Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 33, 1619 - 1626
Homothetic Motions at (E 8) with Bicomplex Numbers (C3) ˘ Faik BABADAG Department of Mathematics, Faculty of Science Ankara University 06100, Tando˘gan-Ankara, Turkey
[email protected] Yusuf YAYLI Department of Mathematics, Faculty of Science Ankara University 06100, Tando˘gan-Ankara, Turkey Nejat EKMEKC´I Department of Mathematics, Faculty of Science Ankara University 06100, Tando˘gan-Ankara, Turkey Abstract In this paper, a matrix similar to Hamilton operators has been defined for bicomplex numbers in E8 and a new motion has been by this matrix.It is shown that this is a homothetic motion.Furthermore, it has been found that the motion defined by regular curve of order r has only one acceleration centre of order (r − 1) at every t−instant.
Mathematics Subject Classification: 53A05, 53A17 Keywords: Bicomplex number, Homothetic motion, Hyper surface, Pole points
1
Introduction
In 2006, D. Rochon and S. Tremblay , presented a paper based on II. The Hilbert Space [1]. The bicomplex (hyperbolic) numbers are given in this paper from a number of different points of view of Hilbert Space for quantum mechanics. In this study the new operator similar to Hamilton operator [4] has been given for bicomplex numbers (C3 ). In [3], homothetic motion has been defined
˘ Y. YAYLI and N. EKMEKC´I F. BABADAG,
1620
and investigated in four dimension Euclidean space E4 . It is shown that this study can be repeated for bicomplex numbers(C3 ), that is a homothetic motion in eight dimension Euclidean space E8 and this homothetic motion holds all of the properties [5, 6].
2
Bicomplex Numbers(C2)
A bicomplex number is defined as a complex number depending on four units +1, i, j, ij : X = x1 + ix2 + jx3 + ijx4 where x1 , x2 , x3 , x4 are real called the components of X and +1, i, j, ij are arbitrary ‘units’ which satisfy the relations i2 = j 2 = −1 and ij = ji. Write x1 + ix2 + jx3 + ijx4 as (x1 + ix2 ) + j(x3 + ix4 ) = w + jz ,where w, z are complex numbers and j 2 = −1. The bicomplex numbers are thus viewed as pairs of complex numbers. Let C2 = {x1 + ix2 + jx3 + ijx4 , xk ∈ R and 1 ≤ k ≤ 4} = {w + jz : w, z ∈ C} . The conjugation of X bicomplex number (C2 ) [1] is in the form of X = w + jz = w − j z = x1 − ix2 − jx3 + ijx4 where, −
X.X = x21 + x22 + x23 + x24 + 2ij(x1 x4 − x2 x3 ) and x1 x4 − x2 x3 = 0 i.e.,
−
X.X = x21 + x22 + x23 + x24 ∈ R.
3
Bicomplex Numbers(C3)
Let C3 = {a + bk : a, b ∈ C2 } . Where k is an arbitrary unit with k 2 = −1. Defined the product in C3 as β1 β2 = (a1 + kb1 )(a2 + kb2 )
Homothetic motions at (E 8 ) with bicomplex numbers = (a1 a2 − b1 b2 ) + k(b1 a2 + b2 a1 ) ; ai , bi ∈ C2 ; ki ∈ C3
1621 (1)
The multiplication in C3 are closed, associative, commutative and distributive with respect to addition. Thus, C3 is a Banach algebra [2]. It is possible to give the production in C3 similar to the Hamilton operators which has defined [1]. If β = a + bk a = x1 + ix2 + jx3 + ijx4 ∈ C2 b = x5 + ix6 + jx7 + ijx8 ∈ C2 xi ∈ R is bicomplex number ⎡ x1 −x2 ⎢ x2 x1 ⎢ ⎢ x3 −x4 ⎢ ⎢ x4 x3 N(β) = ⎢ ⎢ x5 −x6 ⎢ ⎢ x6 x5 ⎢ ⎣ x7 −x8 x8 x7
in C3 and then N + and N − is defined as ⎤ −x3 x4 −x5 x6 x7 −x8 −x4 −x3 −x6 −x5 x8 x7 ⎥ ⎥ x1 −x2 −x7 x8 −x5 x6 ⎥ ⎥ x2 x1 −x8 −x7 −x6 −x5 ⎥ ⎥ −x7 x8 x1 −x2 −x3 x4 ⎥ ⎥ −x8 −x7 x2 x1 −x4 −x3 ⎥ ⎥ x5 −x6 x3 −x4 x1 −x2 ⎦ x6 x5 x4 x3 x2 x1
N(β) =
N(a) −N(b) N(b) N(a)
(2)
where N are the same Hamilton operators and N is a 4x4 matrix (see [4] for N). Using the definition of N, the multiplication of two bicomplex numbers a and b is given by ab = N(a)b,
a, b ∈ C2 .
