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Generalized galilean transformations and dual Quaternions.

[section]1. Introduction

A unit real quaternion is a rotation operator on rigid body motion in Euclidean space. Unit dual quaternions are also used both as rotation and screw operators. Unit dual quaternions are seen as screw operator especially in Mechanics and Kinematics. Galilean geometry of motions was studied in [3]. n-dimensional dual complex numbers was given in [4]. These numbers are viewed as analysis. Galilean transformations are given as shear motion on plane [2]. Shear motion in Galiean space [G.sub.3] was given [2]. Moreover, union of shear motion and Euclidean motions was introduced. Galilean transformation (shear motion) was given by quaternions (in dual quaternion form) [1]. Here, we redefine dual quaternions in a new way for the first time. We work out Majernik's work in a new point of view by using structures of Lie groups and algebras. These are subgroups of Heisenberg Lie groups. We obtain elements of groups by the exponential expansion of quaternion forms of an element of Lie algebra. And we extended the work to the Galilean space [G.sub.n]. Finally, we give Galilean transformation as dual quaternion operators.

[section]2. Galilean transformations in galilean space [G.sub.2]

Galilean transformations were examined widely in [2]. Let X [member of] [R.sup.n] and [G.sub.n] be Galilean space ([R.sup.n], || ||) with

||X|| = {|[x.sub.1]|, [x.sub.1] [not equal to] 0 [square root of [x.sup.2.sub.2] + [x.sup.2.sub.3] + ... + [x.sup.2.sub.n]], [x.sub.1] = 0

for n = 2, 3, 4. We redefine Galilean transformation by using quaternion operators. For any X = (x, y) [member of] [G.sub.2], Galilean transformation (shear motion) is defined as the following:

f : [G.sub.2] [right arrow] [G.sub.2], X [right arrow] f(X) = (x, vx + y),

f is a linear transformation, so f has the matrices form as the following

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 2.1. Let f be a linear transformation. Then f is a Galilean transformation, where

f : [G.sub.2] [right arrow] [G.sub.2], X [right arrow] f(X) = (x, vx + y).

Proof. For x [not equal to] 0 and x = 0, we have

||X|| = |x| = ||f(X)||

and

||X|| = |y| = ||f(X)||.

f is a Galilean transformation, because the linear function f is a isometry.

Theorem 2.1. Let Gal(2) be a Lie group. Then Gal(2) and g(2) are Lie algebras of Gal(2), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[section]3. Dual numbers and galilean transformation in [G.sub.2]

Every vector X = (x, y) in [R.sup.2] can be written as X = x+[epsilon]y with [[epsilon].sup.2] = 0. So sp {1, [epsilon]} = [R.sup.2]. The form of x + [epsilon]y is called dual quaternion form of X.

Lemma 3.1. Dual quaternion Q = 1 + [epsilon]v is a Galilean transformation in [G.sub.2].

Proof. Since Q = 1 + [epsilon]v, we have

QX = (1 + [epsilon]v)(x + [epsilon]y) = x + [epsilon](y + vx)

and

||QX|| = ||X||.

So, Q is a Galilean transformation in [G.sub.2]. By using exponential map from Lie algebra to Lie group, on g [member of] g(2) as in g = [[epsilon]v.sub.1] form

e : g(2) [right arrow] Gal(2) g [right arrow] [e.sup.g] = [e.sup.[[epsilon]v.sub.1]] = 1 + [[epsilon]v.sub.1].

Corollary 3.1. Q = [e.sup.g] = 1 + [[epsilon]v.sub.1] is a dual quaternion operator. Thus dual quaternion operator Q is a Galilean transformation.

[section]4. Galilean transformation(shear motion) in galilean space [G.sub.3]

In this section shear motion on Galilean spaces [G.sub.3], Lie group structure of this motion and exponential form are given.

Theorem 4.1. f is a Galilean transformation(shear motion), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. For x [not equal to] 0 and x = 0, we have

||f(X)|| = |x| = ||X||

and

||f(X)|| = [square root of [y.sup.2] + [z.sup.2]] = ||X||.

So, f is a Galilean transformation.

Theorem 4.2. Gal(3) is a Lie group and the g(3) is a Lie algebra of Gal(3), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[section]5. Heisenberg lie group

The set of matrices

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a Lie group with repect to the matrix multiplication. This Heisenberg group has many important applications on Sub-Riemannian geometry and has very important role in physics.

