⎠ ⎝ 0 KK r ⎠ Hurwitz (5) showed that we can only build orthogonal and antisymmetric matrix which lines are a of components of a vector only if n=1, 2,4 or 8.

Consequently the matrix (H n ) or Hurwitz matrix is orthogonal if n+1=8, it results from it that dim ( R n ) = 1, 3 or 7. i=7 The matrix (H ) is the matrix representation of the octonion = exh . 8 ∑i=1 ii The generators of the algebra satisfy: 2 ie =−= 7,1,1 i K (8) −= eeee ijji We can obtain this matrix by hand easily, which will be the object of paragraph 6.

5. Dimension of R n and cross product The problem of the research of the dimension of space where we define the cross product is known for a long time. We know that dim=3 or 7 and we will simply find all these nn − )1( results starting from number of parameters of (V ) which is . Moreover lines of n 2 n rr the matrix (V) are made of components of vector = ∑ i exr . i=1 1- In the case where the vector is without component nn − )1( = 0 the solution is: n=0 and n=1 therefore dim ( R n ) =1. 2 2- In the case where the vector have n components nn − )1( = n the solution is: n=0 and n=3 therefore dim ( R n ) =3. 2 3- In the case where n>3 the vector has n components that are subjected to the constraints conditions: r r r r r r × = iii × ⋅ rreere = 0)().( , The number of the constraints is 2n then it results from it that nn − )1( += 2nn the solution is: n=0 and n=7 therefore dim ( R n ) =7. 2 We thus checked in a simple way the theorem of Eckmann (6-7).

6. Hurwitz’s Transformation and it’s matrix representation. To determine the matrices (H) we must notice that these matrices are antisymmetric and orthogonal. Moreover if ωr r == ur r we find the relation on sums of squares (5), that we write t = uZZ r )()()( 22 with r r 2 2 2 = 1 n ),...... ,( uuetzzZ 1 ++= u N ...... In what follows, we will expose by two simple methods of recurrences (8) the determination of the matrices (V). The first one takes its starting point the transformation of Levi-Civita and the of the matrices (H), the second is based on the law of composition algebra of Cayley-Dickson.

6.1 Levi-Civita Transformation For n=2 Levi-Civita introduced the conformal transformation which is an application of → RR 22 . 2 2 11 2 =−= 2, uuzuuz 212 ⎛ z ⎞ ⎛ uu ⎞⎛ u ⎞ ⎜ 1 ⎟ ⎜ 21 ⎟⎜ 1 that is written ⎜ ⎟ = ⎜ ⎟⎜ ⎟ = UH 22 ))(( (9) ⎝− z2 ⎠ ⎝− uu 12 ⎠⎝− u2 ⎠

6.2 Hurwitz’s Transformations For the generalization of the transformations of Levi-civita we pose ⎛ z ⎞ ⎛ uu ⎞⎛u ⎞ ⎜ 1 ⎟ ⎜ 21 ⎟⎜ 3 ⎜ ⎟ = 2⎜ ⎟⎜ ⎟ = UH 22 )')((2 ⎝z 2 ⎠ ⎝− uu 12 ⎠⎝u 4 ⎠

Using the orthogonality of H 2 )( we find 2 2 2 2 2 2 1 2 1 2 3 ++=+ uuuuzz 4 ))((2 2 2 2 2 And if we put 13 2 3 +−+= uuuuz 4 )()( We write ⎛ z ⎞ ⎛ uuuu ⎞⎛ u ⎞ ⎜ 1 ⎟ ⎜ 3 214 ⎟⎜ 1 ⎟ ⎜− z2 ⎟ ⎜− 34 2 − uuuu 1 ⎟⎜u2 ⎟ = = UH ))(( (10) ⎜ − z ⎟ ⎜ −− uuuu ⎟⎜u ⎟ 44 ⎜ 3 ⎟ ⎜ 1 432 ⎟⎜ 3 ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ 0 ⎠ ⎝− 12 − uuuu 34 ⎠⎝u4 ⎠ Thus we find the transformation of 4 → RR 3 known by the transformation of Kustaanheimo-steifel.

To obtain (H 8 ) and (H16 ) we repeat the same process while replacing t 22 )'(,)( zandUH 3 by H 4 )( , 4 = K 85 ),,()'( zanduuU 5 we deduce (H8 ) then

we adopt the same way for (H16 ) .

6.3 Hurwitz’s Transformations and Cayley-Dickson algebra We determine the matrix (H8 ) using the result of our work (9) on the transformation of Hurwitz in the theory of the and which consists in posing

⎛ yx ⎞ ⎛ − vv 21 ⎞⎛ vv 43 ⎞ ⎜ ⎟ = 2⎜ ⎟⎜ ⎟ (11) ⎝− xy ⎠ ⎝ vv 12 ⎠⎝− vv 34 ⎠

= + 21 , = + izzyizzx 43 , , +=+= iuuviuuv With 432211 (12) , +=+= iuuviuuv 874653

()( +−+= vvvvvvvvz 443322115 )

izz 21 (2 ν ν +=+ ν ν 4231 ), + izz 43 = (2 ν ν −ν ν 3241 ), We obtain 2 2 2 2 z 15 2 3 −−+= νννν 4 .

