Advances in Pure Mathematics
Vol.05 No.01(2015), Article ID:53475,8 pages
10.4236/apm.2015.51005
Time Scale Approach to One Parameter Plane Motion by Complex Numbers
Hatice Kusak Samanci1, Ali Caliskan2
1Department of Mathematics, Faculty of Sciences, Bitlis Eren Üniversitesi, Bitlis, Turkey
2Department of Mathematics, Faculty of Sciences, Ege Üniversitesi, Izmir, Turkey
Email: ecitah_tamus@yahoo.com , ali.caliskan@ege.edu.tr
Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 5 January 2015; revised 20 January 2015; accepted 23 January 2015
ABSTRACT
This paper presents building one-parameter motion by complex numbers on a time scale. Firstly, we assumed that
and
were moving in a fixed time scale complex plane and
and
were their orthonormal frames, respectively. By using complex numbers, we investigated the delta calculus equations of the motion on
. Secondly, we gave the velocities and their union rule on the time scale. Finally, by using the delta-derivative, we got interesting results and theorems for the instantaneous rotation pole and the pole curves (trajectory). In kinematics, investigating one-parameter motion by complex numbers is important for simplifying motion calculation. In this study, our aim is to obtain an equation of motion by using complex numbers on the time scale.
Keywords:
Complex Numbers, Kinematic, Time Scales, Pole Curve
1. Introduction
The calculus on time scales was initiated by B. Aulbach and S. Hilger in order to create a theory that can unify discrete and continuous analysis, [1] . Some preliminary definitions and theorems about delta derivative can be found in the references [2] - [4] .
In this study, some properties of motion in references [5] - [7] are investigated by using time scale complex planes. We find delta calculus equations of the motion and finally we get some results about the pole curves.
2. Preliminaries
A time scale is an arbitrary nonempty closed subset of the real numbers.
Definition 2.1. Let
be any time scale. The forward jump operator
is defined by
and the backward jump operator
is defined by
In this definition, we put
(i.e.
, if
has a maximum t) and
(i.e.
, if
has a minimum t), where
denotes the empty set. If
, we say that t is right-scat- tered, while if
we say that t is left-scattered. Points that are right-scattered and left-scattered at the same time are called isolated. Also if
and
, then t is called right-dense, and if
and
, then t is called left-dense. Points that are right-dense and left-dense at the same time are called dense.
Finally, the graininess function
is defined by
If
is a function, then we define the function
by
Let us define the interior of
relative to
which is a function that maps
into
to be the set
Definition 2.2. Assume
is a function and let
. Then we define
to be the number (pro- vided it exists) with the property that given any ε > 0, there is a neighborhood U of t (i.e.
for some
) such that
We call
the delta (or Hilger) derivative of f at t. Moreover, we say that f is delta (or Hilger) differentiable on
provided
exists for all
.
Theorem 2.1. Assume
is a function and let
. Then we have the following:
1) If f is differentiable at t, then f is continuous at t.
2) If f is continuous at t and t is right-scattered, then f is differentiable at t with
3) If t is right-dense, then f is differential at t if the limit
exists as a finite number. In this case a given
4) If f is differentiable at t then
Theorem 2.2. Assume
are differentiable at
. Then:
1) The sum
is differentiable at t with
2) For any constant,
is differentiable at t with
3) The product
is differentiable at t with
4) If
then
is differentiable at t with
5) If, then
is differentiable at t with
In the reference [3] , the chain rule on time scales is given for various cases.
Theorem 2.3. Assume
is continuous,
is delta differentiable on
, and
is continuously differentiable. Then, there exists
in the real interval
with
Theorem 2.4. Let
be continuously differentiable and suppose
is delta differentiable. Then
is delta differentiable and the formula
holds.
Theorem 2.5. Assume that
is strictly increasing function and
is a time scale. Let
. If
and
exist for
, then
(2.1)
Definition 2.3. For the given time scales
and
, let us set
(2.2)
where
is the imaginary unit. The set
is called the time scale complex plane.
Definition 2.4. For, we define the cylinder transformation
by
and for, let
.
Definition 2.5. If, then we define the exponential function by
where the cylinder transformation
is introduced in Definition 2.4.
Theorem 2.6. If
then
1)
and
;
2);
3);
4);
5);
6);
7);
8);
Theorem 2.7. Assume
for
Theorem 2.8. If
then
Theorem 2.9. If
and
then
3. One Parameter Motion and Hilger Complex Numbers on a Time Scale
Assume that
is a time scale. Let us set the time scale complex plane for as
(3.1)
Here, let
and
be moving in a fixed time scale complex plane. The motion is called as one-parameter planar motion by the complex numbers on the time scale and denoted as
for a planar motion of E relative to E′.
and
be their orthonormal frames, respectively. We suppose that
is fixed, then we say that
moves with respect to
,
,
are the functions of a time scale parameter t. Let
and
be the position vectors of a point X in the plane, as following we can write the coordinates of the point X by using complex numbers on the time scale with respect to a fixed or moving plane
and
, respectively. So:
The translation vector
can be written as the following equation on a fixed plane
:
by using the definition of the time scale complex plane. The translation vector is more suitable as
for doing the formulas symmetric on the moving plane.
