Wednesday, August 9, 2017

The Change Depends on the Direction of the Motion: The Symmetric Group Action (2)

Following to article of “The Change Depends on the Direction of the Motion: The Symmetric Group Action (1)” posted on link: https://emfps.blogspot.com/2017/08/the-change-depends-on-direction-of.html, the purpose of this article is to introduce some properties of operators and transformations which are formed by moving three points on circle and sphere.
 But, before starting of this article, let me tell you more explanations about theorems mentioned in previous article as follows:

1. All theorems in previous article denote to get the maximum and minimum for vectors in all directions but if we need to have the maximum and minimum of each point on surface or space exchanged by operators and transformations, all equations should be changed as follows:

Theorem (1): │V│max = (2^0.5).r1.r2              and              │V│min = 0

Theorem (4): │V│max = r2.max (z, r1)*(2^0.5)   and 

│V│min = r2.min (z, r1)*(2^0.5) 
 
Where:

r1 = radius in operator or transformation matrix

r2 = radius of each point on surface or space in accordance with its polar coordinates

2. In Theorem (4), if z = r1 then we can say this transformation matrix maps a random point on surface to a random point on a sphere with radius equal to:

  R = (2^0.5).r1.r2 

R = radius of the sphere           

An operator or transformation matrix formed by three points on circle


Suppose three points on a circle are rotating in which the distance among all three points are the same and equal just like below figure: 



For reaching to above conditions, below polar coordinates for each point should be established:

A:
x = r cos θ
y = r sin θ

B:
x = - r sin (θ + 30)
y = r cos (θ +30) 

C:
x = - r sin (30 – θ)
y = - r cos (30 – θ)

By considering any random number for “r” and “θ”, you can see not only all distances are equal but also all three points are on a circle. 

Example:

r = 23   and   θ = 41 degree

AO = BO = CO = 23
AB = BC = CA = 39. 83716857

Above polar coordinates give us a transformation matrix 3*2 as follows:


The properties of transformation matrix 3*2 for R^3 to R^2

By multiplying matrix M by any 3D vectors in the space, we can extract the properties of this transformation matrix as follows:

Theorem (6): The maximum magnitude among 2D vectors produced by three point’s transformation matrix M is calculated by using below equation:

│V│max = 0.5. (6^0.5).r1.r2 

Where:

r1 = radius in transformation matrix M
r2 = radius of each point in the space (3D) in accordance with its polar coordinates

The minimum magnitude is obtained by using below equation:

  │V│min = r1.r2 / Ф

Ф = the constant coefficient equal to 176.943266509085

Here is a very interesting property:

Theorem (7): Always there are six points or six 2D vectors produced by three point’s transformation matrix M which give us the maximum magnitude while there is only one point or one 2d vector which gives us the minimum magnitude in which the direction of all points or 2D vectors is between 0 degree to 180 degree.

The property of transformation matrix 2*3 for R^2 to R^3

It is transpose of above matrix in which we will have below matrix:





Theorem (8): This transformation matrix maps a random point on surface to a random point on a sphere with radius equal to:  R = 0.5. (6^0.5).r1.r2 

Where:

R = radius of the sphere           
r1 = radius in transformation matrix M
r2 = radius of each point on the surface (2D) in accordance with its polar coordinates



The properties of an operator 3*3 

If we want to study these three points in 3D space rotating on circle or sphere, we will have an operator 3*3. In this case, there are several statements where I have started three forms as follows:

1. I added a constant coordinate (z) for each point and matrix will be:




Theorem (9): Maximum and minimum magnitudes among 3D vectors produced by operator M are calculated by using below equations and conditions:

If    r1 /z > 2^0.5   Then   │V│max = 0.5. (6^0.5).r1.r2     and

│V│min = (3^0.5).r2 .min (z, r1)

If    r1 /z < 2^0.5   Then   │V│max = (3^0.5).r2 .max (z, r1)   and

│V│min = 0.5. (6^0.5).r1.r2     

Theorem (10): If r1 /z = 2^0.5 then this operator maps a random point in the space to a random point on a sphere with radius equal to:

 R = 0.5. (6^0.5).r1.r2       or

R = (3^0.5).r2 .min (z, r1)

Where:

R = radius of the sphere           
r1 = radius in operator M
r2 = radius of each point in the space (3D) in accordance with its polar coordinates



Theorem (11): If    r1 /z > 2^0.5   Then,   Always there are six points in the space or six 3D vectors produced by three point’s operator M which give us the maximum magnitude while there is only one point or one 3D vector which gives us the minimum magnitude.

