Dave's Solution to the Three-Body Problem

Dave's Solution to the Three-Body Problem

Postby Guest » Tue Sep 10, 2019 7:36 pm

"Simple seeks simplest (best) solution." -- Dave.

Relevant Reference Link:

https://en.wikipedia.org/wiki/Three-body_problem.

In our closed system, the total and constant Energy (E) = total Kinetic Energy (K) + total Gravitational Energy (U) of our three-body system.

We recall our system equations with regards to acceleration (a) due to gravitational energy:

1a. [tex]\frac{d^{2}x_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{x_{1 }-x_{2 }}{r_{12}^{3}} + m_{3 }* \frac{x_{1 }-x_{3 }}{r_{13}^{3}})[/tex];

2a. [tex]\frac{d^{2}y_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{y_{1 }-y_{2 }}{r_{12}^{3}} + m_{3 }* \frac{y_{1 }-y_{3 }}{r_{13}^{3}})[/tex];

3a. [tex]\frac{d^{2}z_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{z_{1 }-z_{2 }}{r_{12}^{3}} + m_{3 }* \frac{z_{1 }-z_{3 }}{r_{13}^{3}})[/tex];

4a. [tex]\frac{d^{2}x_{2 }}{dt^{2}} = -G* (m_{1 }* \frac{x_{2 }-x_{1 }}{r_{21}^{3}} + m_{3 }* \frac{x_{2 }-x_{3 }}{r_{23}^{3}})[/tex];

5a. [tex]\frac{d^{2}y_{2 }}{dt^{2}} = -G* ( m_{1 }* \frac{y_{2 }-y_{1 }}{r_{21}^{3}} + m_{3 }* \frac{y_{2 }-y_{3 }}{r_{23}^{3}})[/tex];

6a. [tex]\frac{d^{2}z_{2 }}{dt^{2}} = -G* ( m_{1 }* \frac{z_{2 }-z_{1 }}{r_{21}^{3}} + m_{3 }* \frac{z_{2 }-z_{3 }}{r_{23}^{3}})[/tex];

7a. [tex]\frac{d^{2}x_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{x_{3 }-x_{1 }}{r_{31}^{3}} + m_{2 }* \frac{x_{3 }-x_{2 }}{r_{32}^{3}})[/tex];

8a. [tex]\frac{d^{2}y_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{y_{3 }-y_{1 }}{r_{31}^{3}} + m_{2 }* \frac{y_{3 }-y_{2 }}{r_{32}^{3}})[/tex];

9a. [tex]\frac{d^{2}z_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{z_{3 }-z_{1 }}{r_{31}^{3}} + m_{2 }* \frac{z_{3 }-z_{2 }}{r_{32}^{3}})[/tex]

with [tex]r_{ij} = \sqrt{(x_{i }-x_{j })^{2} + (y_{i }-y_{j })^{2} + (z_{i }-z_{j })^{2} }[/tex] where G is the gravitational constant.

Now, we generate our final system equations corresponding to the energy conservation law for our closed system:

1. [tex]\frac{1}{2} * m_{1 }*(\frac{dx_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{x_{1 }-x_{2 }}{r_{12}^{3}} + m_{3 }* \frac{x_{1 }-x_{3 }}{r_{13}^{3}})*\triangle r_{1 } =E_{x_{1 }}[/tex];

2. [tex]\frac{1}{2} * m_{1 }*(\frac{dy_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{y_{1 }-y_{2 }}{r_{12}^{3}} + m_{3 }* \frac{y_{1 }-y_{3 }}{r_{13}^{3}})*\triangle r_{1 } =E_{y_{1 }}[/tex];

3. [tex]\frac{1}{2} * m_{1 }*(\frac{dz_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{z_{1 }-z_{2 }}{r_{12}^{3}} + m_{3 }* \frac{z_{1 }-z_{3 }}{r_{13}^{3}})*\triangle r_{1 } =E_{z_{1 }}[/tex];

4. [tex]\frac{1}{2} * m_{2 }*(\frac{dx_{2 }}{dt})^{2} - G* m_{2 }*( m_{1 }* \frac{x_{2 }-x_{1 }}{r_{21}^{3}} + m_{3 }* \frac{x_{2 }-x_{3 }}{r_{23}^{3}})*\triangle r_{2 } =E_{x_{2 }}[/tex];

5. [tex]\frac{1}{2} * m_{2}*(\frac{dy_{2 }}{dt})^{2} - G* m_{2 }*( m_{2 }* \frac{y_{2 }-y_{1 }}{r_{21}^{3}} + m_{3 }* \frac{y_{2 }-y_{3 }}{r_{23}^{3}})*\triangle r_{2 } =E_{y_{2 }}[/tex];

6. [tex]\frac{1}{2} * m_{2 }*(\frac{dz_{2 }}{dt})^{2} - G* m_{2 }*( m_{1 }* \frac{z_{2 }-z_{1 }}{r_{21}^{3}} + m_{3 }* \frac{z_{2 }-z_{3 }}{r_{23}^{3}})*\triangle r_{2 } =E_{z_{2 }}[/tex];

7. [tex]\frac{1}{2} * m_{3 }*(\frac{dx_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{x_{3 }-x_{1 }}{r_{31}^{3}} + m_{2 }* \frac{x_{3 }-x_{2 }}{r_{32}^{3}})*\triangle r_{3 } =E_{x_{3 }}[/tex];

8. [tex]\frac{1}{2} * m_{3 }*(\frac{dy_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{y_{3 }-y_{1 }}{r_{31}^{3}} + m_{2 }* \frac{y_{3 }-y_{2 }}{r_{32}^{3}})*\triangle r_{3 } =E_{y_{3 }}[/tex];

9. [tex]\frac{1}{2} * m_{3 }*(\frac{dz_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{z_{3 }-z_{1 }}{r_{31}^{3}} + m_{2 }* \frac{z_{3 }-z_{2 }}{r_{32}^{3}})*\triangle r_{3 } =E_{z_{3 }}[/tex]

where [tex]\triangle r_{k } = \sqrt{(x_{k}(t + \triangle t )-x_{k }(t))^{2} + (y_{k}(t + \triangle t)-y_{k }(t))^{2} + (z_{k}(t + \triangle t)-z_{k }(t))^{2} }[/tex].