After identification and a rather simple arrangement we obtain: ⎛ z ⎞ ⎛ uuuuuuuu ⎞⎛ u ⎞ ⎜ 1 ⎟ ⎜ 5 6 7 8 1 2 3 4 ⎟⎜ 1 ⎟ ⎜− z2 ⎟ ⎜− 56 8 − 27 1 −− uuuuuuuu 34 ⎟⎜u2 ⎟ ⎜ − z ⎟ ⎜ −− −− uuuuuuuu ⎟⎜u ⎟ ⎜ 3 ⎟ ⎜ 7 58 6 3 4 1 2 ⎟⎜ 3 ⎟ ⎜− z4 ⎟ ⎜ − 78 − 56 4 − 23 − uuuuuuuu 1 ⎟⎜u4 ⎟ ⎜ ⎟ = ⎜ ⎟⎜ ⎟ (13) − z −− − uuuuuuuu u ⎜ 5 ⎟ ⎜ 1 2 54 6 7 8 ⎟⎜ 5 ⎟

⎜ 0 ⎟ ⎜− 12 − 34 − 56 − uuuuuuuu 78 ⎟⎜u6 ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ 0 ⎟ ⎜ − 43 1 2 −− 87 5 − uuuuuuuu 6 ⎟⎜u7 ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ 0 ⎠ ⎝ 4 −− 23 1 8 −− 67 uuuuuuuu 5 ⎠⎝u8 ⎠ n rr Finally if we write = ∑ i exr i we deduce the matrix(V7 ) : i=1 ⎛ 0 −− − xxxxxx ⎞ ⎜ 7 6 45 3 2 ⎟ ⎜− x7 0 − 65 3 − xxxxx 14 ⎟ ⎜ xx 0 −−− xxxx ⎟ ⎜ 6 5 7 2 1 4 ⎟ V7 )( = ⎜ 5 6 −− xxx 7 0 − 21 xxx 3 ⎟ (14) ⎜ −− xxxx 0 − xx ⎟ ⎜ 4 23 1 7 6 ⎟

⎜ − 143 2 −− xxxxx 7 0 x5 ⎟ ⎜ ⎟ ⎝ 2 − 41 − 63 − xxxxxx 5 0 ⎠

To determine the matrices n nH = ,,5,4),( we suppose that elements x, y, 2 K

,, vvv 321 and v4 are defined on H, O and more generally they may be elements of the algebra of Cayley-Dickson (9). Then we adopt the same method of recurrence as above.

7. Properties of the matrix (V)

In the case where n=7 we find for the matrix (V7 ) analogue properties of the matrix

V3 )( as follows: 3 −= r 2 VrV )()()( 3 3 (15) 3 r 2 7 −= VrV 7 )()()( If the vector is unitary the following expression is valid for n=3 and n=7. 2 θ n −=− θ ViViExp n −− θ Vn ))(cos1()(sin1)]([ (16) Consequently we find a striking analogy between the two cases: 2 3 −=−= VVIH )()(,)( 4 3 3 (17) 2 3 8 7 −=−= VVIH 7 )()(,)(

8. Hurwitz transformation and spinor theory There is a close link between the Hurwitz transformations and spinor theory. t In this regard, we put zi in quadratic form in terms of () (vandv ) 8.1 Transformation 8 → RR 2 . t zi = σ i vv ))(()( t With = ()( vvv 21 ) and (σ i ) denotes the (11). ⎛ 10 ⎞ ⎛0 − i⎞ ⎛ 01 ⎞ σ 1 )( = ⎜ ⎟ , σ 2 )( = ⎜ ⎟ , σ 3 )( = ⎜ ⎟ ⎝ 01 ⎠ ⎝ i 0 ⎠ ⎝ −10 ⎠ 8.2 Transformation 8 → RR 5 . t Put = ()( vvvvv 4321 ) , then by explicit calculation we find t 5 t 2 1 = γ 2 = γ vvizvvz )()(),()( t 2 t 1 3 = γ 4 = γ vvizvviz )()(),()( t 0 4 = γ vvz )()( It is clear that γ-matrices are the famous Dirac representation.

0 ⎛ I 0⎞ i ⎛ 0 σ i ⎞ 5 ⎛0 I ⎞ γ = ⎜ ⎟, γ = ⎜ ⎟, γ = ⎜ ⎟ ⎝0 I ⎠ ⎝−σ i 0 ⎠ ⎝ I 0⎠

Finally we can change the Euclidean by a pseudo-Euclidean space (11) which doesn’t affect our treatment. Moreover the study of Hurwitz transformations will be the subject of another paper.

7. References [1] B.Eckmann,”Stetige Lösungen 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 top Équation,” hep-th/9710079. [4] J. Bass, ”Cours de Mathématiques 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). [8] M. Hage Hassan; “Rapport de Recherche “ Université 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 in Sciences Eds. B. Gruber and M. Ramek (Plenum Press, New York, 1977))