The conjugation of β(i, j, k),[1], β = a + kb = a − kb = [(x1 − ix2 ) − j(x3 − ix4 )] − k [(x5 − ix6 ) − j(x7 − ix8 )] = x1 − ix2 − jx3 + ijx4 − kx5 + kix6 + kjx7 − kijx8 ). where x21 + x22 + x23 + x24 + x25 + x26 + x27 + x28 +2ij(x1 .x4 + x5 .x8 − x2 .x3 − x6 .x7 ) β.β = +2ki(x1 .x6 + x3 .x8 − x2 .x5 − x4 .x7 ) +2kj(x1 .x7 + x2 .x8 − x3 .x5 − x4 .x6 )
˘ Y. YAYLI and N. EKMEKC´I F. BABADAG,
1622 and
⎡
⎤4 x21 + x22 + x23 + x24 + x25 + x26 + x27 + x28 ⎢ ⎥ +2(x1 x7 + x2 .x8 − x3 x5 − x4 x6 ) ⎥ . det N = ⎢ ⎣ ⎦ +2(x1 x4 + x5 x8 − x2 x3 − x6 x7 ) +2(x1 x6 + x3 x8 − x2 x5 − x4 x7 )
Let G be the set of matrix N(β). Then G is a commutative algebra with respect to matrix addition and product. Lemma 3.1 G and C3 are isomorphic algebras. Proof. Let us define f : C3 → G , by β → f (β) = N(β). Since f (β1 .β2 ) = f (β1 ).f (β2 ) = N(β1 ).N(β2 ) and f (β1 + β2 ) = f (β1 ) + f (β2 ) and f (λβ) = λ.f (β), then f is a algebra isomorphism.
4
Homothetic Motions at E8
Let β = (x1 , x2 , x3 , x4, x5 , x6 , x7 , x8 ) and 0} ⊂ E 8 M1 = {x1 .x7 + x2 .x8 − x3 .x5 − x4 .x6 = 0, β = M2 = {x1 .x4 + x5 .x8 − x2 .x3 − x6 .x7 = 0, β = 0} ⊂ E 8 M3 = {x1 .x6 + x3 .x8 − x2 .x5 − x4 .x7 = 0, β = 0} ⊂ E 8 M = M1 ∩ M2 ∩ M3 be a hyper surface in E 8
S 7 = x21 + x22 + x23 + x24 + x25 + x26 + x27 + x28 = 1 be a unit sphere. Let us consider the following curve: α
:
I ⊂ R → M ⊂ E 8 defined by
α(t) = (x1 (t), x2 (t), x3 (t), x4 (t), x5 (t), x6 (t), x7 (t), x8 (t)) for every t ∈ I. We suppose that the curve α(t) is differentiable regular The operator B, corresponding to α(t) is defined by the ⎡ x4 −x5 x6 x1 −x2 −x3 ⎢ x2 x1 −x4 −x3 −x6 −x5 ⎢ ⎢ x3 −x4 x1 −x2 −x7 x8 ⎢ ⎢ x4 x3 x2 x1 −x8 −x7 B = N(α) = ⎢ ⎢ x5 −x6 −x7 x8 x1 −x2 ⎢ ⎢ x6 x5 −x8 −x7 x2 x1 ⎢ ⎣ x7 −x8 x5 −x6 x3 −x4 x8 x7 x6 x5 x4 x3
the curve of order r. following . ⎤ x7 −x8 x8 x7 ⎥ ⎥ −x5 x6 ⎥ ⎥ −x6 −x5 ⎥ ⎥ (3) −x3 x4 ⎥ ⎥ −x4 −x3 ⎥ ⎥ x1 −x2 ⎦ x2 x1
Homothetic motions at (E 8 ) with bicomplex numbers
1623
Let α(t) be a unit velocity. If α(t) does not pass through the origin, the above matrix can be represented as ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ B = h⎢ ⎢ ⎢ ⎢ ⎢ ⎣
x1 /h x2 /h x3 /h x4 /h x5 /h x6 /h x7 /h x8 /h