Next, we list the important position functions corresponding to our spatial components:

10. [tex]x_{1 } = x_{1 }(t) = \frac{1}{2} * a_{11 }*t^{2} + b_{11 }*t + c_{11 }[/tex];

11. [tex]y_{1 } = y_{1 }(t) = \frac{1}{2} * a_{12 }*t^{2} + b_{12 }*t + c_{12 }[/tex];

12. [tex]z_{1 } = z_{1 }(t) = \frac{1}{2} * a_{13 }*t^{2} + b_{13 }*t + c_{13 }[/tex];

13. [tex]x_{2} = x_{2 }(t) = \frac{1}{2} * a_{21 }*t^{2} + b_{21 }*t + c_{21 }[/tex];

14. [tex]y_{2 } = y_{2 }(t) = \frac{1}{2} * a_{22 }*t^{2} + b_{22 }*t + c_{22 }[/tex];

15. [tex]z_{2 } = z_{2 }(t) = \frac{1}{2} * a_{23 }*t^{2} + b_{23 }*t + c_{23 }[/tex];

16. [tex]x_{3 } = x_{3 }(t) = \frac{1}{2} * a_{31 }*t^{2} + b_{31 }*t + c_{31 }[/tex];

17. [tex]y_{3 } = y_{3 }(t) = \frac{1}{2} * a_{32 }*t^{2} + b_{32 }*t + c_{32 }[/tex];

18. [tex]z_{3 } = z_{3 }(t) = \frac{1}{2} * a_{33 }*t^{2} + b_{33 }*t + c_{33 }[/tex]

where [tex]a_{ij }[/tex] is the acceleration variable; [tex]b_{ij }[/tex] is the speed variable; [tex]c_{ij }[/tex] is the initial position variable.

Therefore, there are 27 (1a - 9a plus 1-18) system equations with 36 unknowns (independent variables) which helps to explain the rich diversity of possible orbits for the three-body problem... Of course, if we select the energy associated with each spatial component, then we have 27 system equations with 27 unknowns (independent variables).

And if the position functions, 10 - 18, are generally correct, then we have solved the three-body problem generally.

David Cole,

https://www.researchgate.net/profile/David_Cole29.

Go Blue! :D
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Tue Sep 10, 2019 9:50 pm

FYI:

Mass one, [tex]m_{1 }[/tex], is at point, [tex]p_{1 }[/tex] = [tex](x_{1 }, y_{1 }, z_{1 }[/tex]);

Mass two, [tex]m_{2 }[/tex], is at point, [tex]p_{2 }[/tex] = [tex](x_{2 }, y_{2 }, z_{2 }[/tex]);

Mass three, [tex]m_{3 }[/tex], is at point, [tex]p_{3 }[/tex] = [tex](x_{3 }, y_{3 }, z_{3 }[/tex]).
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Wed Sep 11, 2019 11:22 am

Some Computations:

[tex]\frac{d^{2}x_{1 }}{dt^{2}} = a_{11 }[/tex], [tex]\frac{d^{2}y_{1 }}{dt^{2}} = a_{12 }[/tex], [tex]\frac{d^{2}z_{1 }}{dt^{2}} = a_{13 }[/tex];

[tex]\frac{d^{2}x_{2 }}{dt^{2}} = a_{21 }[/tex], [tex]\frac{d^{2}y_{2 }}{dt^{2}} = a_{22 }[/tex], [tex]\frac{d^{2}z_{2 }}{dt^{2}} = a_{23 }[/tex];

[tex]\frac{d^{2}x_{3 }}{dt^{2}} = a_{31 }[/tex], [tex]\frac{d^{2}y_{3 }}{dt^{2}} = a_{32 }[/tex], [tex]\frac{d^{2}z_{3 }}{dt^{2}} = a_{33 }[/tex].

[tex]\frac{dx_{1 }}{dt} = a_{11 } * t + b_{11 }[/tex], [tex]\frac{dy_{1 }}{dt} = a_{12 } * t + b_{12 }[/tex], [tex]\frac{dz_{1 }}{dt} = a_{13 } * t + b_{13 }[/tex];

[tex]\frac{dx_{2 }}{dt} = a_{21 } * t + b_{21 }[/tex], [tex]\frac{dy_{2 }}{dt} = a_{22 } * t + b_{22 }[/tex], [tex]\frac{dz_{2 }}{dt} = a_{23 } * t + b_{23 }[/tex];

[tex]\frac{dx_{3 }}{dt} = a_{31 } * t + b_{31 }[/tex], [tex]\frac{dy_{3 }}{dt} = a_{32 } * t + b_{32 }[/tex], [tex]\frac{dz_{3 }}{dt} = a_{33 } * t + b_{33 }[/tex].
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Wed Sep 11, 2019 1:23 pm

Guest wrote:Some Computations:

[tex]\frac{d^{2}x_{1 }}{dt^{2}} = a_{11 }[/tex], [tex]\frac{d^{2}y_{1 }}{dt^{2}} = a_{12 }[/tex], [tex]\frac{d^{2}z_{1 }}{dt^{2}} = a_{13 }[/tex];

[tex]\frac{d^{2}x_{2 }}{dt^{2}} = a_{21 }[/tex], [tex]\frac{d^{2}y_{2 }}{dt^{2}} = a_{22 }[/tex], [tex]\frac{d^{2}z_{2 }}{dt^{2}} = a_{23 }[/tex];

[tex]\frac{d^{2}x_{3 }}{dt^{2}} = a_{31 }[/tex], [tex]\frac{d^{2}y_{3 }}{dt^{2}} = a_{32 }[/tex], [tex]\frac{d^{2}z_{3 }}{dt^{2}} = a_{33 }[/tex].

