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