−x2 /h x1 /h −x4 /h x3 /h −x6 /h x5 /h −x8 /h x7 /h
−x3 /h −x4 /h x1 /h x2 /h −x7 /h −x8 /h x5 /h x6 /h
x4 /h −x3 /h −x2 /h x1 /h x8 /h −x7 /h −x6 /h x5 /h
−x5 /h −x6 /h −x7 /h −x8 /h x1 /h x2 /h x3 /h x4 /h
x6 /h −x5 /h x8 /h −x7 /h −x2 /h x1 /h −x4 /h x3 /h
x7 /h x8 /h −x5 /h −x6 /h −x3 /h −x4 /h x1 /h x2 /h
−x8 /h x7 /h x6 /h −x5 /h x4 /h −x3 /h −x2 /h x1 /h
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4)
B = h.A where, h : I ⊂ R → R , t → h(t) = α(t) =
x21 + x22 + x23 + x24 + x25 + x26 + x27 + x28
and α(t) = 0. Theorem 4.1 In the eguation B = hA, the matrix A is a reel ortogonal matrix. Proof. Since α(t) M and x1 .x7 + x2 .x8 − x3 .x5 − x4 .x6 = 0 x1 .x4 + x5 .x8 − x2 .x3 − x6 .x7 = 0 x1 .x6 + x3 .x8 − x2 .x5 − x4 .x7 = 0. Using equation (4), In the equation B = hA, we find AT A = AAT = I8 and detA=1. Theorem 4.2 If α(t) ∈ S 7 ∩ M , B is a reel ortogonal matrix. Proof. If α(t) ∈ M, h = 1 .
5
A Motion with One Parameter
We let the fixed space and the motional space be ,respectively, R0 and R. In this case, one- parametric motion of R0 with respect to R will be denoted by R0 R . This motion can be expressed as follow: X hA C X0 = . (5) 1 0 1 1
˘ Y. YAYLI and N. EKMEKC´I F. BABADAG,
1624
where, X and X0 represent position vectors of any point respectively in R and R0 , and C represent any translation vector. Theorem 5.1 The motion defined by the equation in (5) in eight dimension Euclidean space E 8 is a homothetic motion. Proof. See [5, 6] for the proof. ·
Theorem 5.2 Let α(t) be a unit velocity curve and α(t) ∈ M, then the derivation operator ·
B of B = hA is a reel orthogonal matrix in E 8 . ·2 ·2 ·2 ·2 ·2 ·2 ·2 ·2 Proof. Let α(t) is a unit velocity curve, x1 +x2 +x3 +x4 +x5 +x6 +x7 +x8 = 1 · and since α(t) ∈ M, ·
·
·
·
·
·
·
·
x1 x7 + x2 x8 − x3 x5 − x4 x6 = 0 · · · · · · · · x1 x4 + x5 x8 − x2 x3 − x6 x7 = 0 · · · · · · · · x1 x6 + x3 x8 − x2 x5 − x4 x7 = 0. ·
· T
·
Thus, BεB = I8 and det B = 1. ·
Theorem 5.3 If α(t) is a unit velocity curve and α(t) ∈ M, the motion is a regular motion and it is independent of h . ·
·
Proof. This motion is regular as detB = 1 ;also ,the value of det B is independent of h.