[tex]\frac{dx_{1 }}{dt} = a_{11 } * t + b_{11 }[/tex], [tex]\frac{dy_{1 }}{dt} = a_{12 } * t + b_{12 }[/tex], [tex]\frac{dz_{1 }}{dt} = a_{13 } * t + b_{13 }[/tex];

[tex]\frac{dx_{2 }}{dt} = a_{21 } * t + b_{21 }[/tex], [tex]\frac{dy_{2 }}{dt} = a_{22 } * t + b_{22 }[/tex], [tex]\frac{dz_{2 }}{dt} = a_{23 } * t + b_{23 }[/tex];

[tex]\frac{dx_{3 }}{dt} = a_{31 } * t + b_{31 }[/tex], [tex]\frac{dy_{3 }}{dt} = a_{32 } * t + b_{32 }[/tex], [tex]\frac{dz_{3 }}{dt} = a_{33 } * t + b_{33 }[/tex].


1. [tex]\frac{1}{2} * m_{1 }*( a_{11 } * t + b_{11 })^{2} - m_{1 } * a_{11 } * \triangle r_{1 } =E_{x_{1 }}[/tex];

2. [tex]\frac{1}{2} * m_{1 }*( a_{12 } * t + b_{12 })^{2} - m_{1 } * a_{12 } * \triangle r_{1 } =E_{y_{1 }}[/tex];

3. [tex]\frac{1}{2} * m_{1 }*( a_{13 } * t + b_{13 })^{2} - m_{1 } * a_{13 } * \triangle r_{1 } =E_{z_{1 }}[/tex];

4. [tex]\frac{1}{2} * m_{2 }*( a_{21 } * t + b_{21 })^{2} - m_{2 } * a_{21 } * \triangle r_{2 } =E_{x_{2 }}[/tex];

5. [tex]\frac{1}{2} * m_{2 }*( a_{22 } * t + b_{22 })^{2} - m_{2 } * a_{22 } * \triangle r_{2 } =E_{y_{2}}[/tex];

6. [tex]\frac{1}{2} * m_{2 }*( a_{23 } * t + b_{23 })^{2} - m_{2 } * a_{23} * \triangle r_{2 } =E_{z_{2 }}[/tex];

7. [tex]\frac{1}{2} * m_{3 }*( a_{31 } * t + b_{31 })^{2} - m_{3 } * a_{31 } * \triangle r_{3 } =E_{x_{3 }}[/tex];

8. [tex]\frac{1}{2} * m_{3 }*( a_{32 } * t + b_{32 })^{2} - m_{3 } * a_{32 } * \triangle r_{3} =E_{y_{3 }}[/tex];

9. [tex]\frac{1}{2} * m_{3 }*( a_{33 } * t + b_{33 })^{2} - m_{3 } * a_{33 } * \triangle r_{3 } =E_{z_{3 }}[/tex];

where [tex]\triangle r_{k } = \sqrt{(x_{k}(t + \triangle t )-x_{k }(t))^{2} + (y_{k}(t + \triangle t)-y_{k }(t))^{2} + (z_{k}(t + \triangle t)-z_{k }(t))^{2} }[/tex].
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Wed Sep 11, 2019 9:15 pm

FYI: E is the total energy for our three-body system, and it must remain constant while the energy associated with each spatial component may vary.

[tex]E = E_{x_{1 }} + E_{y_{1 }} + E_{z_{1 }} + E_{x_{2 }} + E_{y_{2 }} + E_{z_{2 }} + E_{x_{3 }} + E_{y_{3}} + E_{z_{1 }}[/tex].
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Wed Sep 11, 2019 9:16 pm

Guest wrote:FYI: E is the total energy for our three-body system, and it must remain constant while the energy associated with each spatial component may vary.

[tex]E = E_{x_{1 }} + E_{y_{1 }} + E_{z_{1 }} + E_{x_{2 }} + E_{y_{2 }} + E_{z_{2 }} + E_{x_{3 }} + E_{y_{3}} + E_{z_{3 }}[/tex].
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Thu Sep 12, 2019 12:28 am

FYI: E is the total energy for our three-body system, and it must remain constant while the energy associated with each spatial component may vary.

[tex]E = E_{x_{1 }} + E_{y_{1 }} + E_{z_{1 }} + E_{x_{2 }} + E_{y_{2 }} + E_{z_{2 }} + E_{x_{3 }} + E_{y_{3}} + E_{z_{3 }}[/tex].

FYI: The Important Noether's Theorem is applicable here.

"Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law...",

https://en.wikipedia.org/wiki/Noether%27s_theorem#Example_1:_Conservation_of_energy.