Lemma 5.1. Gal(3) is a subgroup of Heisenberg Lie group.

Proof. Gal(3) is obtained from Heisenberg Lie group by taking [x.sub.2] = 0.

[section]6. Dual Quaternions and galilean transformation in [G.sub.3]

Every vector X in [R.sup.3] can be written as X = x + iy + jz, where [i.sup.2] = [j.sup.2] = ij = ji = 0 and sp {1, i, j} = [R.sup.3]. The form X = x + iy + jz is called as dual quaternion form of X [member of] [R.sup.3].

Lemma 6.1. Q = 1 + ai + bj is a Galilean transformation in [G.sub.3].

Proof. Since Q = 1 + ai + bj, we have

QX = (1 + ai + bj)(x + iy + jz) = x + (ax + y)i + (bx + z)j

and

||QX|| = ||X||.

Thus the Q is a Galilean transformation. Furthermore, we can write g [member of] g(3) as g = [a.sub.1]i+[b.sub.1]j. So we can write an exponential map as follows:

e : g(3) [right arrow] Gal(3) g [right arrow] [e.sup.g] = [e.sup.[a.sub.1]i+[b.sub.1]j] = 1 + [a.sub.1]i + [b.sub.1]j.

Corollary 6.1. [e.sup.g] = Q = 1 + [a.sub.1]i + [b.sub.1]j is a dual quaternion operator. Thus the dual quaternion operator Q is a Galilean transformation.

[section]7. Galilean transformations in galilean space [G.sub.4]

In this part shear motion is defined in Galilean spaces [G.sub.4]. Structure of Lie group of this motion and exponential form are given.

Theorem 7.1. f is a Galilean transformation(shear motion), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. For x [not equal to] 0 and x = 0, we have

||f(X)|| = |x| = ||X||

and

||f(X)|| = [square root of [y.sup.2] + [z.sup.2] + [l.sup.2]] = ||X||.

So, f is isometry and Galilean transformation.

Theorem 7.2. Gal(4) is a Lie group and g(4) is a Lie algebra of the Gal(4), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[section]8. Dual Quaternions and galilean transformations in [G.sub.4]

In this part, we reviewed study in [1]. Every vector X in [R.sup.3] can be written as the form X = x + iy + jz + kl, where

[i.sup.2] = [j.sup.2] = [k.sup.2] = ij = ji = ik = ki = kj = jk = 0

and

sp {1,i,j,k} = [R.sup.4].

The form X = x + iy + jz + kl is called as dual quaternion form of X [member of] [R.sup.4].

Lemma 8.1. Q = 1 + ai + bj + dk is a Galilean transformation in [G.sub.4].

Proof. Since Q = 1 + ai + bj + dk, we have

QX = (1 + ai + bj + dk)(x + iy + jz + kl) = x + (ax + y)i + (bx + z)j + (dx + l)k

and

||QX|| = ||X||.

Thus, Q is a Galilean transformation. Furthermore, we can write g [member of] g(3) as g = [a.sub.1]I + [b.sub.1]j + [d.sub.1]k So we can write an exponential map as follows:

e : g(4) [right arrow] Gal(4) g [right arrow] [e.sup.g] = [e.sup.[a.sub.1]i+[b.sub.1]j+[d.sub.1]k] = 1 + [a.sub.1]i + [b.sub.1]j + [d.sub.1]k.

Corollary 8.1. Q = [e.sup.g] = 1 + [a.sub.1]i + [b.sub.1]j + [d.sub.1]k is a dual quaternion operator. Thus dual quaternion operator Q is a Galilean transformation.

[section]9. Galilean transformations in galilean space [G.sub.n]

In this part, Galilean transformation generalized to Galilean spaces [G.sub.n], by using Galilean transformation in spaces [G.sub.2], [G.sub.3] and [G.sub.4].

Theorem 9.1. f is a Galilean transformation, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. For [x.sub.1] [not equal to] 0 and [x.sub.1] = 0, we have

||f(X)|| = |[x.sub.1]| = ||X||

and

||f(X)|| = [square root of [x.sub.2.sup.2] + [x.sub.3.sup.2] + [x.sub.n.sup.2]] = ||X||.

Then, f is isometry, thus f is a Galilean transformation.

