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

The Change Depends on the Direction of the Motion: The Angles in Shadow

In the reference with my article of “The Change Depends on the Direction of the Motion: Generating All Directionsin 3D Space”, before going to any analysis in physics and engineering subjects, you have to calibrate and filter your template by using the method mentioned in this article. In fact, I had to make the necessary amendments to above article because maybe you will take some mistakes in your analysis when you utilize the template stated in my previous article (Generating All Directions in 3D Space).

What is the case?

The case is to be produced many angles which show the same direction. I name these angles:

“The Angles in Shadow”

What is the meaning “The angles in Shadow”?

Let me tell you an example to illustrate this concept.

Assume you are fixing your telescope on four points in the space. First you turn your telescope on Horizontal angle 104 degree (θ = 104) and Vertical angle 253 (β = 253). Then you turn it on Horizontal angle 284 degree (θ = 284) and Vertical angle 287 (β = 287). Then you turn it on Horizontal angle 104 degree (θ = 104) and Vertical angle 287 (β = 287) and finally you turn it on Horizontal angle 284 degree (θ = 284) and Vertical angle 253 (β = 253). If you use the equations stated in my previous article, you can calculate all directions as follows:

 As you can see, you are really looking at two points instead of four points. In fact, state number 1 and 2 are in the same direction and state number 3 and 4 are in the same direction.

How can we find the angles in shadow?

Here I am willing to introduce to you two methods. The method (1), which uses some trigonometric equations while method (2) follows the same method mentioned in article of “Can We Solve a Nonlinear Equation with Many Variables?” posted on link: http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with.html

At the first, we should bear in mind that below conditions should be established for both methods:

Condition (1): For Vertical angles,

 Sin β = Sin β'   and    Cos β = - Cos β'

Condition (2):  For Horizontal angles, 

Sin θ = - Sin θ'   and   Cos θ = - Cos θ'

 Method (1):

We as well as know Trigonometric reduction formulas which are as follows:

Sin (90 – α) = Cos α    
   
Cos (90 – α) = Sin α   

Sin (90 + α) = Cos α   

Cos (90 + α) = - Sin α

To obtain condition (1), I use below tricks:

Sin (90 – α) = Cos α = Sin β         

Cos (90 – α) = Sin α = Cos β

Sin (90 + α) = Cos α = Sin β'        

 Cos (90 + α) = - Sin α = - Cos β'

And so, we have below formulas in Trigonometric:

Cos (180 + α) = - Cos α             

Sin (180 + α) = - Sin α

To get condition (2), I also use below formulas:

Cos (180 + α) = - Cos α = - Cos θ'       

 Sin (180 + α) = - Sin α = - Sin θ'

θ = α    and   θ' = 180 + α

According to above relationships, I can write a simple algorithm to generate all direction including symmetry direction and others as follows:

As you can see, I have fixed angle of (α) and have copied and pasted all angles on green, red, blue and yellow colors that if you change only angle of (α), you can easily get all the same directions just like below examples:

You can find the results for all 360 degrees by using this algorithm and a sensitivity analysis between α and θ, θ', β, and β'.

Method (2):

If we apply all 360 degrees for algorithm method (1), we will have 180 states for θ and θ' and 181 states for β and β' where we can not find the angles mentioned in above example (θ = 104 and β = 253). It means that method (1) gives us incomplete results. But, by using the method mentioned in article of “Can We Solve a Nonlinear Equation with Many Variables?” posted on link: http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with.html, we can generate the complete results as follows:

For condition (1), we have:  tg β = - tg β'

For condition (2), we have:  tg θ = tg θ'

For establishing condition (1), we have to solve below equation:

tg β + tg β' = 0

This is an equation with two independent variables which can be solved with the method mentioned in article of “Can We Solve a Nonlinear Equation with Many Variables?” posted on link: http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with.html,.

For establishing condition (2), we have to solve below equation:

tg β - tg β' = 0

This is an equation with two independent variables which can be also solved with above method.

By using method (2), we will have 360 states for θ and θ' and 358 states for β and β' in which the angles mentioned in above example (θ = 104 and β = 253) are also included in our results just like below cited:

Finally, method (2) says to us that there are (360 * 357) / 2 = 64260 repetitions which should be deducted from the total 130322 states where we should take our analysis in accordance with 66062 states.

 By analysis of  the results extracted from the method (2), we can find the general formulas as follows:

θ' = θ +180    If     0 ≤ θ ≤ 180

   

β' = 180 – β   If     0 ≤ β ≤ 180

θ' = θ -180     If    180 ≤ θ ≤ 360   

β' = 540 – β   If    180 ≤  β ≤ 360

θ and β = degree

Example (1):

Suppose θ = 56 degree and β = 112 degree. According to above formulas, we should use below equations:

θ' = θ +180    If     0 ≤ θ ≤ 180

   

β' = 180 – β   If     0 ≤ β ≤ 180

Then we have:

θ' = 180 +56 = 236 degree

β' = 180- 112 = 68 degree

By using equations mentioned 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, we will have below directions:

Direction V =(cos β * cos θ) i + (cos β * sin θ) j + (sin β) k

 For θ = 56   and    β = 112   , Direction V is:

Direction V = -0.20948 i -0.31056 j + 0.927184 k

For θ' = 236   and    β' = 68   , Direction V is:

Direction V = -0.20948 i -0.31056 j + 0.927184 k

You can see that the direction both of them are the same.