If    r1 /z < 2^0.5   Then,  Always there are six points in the space or six 3D vectors produced by three point’s operator M which give us the minimum magnitude while there is only one point or one 3D vector which gives us the maximum magnitude.   

2. I replaced the constant coordinate (y) instead of (z):


The properties of this operator are similar to theorems (9), (10) and (11).

3. Suppose that three points in the space which have the same distance are rotating on a sphere. In this case, we will have below polar coordinates for all three points as follows:

 Point A:

x = r * cos β * cos θ
y = r * cos β * sin θ
z = r * sin β

Point B:

x = -r * cos (60 – β) * cos θ
y = -r * cos (60 – β) * sin θ
z = r * sin (60 – β)

Point C:

x = -r * cos (60 + β) * cos θ
y = -r * cos (60 + β) * sin θ
z = -r * sin (60 + β)

Example:

Assume θ = 56, β = 17 and    r = 31

According to above coordinates, we can calculate the distance among points and also the distance between points and center of sphere which answers are:

AO = BO = CO = 31
AB = BC = CA = 53.69358

The properties of operator 3*3

Above coordinates of points A and B and C make an operator 3*3 as follows:



Note: θ and β have been introduced in my previous article of “The Change Depends on the Direction of the Motion: Generating All Directions in 3D Space” posted on link: https://emfps.blogspot.com/2017/05/the-change-depends-on-direction-of.html

The property of this transformation matrix is similar to theorems (6) and (7).

An operator or transformation matrix formed by four points on circle


 To be continued….


Wednesday, August 2, 2017

The Change Depends on the Direction of the Motion: The Symmetric Group Action (1)

As you can see, in article of “The Change Depends on the Direction of the Motion: Generating All Directions in 3D Space” posted on link: https://emfps.blogspot.com/2017/05/the-change-depends-on-direction-of.html, only one particle (or one point) was moving on a circle or a sphere to produce some formulas. But, if we increase the number of the particles (points) by two, three, four, five and more in which these points are symmetrically and simultaneously travelling on a circle or a sphere, we will have a symmetric group action. In fact, the results of the particles ’motions make some matrices n*n as operators and some matrices m*n as transformation matrices where all these matrices as symmetric groups actions have very interesting properties. In real world, we can see the applications of these symmetric groups actions every day. For example, an airscrew, screw propeller, ceiling fan, turbine, rotary machines, rotary heat exchangers, helicopter blade, vibrations of a circular drum and also in quantum physics, Hamiltonian operator for Schrodinger’s equation, in chemistry the methane molecule (CH4) is a symmetric group action by four points (particles), all are the ideas inferred from the great theory in mathematics which is the Group Theory.
Even though these operators and transformations have many properties, my focus in this article is only to generate the fields (orbits) and its magnitude. Thus, the purpose of this article is firstly to make many operators and transformations matrices for two, three, four and five points which are rotating and secondly to find out one the most important properties which is the fields (orbits) and its magnitude.

Introduction
According to Tom Davis (2006) on paper of “Group Theory via Rubik’s Cube”, he stated:
A group is a mathematical object of great importance, but the usual study of group theory is highly abstract and therefore difficult for many students to understand. Very important classes of groups are so-called permutation groups which are very closely related to Rubik’s cube. Thus, in addition to being a fiendishly difficult puzzle, Rubik’s cube provides many concrete examples of groups and of applications of group theory.”

Therefore, it is better for the beginners to perceive the concepts and applications of the Group theory, provide a Rubik’s Cube.



The most important operators in Physics and Engineering are the Gradient, Curl and Divergence. In fact, an operator is a mathematical object that maps one state vector into another one in which it can be written in matrix form and is considered as a group action which is continuous such as rotation of a circle or discrete such as reflection of a bilaterally symmetric figure. The group theory develops the significant features in the formulation of physics similar to chemistry where the group theory is utilized to illustrate symmetries of crystal and molecular structures. It means that the applications of the group theory are endless.
In the next part, I will continue the topic by rotating two points on diameter of a circle then I develop it by using two points on dimeter of a sphere.