Therefore, we expect symmetrical orbits in our solutions to the three-body problem.
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Thu Sep 12, 2019 10:49 am

FYI: 'Weird Orbits - the three-body problem',

https://www.youtube.com/watch?v=eqSPvyaxMI8.
Guest
 

Re26: Dave's Solution to the Three-Body Problem

Postby Guest » Thu Sep 12, 2019 11:38 am

FYI: '3-Body Problem - Periodic Solutions',

https://www.youtube.com/watch?v=8_RRZcqBEAc.
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Thu Sep 12, 2019 1:57 pm

Relevant Reference Link:

'On a General Solution of the Three-Body Problem',

https://www.math10.com/forum/viewtopic.php?f=63&t=8398.
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Fri Sep 13, 2019 1:21 am

Guest wrote:
Guest wrote:Some Computations:

1. [tex]\frac{1}{2} * m_{1 }*( a_{11 } * t + b_{11 })^{2} - m_{1 } * a_{11 } * \triangle r_{1 } =E_{x_{1 }}[/tex];

2. [tex]\frac{1}{2} * m_{1 }*( a_{12 } * t + b_{12 })^{2} - m_{1 } * a_{12 } * \triangle r_{1 } =E_{y_{1 }}[/tex];

3. [tex]\frac{1}{2} * m_{1 }*( a_{13 } * t + b_{13 })^{2} - m_{1 } * a_{13 } * \triangle r_{1 } =E_{z_{1 }}[/tex];

4. [tex]\frac{1}{2} * m_{2 }*( a_{21 } * t + b_{21 })^{2} - m_{2 } * a_{21 } * \triangle r_{2 } =E_{x_{2 }}[/tex];

5. [tex]\frac{1}{2} * m_{2 }*( a_{22 } * t + b_{22 })^{2} - m_{2 } * a_{22 } * \triangle r_{2 } =E_{y_{2}}[/tex];

6. [tex]\frac{1}{2} * m_{2 }*( a_{23 } * t + b_{23 })^{2} - m_{2 } * a_{23} * \triangle r_{2 } =E_{z_{2 }}[/tex];

7. [tex]\frac{1}{2} * m_{3 }*( a_{31 } * t + b_{31 })^{2} - m_{3 } * a_{31 } * \triangle r_{3 } =E_{x_{3 }}[/tex];

8. [tex]\frac{1}{2} * m_{3 }*( a_{32 } * t + b_{32 })^{2} - m_{3 } * a_{32 } * \triangle r_{3} =E_{y_{3 }}[/tex];

9. [tex]\frac{1}{2} * m_{3 }*( a_{33 } * t + b_{33 })^{2} - m_{3 } * a_{33 } * \triangle r_{3 } =E_{z_{3 }}[/tex];

where [tex]\triangle r_{k } = \sqrt{(x_{k}(t + \triangle t )-x_{k }(t))^{2} + (y_{k}(t + \triangle t)-y_{k }(t))^{2} + (z_{k}(t + \triangle t)-z_{k }(t))^{2} }[/tex].



Hmm. We are doubtful about equations, 1 - 9, above. The value, [tex]\triangle r_{k }[/tex], seems incorrect (too big or too small or inexact) for equations, 1-9. And we have more doubts too.
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Fri Sep 13, 2019 1:56 am

An Update:

Some Computations:

1. [tex]\frac{1}{2} * m_{1 }*( a_{11 } * t + b_{11 })^{2} - m_{1 } * a_{11 } * |x_{1}( t + \triangle t) - x_{1}(t)|= E_{x_{1}}[/tex];

2. [tex]\frac{1}{2} * m_{1 }*( a_{12 } * t + b_{12 })^{2} - m_{1 } * a_{12 } * |y_{1}( t + \triangle t) -y_{1}(t)|= E_{y_{1}}[/tex];

3. [tex]\frac{1}{2} * m_{1 }*( a_{13 } * t + b_{13 })^{2} - m_{1 } * a_{13 } * |z_{1}( t + \triangle t) - z_{1}(t)|= E_{z_{1}}[/tex];

4. [tex]\frac{1}{2} * m_{2 }*( a_{21 } * t + b_{21 })^{2} - m_{2 } * a_{21 } * |x_{2}( t + \triangle t) - x_{2}(t)|= E_{x_{2}}[/tex];

5. [tex]\frac{1}{2} * m_{2 }*( a_{22 } * t + b_{22 })^{2} - m_{2 } * a_{22 } * |y_{2}( t + \triangle t) -y_{2}(t)|= E_{y_{2}}[/tex];

6. [tex]\frac{1}{2} * m_{2 }*( a_{23 } * t + b_{23 })^{2} - m_{2 } * a_{23} * |z_{2}( t + \triangle t) - z_{2}(t)|= E_{z_{2}}[/tex];

7. [tex]\frac{1}{2} * m_{3 }*( a_{31 } * t + b_{31 })^{2} - m_{3 } * a_{31 } * |x_{3}( t + \triangle t) - x_{3}(t)|= E_{x_{3}}[/tex];

8. [tex]\frac{1}{2} * m_{3 }*( a_{32 } * t + b_{32 })^{2} - m_{3 } * a_{32 } * |y_{3}( t + \triangle t) - y_{3}(t)|= E_{y_{3}}[/tex];

9. [tex]\frac{1}{2} * m_{3 }*( a_{33 } * t + b_{33 })^{2} - m_{3 } * a_{33 } * |z_{3}( t + \triangle t) - z_{3}(t)|= E_{z_{3}}[/tex].
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Fri Sep 13, 2019 9:37 am

An Update:

"Simple seeks simplest (best) solution." -- Dave.

In our closed system, the total and constant Energy (E) = total Kinetic Energy (K) + total Gravitational Energy (U) of our three-body system.