Example (2):

Suppose θ = 221 degree and β = 295 degree. According to above formulas, we should use below equations:

θ' = θ -180     If    180 ≤ θ ≤ 360   

β' = 540 – β   If    180 ≤  β ≤ 360

Then we have:

θ' = 221 - 180 = 41 degree

β' = 540 - 295 = 245 degree

Direction V =(cos β * cos θ) i + (cos β * sin θ) j + (sin β) k

 For θ = 221   and    β = 295   , Direction V is:

Direction V = -0.31895 i -0.27727 j – 0.90631 k

For θ' = 41   and    β' = 245   , Direction V is:

Direction V = -0.31895 i -0.27727 j – 0.90631 k

You can see that the direction both of them are the same.

In the reference with above formulas, we reach to a constant number equal to 64442 rows on spreadsheet of excel which shows us all directions in 3D space.

Conclusion

When we encounter the big data on our spreadsheet, the most crucial thing to bear in mind is to deduct the data in which there will not be any difference in our final results and analysis because it is possible that our laptop and computer will not be able to process the big data due to its technical characteristics.

The result of this article says to us that we can decrease 130321 rows on our spreadsheet to 64442 rows while the results and analysis will be finally the same.

Example:

Suppose you have a curve in 3D space as follows and the point P (x, y, z) wants to move from P0 (-1, 3, 2) on this curve toward all directions in amount of 4 unit (∆s = 4 unit). The question is:

What is the maximum change of below function when point P (x, y, z) moves in all directions? Which direction will the maximum change of the function occur?

Solution:

Suppose above function is a curve of heat equation to analyze thermal conduction in a room. Why do I say, assume this function is a curve of heat equation? Because this function follows the Laplace's equation.

Now, when I try above formula for “i” between 1 and 130321, the answer is the same.

The Images of The Universe

Here is the images of the Universe. Everything is the same

PS: “I think that it is not still the time to release all results around the world”

The Change Depends on the Direction of the Motion: Generating All Directions in 3D Space

In physics, science, and engineering, especially fluid dynamics and electromagnetism fields, we usually need to investigate the changes of a function in different directions. In this case, the best way for analyzing and designing is to have different directions all together in our hands, in which we will be able to compare all results to reach the new theorems and the new physical phenomena. The purpose of this article is to make a spreadsheet on an Excel file by using a new method where you will have all unit vectors in 3D space in different directions.

Let me give you an example:

Suppose you have a curve of f (x, y, z) = x^2 + y^2 - 2z^2 and the point P (x, y, z) wants to move from P0 (-1, 3, 2) on this curve toward all directions in amount of 4 unit (∆s = 4 unit). The question is:

What is the maximum change of the function when point P (x, y, z) moves in all directions? In which direction will the maximum change in the function occur?

Solution:

By making an analytic model in Excel, you will be able to answer the above questions. There is no need for any other software.

Unit Vectors in 2D and 3D

You as well as know, we can easily get all unit vectors in 2D surface by using bellow formula:

U = i cosθ + j sinθ

On the other hand, we can obtain the direction of a vector in 2D and 3D, by using below formula:

Direction of A = A / │A│

 But, how can we prove above formula?

A vector in 3D space can be modelled as the radius of a sphere in which we will have below function:

w = r (x, y, z) = (x^2 + y^2 + z^2) ^0.5

The gradient vector in any point is:

As you can see, the gradient vector proves above formula (formula:

Direction of A = A / │A│) where vector A = xi + yj + zk because we have:

Generating all directions in 3D space

 Consider a particle starts its circular motion on surface XY and simultaneously has a circular motion on surface XZ perpendicular to surface XY. In this case, this particle will produce a sphere where its circular angle on surface XY is “θ” and its circular angle on surface XZ is “β”.

According to above circular angles and radius of sphere (r), we can calculate coordination of point P (x, y, z) on sphere by using below equations:

x = r * cos β * cos θ

y = r * cos β * sin θ

z = r * sin β

Example:

Solve equation:    x^2 + y^2 + z^2 = 225

If θ = 25 degree and β = 56 degree

Answer:

We have:  r = (225) ^0.5 = 15

x = 15 * cos 56 * cos 25 = 7.602013

y = 15 * cos 56 * sin 25 = 3.544877

z = 15 * sin 56 = 12.43556

(7.602013 ^2) + (3.544877 ^2) + (12.43556 ^2) = 225

In fact, we have below vector:

V = xi + yj + zk

V = (r * cos β * cos θ)i + (r * cos β * sin θ)j + (r * sin β)k

For r = 1, we have below unit vector in 3D space:

U = (cos β * cos θ)i + (cos β * sin θ)j + (sin β)k

It is clear; the range of changes for “θ” and “β” is between 0 and 360 degree.

Therefore, for making a spreadsheet included all directions, we should go below steps:

- Choose ∆θ and ∆β between 0 and 360 degree. For instance, I considered ∆θ = ∆β = 1

- By using the method stated in article of “Can We Solve a Nonlinear Equation with Many Variables?” posted on link: http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with.html, we should find all combinations of “θ” and “β” for ∆θ and ∆β between 0 and 360 degree. For example, when I considered ∆θ = ∆β = 1, I will have 130321 combinations on my excel spreadsheet (361^2 = 130321).

If you choose ∆θ = ∆β = 0.5, you will have 519841 combinations (directions) on your excel spreadsheet (721^2 = 519841).

Anyway, I think that ∆θ = ∆β = 1 is enough.

- Using from above equations for r = 1 and each set of combinations.  In this case, you have 130321 rows that it show you all directions which you need to your analysis.

In the next articles, I will show you how we can utilize this spreadsheet as a template to investigate the changes of some physical functions.