The Distances among The Particles in The Space (2)

Following to the article of  "The Distances among The Particles in The Space (1)", let me continue new conjectures as follows:

  

Conjecture (4):

The distances between the point P0 (a, a, a) and the points mentioned in conjecture (3) are always the same where “a” is a member of real number.

Example:

Suppose we have point P1 with below coordination:

P1 (-3, 15, 34)

According to conjecture (3), the points P2 and P3 will be:

P2 (34, -3, 15)

P3 (15, 34, -3)

The distances among the points P1 and P2 and P3 are equal to 45.32108.

Assume we have point P0 with below coordination:

P0 (56, 56, 56)

The distances between the point P0 (56, 56, 56) and the points P1 and P2 and P3 is the same and 

equal to 75.13987.

d (P0, P1 ) = 75.13987

d (P0, P2) = 75.13987

d (P0, P3 ) = 75.13987

The Mapping A System of Four Particle for Given Gravity Potential Energy

Suppose we have four particles P0, P1, P2 and P3 with the distances among them in infinity   which have the same masses of “m”. If an external force brings all these particles in new location just like 

below figure, how can we map the location of these particles for a constant gravity potential energy?

  As I told you, suppose that mass of all particles is the same. Therefore, for our analysis, we have three independent variables of “m”, “r” and “R”.

Suppose the constant gravity potential energy is equal -1.008E-08 J

U (r) = -1.008E-08 J

  As I stated in my previous article, to find the coordination and mapping the particles, we should take two steps:

Step (1):

To find the mass of particles accompanied by the distances of “r” and “R”, we should apply the method mentioned in article of “Solving a Nonlinear Equation with Many Independent Variables” 

I found 8 answers for three independent variables of the particles:

For example, I choose below three variables from above table:

m	r	R	T_G_P_E
1	0.02	2.8	-1.007949E-08
3	0.21	1.2	-1.007949E-08
3	0.24	0.7	-1.007949E-08

And by using these three set, I start step 2.

Step (2):

Since we should get the coordination of particles by using the distances “r’ and “R”, therefore, we have to solve a system of two nonlinear equations as follows:

By using the model presented in article of “A Model to Track theLocation of a Particle in the Space” , we can easily find the coordination of the particles.

For instance, if we have:

 m = 1kg

r = 0.02 m

R = 2.8 m  

I found 444 models that some of them are as follows:

For testing of these models, I use model (1) in above table and we can see the results as follows:

If we have:

 m = 3kg

r = 0.21 m

R = 1.2 m  

I found 276 models that some of them are as follows:

For testing of these models, I use model (1) in above table and we can see the results as follows:

If we have:

 m = 3kg

r = 0.24 m

R = 0.7 m  

I found 276 models that some of them are as follows:

For testing of these models, I use model (1) in above table and we can see the results as follows:

Conclusion

Suppose you are squeezing a sponge by your hand and assume that the work done by your hand will stay the constant. How can you say that you are controlling the potential energy throughout the sponge when you deform the sponge and change the location of particles of the sponge?

In physics, the people usually use the vector fields, gradient vectors, curl and so on. But the problem is, to encounter with the chaos systems in the nature in which you are not able to find a real function for your subject. In this case, the people usually use the methods of the boundary conditions.

I think that the method mentioned in above article can help us to solve the problems which are defined as the chaos systems.