We recall our system equations with regards to acceleration (a) due to gravitational energy:

1a. [tex]\frac{d^{2}x_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{x_{1 }-x_{2 }}{r_{12}^{3}} + m_{3 }* \frac{x_{1 }-x_{3 }}{r_{13}^{3}})[/tex];

2a. [tex]\frac{d^{2}y_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{y_{1 }-y_{2 }}{r_{12}^{3}} + m_{3 }* \frac{y_{1 }-y_{3 }}{r_{13}^{3}})[/tex];

3a. [tex]\frac{d^{2}z_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{z_{1 }-z_{2 }}{r_{12}^{3}} + m_{3 }* \frac{z_{1 }-z_{3 }}{r_{13}^{3}})[/tex];

4a. [tex]\frac{d^{2}x_{2 }}{dt^{2}} = -G* (m_{1 }* \frac{x_{2 }-x_{1 }}{r_{21}^{3}} + m_{3 }* \frac{x_{2 }-x_{3 }}{r_{23}^{3}})[/tex];

5a. [tex]\frac{d^{2}y_{2 }}{dt^{2}} = -G* ( m_{1 }* \frac{y_{2 }-y_{1 }}{r_{21}^{3}} + m_{3 }* \frac{y_{2 }-y_{3 }}{r_{23}^{3}})[/tex];

6a. [tex]\frac{d^{2}z_{2 }}{dt^{2}} = -G* ( m_{1 }* \frac{z_{2 }-z_{1 }}{r_{21}^{3}} + m_{3 }* \frac{z_{2 }-z_{3 }}{r_{23}^{3}})[/tex];

7a. [tex]\frac{d^{2}x_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{x_{3 }-x_{1 }}{r_{31}^{3}} + m_{2 }* \frac{x_{3 }-x_{2 }}{r_{32}^{3}})[/tex];

8a. [tex]\frac{d^{2}y_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{y_{3 }-y_{1 }}{r_{31}^{3}} + m_{2 }* \frac{y_{3 }-y_{2 }}{r_{32}^{3}})[/tex];

9a. [tex]\frac{d^{2}z_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{z_{3 }-z_{1 }}{r_{31}^{3}} + m_{2 }* \frac{z_{3 }-z_{2 }}{r_{32}^{3}})[/tex]

with [tex]r_{ij} = \sqrt{(x_{i }-x_{j })^{2} + (y_{i }-y_{j })^{2} + (z_{i }-z_{j })^{2} }[/tex] where G is the gravitational constant.

Now, we generate our final system equations corresponding to the energy conservation law for our closed system:

1. [tex]\frac{1}{2} * m_{1 }*(\frac{dx_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{x_{1 }-x_{2 }}{r_{12}^{3}} + m_{3 }* \frac{x_{1 }-x_{3 }}{r_{13}^{3}}) * |x_{1}( t + \triangle t) - x_{1}(t)| =E_{x_{1 }}[/tex];

2. [tex]\frac{1}{2} * m_{1 }*(\frac{dy_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{y_{1 }-y_{2 }}{r_{12}^{3}} + m_{3 }* \frac{y_{1 }-y_{3 }}{r_{13}^{3}})* |y_{1}( t + \triangle t) - y_{1}(t)|= E_{y_{1}}[/tex];

3. [tex]\frac{1}{2} * m_{1 }*(\frac{dz_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{z_{1 }-z_{2 }}{r_{12}^{3}} + m_{3 }* \frac{z_{1 }-z_{3 }}{r_{13}^{3}} * |z_{1}( t + \triangle t) - z_{1}(t)|= E_{z_{1}}[/tex];

4. [tex]\frac{1}{2} * m_{2 }*(\frac{dx_{2 }}{dt})^{2} - G* m_{2 }*( m_{1 }* \frac{x_{2 }-x_{1 }}{r_{21}^{3}} + m_{3 }* \frac{x_{2 }-x_{3 }}{r_{23}^{3}}) * |x_{2}( t + \triangle t) - x_{2}(t)|= E_{x_{2}}[/tex];

5. [tex]\frac{1}{2} * m_{2}*(\frac{dy_{2 }}{dt})^{2} - G* m_{2 }*( m_{2 }* \frac{y_{2 }-y_{1 }}{r_{21}^{3}} + m_{3 }* \frac{y_{2 }-y_{3 }}{r_{23}^{3}}) * |y_{2}( t + \triangle t) - y_{2}(t)|= E_{y_{2}}[/tex];

6. [tex]\frac{1}{2} * m_{2 }*(\frac{dz_{2 }}{dt})^{2} - G* m_{2 }*( m_{1 }* \frac{z_{2 }-z_{1 }}{r_{21}^{3}} + m_{3 }* \frac{z_{2 }-z_{3 }}{r_{23}^{3}}) * |z_{2}( t + \triangle t) - z_{2}(t)|= E_{z_{2}}[/tex];

7. [tex]\frac{1}{2} * m_{3 }*(\frac{dx_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{x_{3 }-x_{1 }}{r_{31}^{3}} + m_{2 }* \frac{x_{3 }-x_{2 }}{r_{32}^{3}}) * |x_{3}( t + \triangle t) - x_{3}(t)|= E_{x_{3}}[/tex];

8. [tex]\frac{1}{2} * m_{3 }*(\frac{dy_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{y_{3 }-y_{1 }}{r_{31}^{3}} + m_{2 }* \frac{y_{3 }-y_{2 }}{r_{32}^{3}}) * y_{3}( t + \triangle t) - y_{3}(t)|= E_{y_{3}}[/tex];

9. [tex]\frac{1}{2} * m_{3 }*(\frac{dz_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{z_{3 }-z_{1 }}{r_{31}^{3}} + m_{2 }* \frac{z_{3 }-z_{2 }}{r_{32}^{3}}) * |z_{3}( t + \triangle t) - z_{3}(t)|= E_{z_{3}}[/tex].