6
pole points and pole curves of the motion
To find the pole points, we have to solve the equation ·
·
BX + C = 0
(6)
Any solution of equation (6) is a pole point of the motion at that instant in ·
R0 . Since, by Theorem 4, we have det B = 1. Hence the equation (6) has only · −1
one solution, i.e. X = (−B theorem can be given.
·
)(C) at every t-instant. In this case the following ·
Theorem 6.1 If α(t) is a unit velocity curve and α(t) ∈ M, then the pole point corresponding · T
·
to each t-instant in R0 is the rotation by B of the speed vektor C of the translation vector at that moment. ·
· T
Proof. Since the matrix B is orthogonal, then the matrix B is orthogonal, too. Thus it makes a rotation.
Homothetic motions at (E 8 ) with bicomplex numbers
1625
Accelaration Centers of Order (r − 1) of a Motion
7
Definition 7.1 The set of the zeros of sliding acceleration of order r is called the acceleration center of order (r-1). By above definition, we have to solve the solution of equation B (r) X + C (r) = 0
(7)
where B (r) =
dr B dr C (r) and C = dtr dtr
Let α(t) be a regular curve of order r and α(r) (t) ∈ M. Then we have (r) (r)
(r) (r)
(r) (r)
(r) (r)
(r) (r) x1 x6 (r) (r) x1 x4
(r) (r) x3 x8 (r) (r) x5 x8
(r) (r) x2 x5 (r) (r) x2 x3
(r) (r) x4 x7 (r) (r) x6 x7
x1 x7 + x2 x8 − x3 x5 − x4 x6
= 0
−
= 0
+ +
−
− −
= 0.
Thus, 8
(r)
(xi )2 = 0
(r)
; xi =
i=1
dr xi . dtr
Also, we have det B (r) =
8
4 (r)
(xi )2
.
i=1
Then det B (r) = 0. Therefore the matrix B (r) has an inverse and by the equation in (7), the acceleration center of order (r − 1) at every t-instant, is −1 −C (r) . X = B (r) Example 7.2 Let α:I ⊂ R → M ⊂ E 8 be a curve given by 1 t → α(t) = (sin t, cos t, sin t, cos t, sin t, cos t, sin t, cos t). 2 · Note that α(t) S 7 and since α(t)= 1 , then α(t) is a unit velocity curve. ·
··
Moreover, α(t) ∈ M, α(t) ∈ M,...,α(r) ∈ M. Thus α(t) satisfies all conditions of the above theorems.
1626
˘ Y. YAYLI and N. EKMEKC´I F. BABADAG,
References [1] D. Rochon and S. Tremblay, Bicomplex Quantum Mechanics: II. The Hilbert Space 135-157. Birkh¨ auser Basel (June, 2006) [2] G.B. Price, An Introduction to Multicomplex Spaces and Functions, Marcel Dekker, Inc: New York. I(1)-44(1).(1991). [3] H. Kabadayı, Y. Yaylı Homothetic Motion at E 4 with Bicomplex Numbers Applied Mathematics Letters(Submitted) [4] O.P. AgrawaL, Hamilton Operators and Dual Number Quaternions in Spatial Kinematics, Mec-Mach Theory (22),569-575(1987). [5] Y. Yayli , B. B¨ uk¸cu ¨ , Homothetic motions at with Cayley Numbers. Mech. Mach. theory vol. 30, No.3 , 417-420 ,(1995). [6] Y. Yayli, Homothetic motions at E 4 . Mech. Mach. Theory 27(3), 303-305 (1992). Received: March, 2009