Theorem 9.2. Gal(n) is a Lie group and g(n) is a Lie algebra of Gal(4), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[section]10. Dual Quaternions and galilean transformation in Gn

In this part, every vector X in [R.sup.n] is written in form X = [x.sub.1] + [x.sub.2][i.sub.1] + [x.sub.3][i.sub.2] + ... + [x.sub.n][i.sub.n-1], where [x.sub.1], [x.sub.2], ... , [x.sub.n] are real numbers. The components of X and [i.sub.1], [i.sub.2], ... , [i.sub.n-1] are units which satisfy the relations [i.sup.2.sub.1] = [i.sup.2.sub.2] = ... = [i.sup.2.sub.n-1] = 0 and [i.sub.j][i.sub.k] = [i.sub.k][i.sub.j] = 0, 1 [less than or equal to] k, j [less than or equal to] n - 1. Also, sp {1, [i.sub.1], [i.sub.2], ... , [i.sub.n-1]} = [R.sup.n], X = [x.sub.1] + [x.sub.2][i.sub.1] + [x.sub.3] [i.sub.2] + ... + [x.sub.n][i.sub.n-1] is called as dual quaternion form of X [member of] [R.sup.n].

Lemma 10.1. Dual quaternion operator Q = 1 + [v.sub.1][i.sub.1] + [v.sub.2][i.sub.2] + ... + [v.sub.n-1][i.sub.n-1] is a Galilean transformation.

Proof. Since Q = 1 + [v.sub.1][i.sub.1] + [v.sub.2][i.sub.2] + ... + [v.sub.n-1][i.sub.n-1], we have

QX = (1 + [v.sub.1][i.sub.1] + [v.sub.2][i.sub.2] + ... + [v.sub.n-1][i.sub.n-1])([x.sub.1] + [x.sub.2][i.sub.1] + [x.sub.3][i.sub.2] + ... + [x.sub.n][i.sub.n-1]) = [x.sub.1] + ([v.sub.1][x.sub.1] + [x.sub.2])[i.sub.1] + ([v.sub.2][x.sub.1] + [x.sub.3])[i.sub.2] + ... ([v.sub.n-1][x.sub.1] + [x.sub.n])[i.sub.n-1]

and

||QX|| = ||X||.

So, the dual quaternion operator Q is a Galilean transformation. For any element from Lie algebra , a [member of] g(n), a = [a.sub.1][i.sub.1] + [a.sub.2][i.sub.2] + ... + a"-lin-1 [member of] g(n) by using exponential map, we have

e : g(n) [right arrow] Gal(ra)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

= 1 + [a.sub.1][i.sub.1] + [a.sub.2][i.sub.2] + ... [a.sub.n-1][i.sup.n.sub.n-1]

Corollary 10.1. The transformation Q is a dual quaternion operator. So, dual quaternion operator Q is a Galilean transformation.

Corollary 10.2. Let a, b [member of] g(n), then Q(a) = [e.sup.a] [member of] Gal(n) and Q(b) = [e.sup.b] [member of] Gal(n).

Furthermore Q(a)Q(b) = [e.sup.a][e.sup.b] = [e.sup.a+b] = Q(a + b). This implies the addition theorem for the velocity on a Galilean transformation.

References

[1] V. Majernik, Quaternion formulation of the Galilean space-time Transformation, Acta Phys, 56 (2006), No. l, 9-14.

[2] I. M. Yaglom, A simple Non-Euclidean geometry and its physical basic, Springer-Verlag Inc, New York, 1979.

[3] M. Nadjafikkah, A.R., Forough, galilean geometry of motions, arXiv:0707.3195v1, math. DG, 2007.

[4] P. Fjelstad, S. G. Gal, N-Dimensional dual complex numbers advances in applied Clifford algebras, 1998, No. 2, 309-322.

Yusuf Yaylit ([dagger]) and Esra Esin Tutuncu ([double dagger])

([dagger]) Ankara University, Faculty of Science, Department of Mathematics, 06100, Tandogan-Ankara-TURKEY

([double dagger]) Rize University, Faculty of Education, Rize-TURKEY

yayli@science.ankara.edu.tr tutuncu.ee@gmail.com
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Author:Yayli, Yusuf; Tutuncu, Esra Esin
Publication:Scientia Magna
Article Type:Report
Geographic Code:7TURK
Date:Jan 1, 2009
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