Next, we list the important position functions corresponding to our spatial components:

10. [tex]x_{1 } = x_{1 }(t) = \frac{1}{2} * a_{11 }*t^{2} + b_{11 }*t + c_{11 }[/tex];

11. [tex]y_{1 } = y_{1 }(t) = \frac{1}{2} * a_{12 }*t^{2} + b_{12 }*t + c_{12 }[/tex];

12. [tex]z_{1 } = z_{1 }(t) = \frac{1}{2} * a_{13 }*t^{2} + b_{13 }*t + c_{13 }[/tex];

13. [tex]x_{2} = x_{2 }(t) = \frac{1}{2} * a_{21 }*t^{2} + b_{21 }*t + c_{21 }[/tex];

14. [tex]y_{2 } = y_{2 }(t) = \frac{1}{2} * a_{22 }*t^{2} + b_{22 }*t + c_{22 }[/tex];

15. [tex]z_{2 } = z_{2 }(t) = \frac{1}{2} * a_{23 }*t^{2} + b_{23 }*t + c_{23 }[/tex];

16. [tex]x_{3 } = x_{3 }(t) = \frac{1}{2} * a_{31 }*t^{2} + b_{31 }*t + c_{31 }[/tex];

17. [tex]y_{3 } = y_{3 }(t) = \frac{1}{2} * a_{32 }*t^{2} + b_{32 }*t + c_{32 }[/tex];

18. [tex]z_{3 } = z_{3 }(t) = \frac{1}{2} * a_{33 }*t^{2} + b_{33 }*t + c_{33 }[/tex]

where [tex]a_{ij }[/tex] is the acceleration variable; [tex]b_{ij }[/tex] is the speed variable; [tex]c_{ij }[/tex] is the initial position variable (coordinate).

Therefore, there are 27 (1a - 9a plus 1-18) system equations with 36 unknowns (independent variables) which helps to explain the rich diversity of possible orbits for the three-body problem... Of course, if we select the energy associated with each spatial component, then we have 27 system equations with 27 unknowns (independent variables). Yes! The time variable, t, is also independent. However, we do not count it as unknown...

And if the position functions, 10 - 18, are generally correct, then we have solved the three-body problem generally.

David Cole,

https://www.researchgate.net/profile/David_Cole29.

Go Blue! :D

P.S. We apologize for any errors posted here.
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Fri Sep 13, 2019 11:18 am

Guest wrote:An Update:

"Simple seeks simplest (best) solution." -- Dave.

In our closed system, the total and constant Energy (E) = total Kinetic Energy (K) + total Gravitational Energy (U) of our three-body system.

We recall our system equations with regards to acceleration (a) due to gravitational energy:

1a. [tex]\frac{d^{2}x_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{x_{1 }-x_{2 }}{r_{12}^{3}} + m_{3 }* \frac{x_{1 }-x_{3 }}{r_{13}^{3}})[/tex];

2a. [tex]\frac{d^{2}y_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{y_{1 }-y_{2 }}{r_{12}^{3}} + m_{3 }* \frac{y_{1 }-y_{3 }}{r_{13}^{3}})[/tex];

3a. [tex]\frac{d^{2}z_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{z_{1 }-z_{2 }}{r_{12}^{3}} + m_{3 }* \frac{z_{1 }-z_{3 }}{r_{13}^{3}})[/tex];

4a. [tex]\frac{d^{2}x_{2 }}{dt^{2}} = -G* (m_{1 }* \frac{x_{2 }-x_{1 }}{r_{21}^{3}} + m_{3 }* \frac{x_{2 }-x_{3 }}{r_{23}^{3}})[/tex];

5a. [tex]\frac{d^{2}y_{2 }}{dt^{2}} = -G* ( m_{1 }* \frac{y_{2 }-y_{1 }}{r_{21}^{3}} + m_{3 }* \frac{y_{2 }-y_{3 }}{r_{23}^{3}})[/tex];

6a. [tex]\frac{d^{2}z_{2 }}{dt^{2}} = -G* ( m_{1 }* \frac{z_{2 }-z_{1 }}{r_{21}^{3}} + m_{3 }* \frac{z_{2 }-z_{3 }}{r_{23}^{3}})[/tex];

7a. [tex]\frac{d^{2}x_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{x_{3 }-x_{1 }}{r_{31}^{3}} + m_{2 }* \frac{x_{3 }-x_{2 }}{r_{32}^{3}})[/tex];

8a. [tex]\frac{d^{2}y_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{y_{3 }-y_{1 }}{r_{31}^{3}} + m_{2 }* \frac{y_{3 }-y_{2 }}{r_{32}^{3}})[/tex];

9a. [tex]\frac{d^{2}z_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{z_{3 }-z_{1 }}{r_{31}^{3}} + m_{2 }* \frac{z_{3 }-z_{2 }}{r_{32}^{3}})[/tex]

with [tex]r_{ij} = \sqrt{(x_{i }-x_{j })^{2} + (y_{i }-y_{j })^{2} + (z_{i }-z_{j })^{2} }[/tex] where G is the gravitational constant.