An operator by two points on circle

Suppose you are rotating two points on a circle which have the distance equal to 2r, just like below figure:

As you can see, actually you are making many ellipses in different directions:


The general formula of these ellipses is as follows:

Where:

r: radius of circle

θ: angle of rotating

x, y: coordinates of point P on each ellipse

2a:  sum distance between point P (on ellipse) and two points on diameter of circle

If we assume two points “A” and “B” on diameter of circle which are rotating in counterclockwise and have the distance equal 2r, above formula will be obtained by using their polar coordinates as follows:

A:
x = r.cosθ
y = r.sinθ

B:
x = -r.cosθ
y = -r.sinθ

or

A:
x = r.sinθ
y = -r.cosθ

B:
x = -r.sinθ
y = r.cosθ

Polar coordinates of points A and B give us a matrix 2*2 which is an operator:

The properties of two point’s operator in surface

Now, I investigate the properties of matrices M or N by multiplying 2D unit vectors in all directions (V * M or V * N)

Theorem (1): The maximum magnitude among vectors produced by two point’s operator (matrix M or N) is equal radius (r) multiply (2^0.5) and minimum magnitude is equal zero.

│V│max = (2^0.5).r              and              │V│min = 0

Theorem (2): The equation of produced vectors is generally:  V = a (i – j) in which

 Vmax = r (i –j) and the direction is: V = 0.7071i – 0.7071j and V = -0.7071i + 0.7071j

Third property is very interesting where it is about Eigenvalue and Eigenvector of above operator.

 Theorem (3): Eigenvalue and Eigenvector of Matrix M or N for the range of angles:
-135 ≤ θ <45 and -360 ≤ θ < -315 for different radiuses (r) are obtained only in directions of 135 degree and 315 degree.

Example: I calculated some samples on excel spreadsheet as follows:


The properties of transformation matrix for two points in 3D space

If we want to study these two points in 3D space rotating on circle, we will have a transformation matrix. In this case, there are several statements where I have started three forms as follows:
1. I added a constant coordinate (z) for each point and matrix will be:


The properties of transformation matrix 2*3 for R^2 to R^3

This matrix transforms 2D unit vectors in all directions to vectors in 3D space.

Theorem (4): Maximum and minimum magnitudes among 3D vectors produced by transformation matrix M are calculated by using below equations:

│V│max = max (z, r)*(2^0.5)   and │V│min = min (z, r)*(2^0.5)  
Thus, if z = r then all magnitudes of 3D vectors will be the same.

Theorem (5): If z > r then the direction of maximum output will be (0, 0, 1) or (0, 0, -1).
If z > r then the direction of minimum output will be (0, 0, 1) or (0, 0, -1).

The properties of transformation matrix 3*2 for R^3 to R^2

It is transpose of previous matrix in which we will have below matrix: 

The property of this matrix is similar to theorems (1) and (2).

2. I replaced the constant coordinate (y) instead of (z):

The property of this transformation matrix is similar to theorems (4) and (5).

3. Suppose that two points in the space which have the distance equal 2r are rotating on a sphere. In this case, we will have below polar coordinates for both of them:

 Point A:
x = r * cos β * cos θ
y = r * cos β * sin θ
z = r * sin β

Point B:
x = -r * cos β * cos θ
y = -r * cos β * sin θ
z = -r * sin β

The properties of transformation matrix 2*3 for R^2 to R^3

Above coordinates of points A and B can make a transformation matrix as follows:

Note: θ and β have been introduced in my previous article of “The Change Depends on the Direction of the Motion: Generating All Directions in 3D Space” posted on link: https://emfps.blogspot.com/2017/05/the-change-depends-on-direction-of.html

The property of this transformation matrix is similar to theorems (1).

The properties of transformation matrix 3*2 for R^3 to R^2

The transpose of above matrix gives us a transformation matrix 3*2 as follows:

The property of this matrix is similar to theorems (1) and (2).

An example in Fluid Mechanics

Assume a fluid is rotating in counterclockwise by an angular velocity ω around z axis. In this case, each particle on a circle with radius r which is perpendicular to z axis is rotating just like below figure:


According to above figure, the magnitude of velocity for the particle will be equal:

│v│= ω.r and also position vector of this article will be: R = xi + yi

Since vector of velocity is perpendicular to position vector, therefore, the equation for velocity vector will be:  v = ω (-yi + xj)

            These are the characteristics of this field. But this filed can be considered as an operator of a whirlpool in 2D on a circle just like below photo:




An operator by three points on circle

Suppose three points on a circle are rotating in which the distance among all three points are the same and equal. For reaching to these conditions, below polar coordinates for each point should be established:

A:
x = r cos θ
y = r sin θ

B:
x = - r sin (θ + 30)
y = r cos (θ +30)

C:
x = - r sin (30 – θ)
y = - r cos (30 – θ)

Above polar coordinates give us a transformation matrix 3*2 as follows:

The properties of transformation matrix 3*2 for R^3 to R^2

To be continued….

Note:
All researchers who are interested in having these models to find out further the properties, do not hesitate to send their requests to my email: soleimani_gh@hotmail.com