Thus,
is equivalent to the vector
. Let
be a rotation angle between the vectors
and
(or the time scale complex planes
and
), in Figure 1. So we can find the equation
Figure 1. One parameter planar motion on time scale.
(3.2)
For any point, the vector
is
(3.3)
By substituting
in the Equation (3.3)
(3.4)
Then, we can obtain the vector
as follows:
Here, assume the functions
are
-differentiable functions and the parameter t is defined as
on the
time scale. We will cal- culate the formulas for a fixed or moving plane.
Definition 3.1. A velocity vector of the point X with respect to E is called
-relative velocity vector of the point X on the time scale. The equation of relative
-velocity vector is
(3.5)
for the moving time scale complex plane.
Definition 3.2. A velocity vector of the point X with respect to E is called
-relative velocity vector of the point X on the time scale. The equation of the relative
-velocity vector is
(3.6)
(3.7)
for the fixed time scale complex plane.
Definition 3.3. A velocity vector of the point X with respect to the time scale complex plane
on the planar motion
which belongs to a curve
of the point
on
is called the
-absolute velocity vector of the point X on the time scale and is denoted by
.
Definition 3.4. On the planar motion, while the point X is fixed on the moving time scale complex plane
(i.e.
), a velocity vector of the point X is called the
-dragging velocity vector of this point on the time scale and is denoted by
.
So, we obtain the
-absolute velocity
, i.e. the velocity of X with respect to the plane
, from the Equation (3.4) using Equation (3.2).
by Theorem 2.5. Also
and using Theorem 2.7, we have
Here,
is called a delta-angular velocity of the motion
on a time scale, and remembering Equations (3.3) and (3.7), we can find the dragging velocity vector
of the point X
(3.8)
(3.9)
with the restriction, from Equation (3.2) by taking the
-derivative with respect to the parameter t, we get the following equation.
and using Equation (3.2), we get
(3.10)
Theorem 3.1. A
-absolute velocity vector is equal to adding a
-relative velocity vector and
-dragging velocity vector on the motion
, i.e.
(3.11)
Proof. By using Equation (3.10) and Equation (3.5), we can get the following equations:
and thus, we get the relation of the velocities:
We have
We will calculate
here using Equation (3.9) and Equation (3.10);
and
(3.12)
Theorem 3.2. There is only one point at which the
-dragging velocity is zero for any instant
, i.e. which is fixed on the both of the planes
and
, with the restriction
on the motion
.
Proof. The points at which the
-dragging velocity vector is zero for any instant
have to stay fixed for not only the plane
, but also for the plane
on the motion
. By taking
for fixed and moving planes, from (3.15) and (3.8):
(3.13)
(3.14)
we can obtain the following complex vectors;
(3.15)
(3.16)
which are given
-instantaneous rotation pole P on both coordinate systems. Because, the affine axioms
,
are the end-points of
,
, respectively.
Definition 3.5. The point
which corresponds to the position vector
is called the for- ward pole or the instantaneous rotation pole or the instantaneous rotation center for the moving plane on the time scale motion
, in Figure 2.
Definition 3.6. The point
which corresponds to the position vector
is called the for- ward pole or the instantaneous rotation pole or the instantaneous rotation center for the fixed plane on the time scale motion
, in Figure 2.
We can get the following equations from Equation (3.15) and Equation (3.16):
(3.17)
(3.18)
By eliminating
and
from Equation (3.13) and Equation (3.14), the dragging velocity becomes as following:
(3.19)
(3.20)
and;
4. Conclusions
Result 4.1. Two results for the
-dragging velocity of the point
on the moving plane can be obtained as follows:
Figure 2. The pole curve on time scale.
1) Since scalar product of the vector is
and the vector
is zero, these vectors are perpendicular.
2) The length of the vector
can be calculated as follows:
here
denotes for the length of
. From this result, we get the following theorem:
Theorem 4.1. On the motion, the points X of the moving plane E draw trajectories on the fixed time scale complex plane
which their normals (trajectory normals) pass from the instantaneous rotation pole
.
Theorem 4.2. Every point of X of the moving plane E is doing rotational movement (instantaneous rotation movement) with a
-centered,
-angular velocity and p factor on instant t.
Since X is an arbitrary point of the time scale complex plane E, we can give the following theorem:
Theorem 4.3. A one-parameter motion consists of rotation with
angular velocity and p factor around the instantaneous rotation pole
of the moving plane E on t instant, i.e. the plane E rotates with the angle
and the factor p around the point
on the time element
.
Theorem 4.4. The velocity vectors of the instantaneous rotation pole
which draws the forward pole curves on the moving and fixed planes is the same vector at each instant t.
Theorem 4.5. On one-parameter planar motion
the moving pole curve
of the plane E rolls onto the fixed pole curve
of the plane
without sliding.
Result 4.2. Without being depended on time, a motion
occurs by rolling, without sliding, the curve
of E onto the curve
of
.
References
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User.
- Bohner, M. and Peterson, A. (2001) Dynamic Equations on Time Scales, An Introduction with Applications, Birkh
User.
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- Bottema, O. and Roth, B. (1990) Theoretical Kinematics. Dover Publications, Mineola.
- Blaschke, W. (1960) Kinematik und Quaternionen. Mathematische Monographien. Springer, Berlin.
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