Now, we generate our final system equations corresponding to the energy conservation law for our closed system:

1. [tex]\frac{1}{2} * m_{1 }*(\frac{dx_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{x_{1 }-x_{2 }}{r_{12}^{3}} + m_{3 }* \frac{x_{1 }-x_{3 }}{r_{13}^{3}}) * |x_{1}( t + \triangle t) - x_{1}(t)| =E_{x_{1 }}[/tex];

2. [tex]\frac{1}{2} * m_{1 }*(\frac{dy_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{y_{1 }-y_{2 }}{r_{12}^{3}} + m_{3 }* \frac{y_{1 }-y_{3 }}{r_{13}^{3}}) * |y_{1}( t + \triangle t) - y_{1}(t)|= E_{y_{1}}[/tex];

3. [tex]\frac{1}{2} * m_{1 }*(\frac{dz_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{z_{1 }-z_{2 }}{r_{12}^{3}} + m_{3 }* \frac{z_{1 }-z_{3 }}{r_{13}^{3}}) * |z_{1}( t + \triangle t) - z_{1}(t)|= E_{z_{1}}[/tex];

4. [tex]\frac{1}{2} * m_{2 }*(\frac{dx_{2 }}{dt})^{2} - G* m_{2 }*( m_{1 }* \frac{x_{2 }-x_{1 }}{r_{21}^{3}} + m_{3 }* \frac{x_{2 }-x_{3 }}{r_{23}^{3}}) * |x_{2}( t + \triangle t) - x_{2}(t)|= E_{x_{2}}[/tex];

5. [tex]\frac{1}{2} * m_{2}*(\frac{dy_{2 }}{dt})^{2} - G* m_{2 }*( m_{2 }* \frac{y_{2 }-y_{1 }}{r_{21}^{3}} + m_{3 }* \frac{y_{2 }-y_{3 }}{r_{23}^{3}}) * |y_{2}( t + \triangle t) - y_{2}(t)|= E_{y_{2}}[/tex];

6. [tex]\frac{1}{2} * m_{2 }*(\frac{dz_{2 }}{dt})^{2} - G* m_{2 }*( m_{1 }* \frac{z_{2 }-z_{1 }}{r_{21}^{3}} + m_{3 }* \frac{z_{2 }-z_{3 }}{r_{23}^{3}}) * |z_{2}( t + \triangle t) - z_{2}(t)|= E_{z_{2}}[/tex];

7. [tex]\frac{1}{2} * m_{3 }*(\frac{dx_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{x_{3 }-x_{1 }}{r_{31}^{3}} + m_{2 }* \frac{x_{3 }-x_{2 }}{r_{32}^{3}}) * |x_{3}( t + \triangle t) - x_{3}(t)|= E_{x_{3}}[/tex];

8. [tex]\frac{1}{2} * m_{3 }*(\frac{dy_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{y_{3 }-y_{1 }}{r_{31}^{3}} + m_{2 }* \frac{y_{3 }-y_{2 }}{r_{32}^{3}}) * y_{3}( t + \triangle t) - y_{3}(t)|= E_{y_{3}}[/tex];

9. [tex]\frac{1}{2} * m_{3 }*(\frac{dz_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{z_{3 }-z_{1 }}{r_{31}^{3}} + m_{2 }* \frac{z_{3 }-z_{2 }}{r_{32}^{3}}) * |z_{3}( t + \triangle t) - z_{3}(t)|= E_{z_{3}}[/tex].

Next, we list the important position functions corresponding to our spatial components:

10. [tex]x_{1 } = x_{1 }(t) = \frac{1}{2} * a_{11 }*t^{2} + b_{11 }*t + c_{11 }[/tex];

11. [tex]y_{1 } = y_{1 }(t) = \frac{1}{2} * a_{12 }*t^{2} + b_{12 }*t + c_{12 }[/tex];

12. [tex]z_{1 } = z_{1 }(t) = \frac{1}{2} * a_{13 }*t^{2} + b_{13 }*t + c_{13 }[/tex];

13. [tex]x_{2} = x_{2 }(t) = \frac{1}{2} * a_{21 }*t^{2} + b_{21 }*t + c_{21 }[/tex];

14. [tex]y_{2 } = y_{2 }(t) = \frac{1}{2} * a_{22 }*t^{2} + b_{22 }*t + c_{22 }[/tex];

15. [tex]z_{2 } = z_{2 }(t) = \frac{1}{2} * a_{23 }*t^{2} + b_{23 }*t + c_{23 }[/tex];

16. [tex]x_{3 } = x_{3 }(t) = \frac{1}{2} * a_{31 }*t^{2} + b_{31 }*t + c_{31 }[/tex];

17. [tex]y_{3 } = y_{3 }(t) = \frac{1}{2} * a_{32 }*t^{2} + b_{32 }*t + c_{32 }[/tex];

18. [tex]z_{3 } = z_{3 }(t) = \frac{1}{2} * a_{33 }*t^{2} + b_{33 }*t + c_{33 }[/tex]

where [tex]a_{ij }[/tex] is the acceleration variable; [tex]b_{ij }[/tex] is the speed variable; [tex]c_{ij }[/tex] is the initial position variable (coordinate).

Therefore, there are 27 (1a - 9a plus 1-18) system equations with 36 unknowns (independent variables) which helps to explain the rich diversity of possible orbits for the three-body problem... Of course, if we select the energy associated with each spatial component, then we have 27 system equations with 27 unknowns (independent variables). Yes! The time variable, t, is also independent. However, we do not count it as unknown...

And if the position functions, 10 - 18, are generally correct, then we have solved the three-body problem generally.

David Cole,

https://www.researchgate.net/profile/David_Cole29.

Go Blue! :D

P.S. We apologize for any errors posted here.
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Fri Sep 13, 2019 11:25 am

An Update:

"Simple seeks simplest (best) solution." -- Dave.

In our closed system, the total and constant Energy (E) = total Kinetic Energy (K) + total Gravitational Energy (U) of our three-body system.

We recall our system equations with regards to acceleration (a) due to gravitational energy:

1a. [tex]\frac{d^{2}x_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{x_{1 }-x_{2 }}{r_{12}^{3}} + m_{3 }* \frac{x_{1 }-x_{3 }}{r_{13}^{3}})[/tex];

2a. [tex]\frac{d^{2}y_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{y_{1 }-y_{2 }}{r_{12}^{3}} + m_{3 }* \frac{y_{1 }-y_{3 }}{r_{13}^{3}})[/tex];

3a. [tex]\frac{d^{2}z_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{z_{1 }-z_{2 }}{r_{12}^{3}} + m_{3 }* \frac{z_{1 }-z_{3 }}{r_{13}^{3}})[/tex];

4a. [tex]\frac{d^{2}x_{2 }}{dt^{2}} = -G* (m_{1 }* \frac{x_{2 }-x_{1 }}{r_{21}^{3}} + m_{3 }* \frac{x_{2 }-x_{3 }}{r_{23}^{3}})[/tex];

5a. [tex]\frac{d^{2}y_{2 }}{dt^{2}} = -G* ( m_{1 }* \frac{y_{2 }-y_{1 }}{r_{21}^{3}} + m_{3 }* \frac{y_{2 }-y_{3 }}{r_{23}^{3}})[/tex];

6a. [tex]\frac{d^{2}z_{2 }}{dt^{2}} = -G* ( m_{1 }* \frac{z_{2 }-z_{1 }}{r_{21}^{3}} + m_{3 }* \frac{z_{2 }-z_{3 }}{r_{23}^{3}})[/tex];

7a. [tex]\frac{d^{2}x_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{x_{3 }-x_{1 }}{r_{31}^{3}} + m_{2 }* \frac{x_{3 }-x_{2 }}{r_{32}^{3}})[/tex];

8a. [tex]\frac{d^{2}y_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{y_{3 }-y_{1 }}{r_{31}^{3}} + m_{2 }* \frac{y_{3 }-y_{2 }}{r_{32}^{3}})[/tex];

9a. [tex]\frac{d^{2}z_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{z_{3 }-z_{1 }}{r_{31}^{3}} + m_{2 }* \frac{z_{3 }-z_{2 }}{r_{32}^{3}})[/tex]

with [tex]r_{ij} = \sqrt{(x_{i }-x_{j })^{2} + (y_{i }-y_{j })^{2} + (z_{i }-z_{j })^{2} }[/tex] where G is the gravitational constant.

Now, we generate our final system equations corresponding to the energy conservation law for our closed system:

1. [tex]\frac{1}{2} * m_{1 }*(\frac{dx_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{x_{1 }-x_{2 }}{r_{12}^{3}} + m_{3 }* \frac{x_{1 }-x_{3 }}{r_{13}^{3}}) * |x_{1}( t + \triangle t) - x_{1}(t)| =E_{x_{1 }}[/tex];

2. [tex]\frac{1}{2} * m_{1 }*(\frac{dy_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{y_{1 }-y_{2 }}{r_{12}^{3}} + m_{3 }* \frac{y_{1 }-y_{3 }}{r_{13}^{3}}) * |y_{1}( t + \triangle t) - y_{1}(t)|= E_{y_{1}}[/tex];

3. [tex]\frac{1}{2} * m_{1 }*(\frac{dz_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{z_{1 }-z_{2 }}{r_{12}^{3}} + m_{3 }* \frac{z_{1 }-z_{3 }}{r_{13}^{3}}) * |z_{1}( t + \triangle t) - z_{1}(t)|= E_{z_{1}}[/tex];

4. [tex]\frac{1}{2} * m_{2 }*(\frac{dx_{2 }}{dt})^{2} - G* m_{2 }*( m_{1 }* \frac{x_{2 }-x_{1 }}{r_{21}^{3}} + m_{3 }* \frac{x_{2 }-x_{3 }}{r_{23}^{3}}) * |x_{2}( t + \triangle t) - x_{2}(t)|= E_{x_{2}}[/tex];

5. [tex]\frac{1}{2} * m_{2}*(\frac{dy_{2 }}{dt})^{2} - G* m_{2 }*( m_{2 }* \frac{y_{2 }-y_{1 }}{r_{21}^{3}} + m_{3 }* \frac{y_{2 }-y_{3 }}{r_{23}^{3}}) * |y_{2}( t + \triangle t) - y_{2}(t)|= E_{y_{2}}[/tex];

6. [tex]\frac{1}{2} * m_{2 }*(\frac{dz_{2 }}{dt})^{2} - G* m_{2 }*( m_{1 }* \frac{z_{2 }-z_{1 }}{r_{21}^{3}} + m_{3 }* \frac{z_{2 }-z_{3 }}{r_{23}^{3}}) * |z_{2}( t + \triangle t) - z_{2}(t)|= E_{z_{2}}[/tex];

7. [tex]\frac{1}{2} * m_{3 }*(\frac{dx_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{x_{3 }-x_{1 }}{r_{31}^{3}} + m_{2 }* \frac{x_{3 }-x_{2 }}{r_{32}^{3}}) * |x_{3}( t + \triangle t) - x_{3}(t)|= E_{x_{3}}[/tex];

8. [tex]\frac{1}{2} * m_{3 }*(\frac{dy_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{y_{3 }-y_{1 }}{r_{31}^{3}} + m_{2 }* \frac{y_{3 }-y_{2 }}{r_{32}^{3}}) * |y_{3}( t + \triangle t) - y_{3}(t)|= E_{y_{3}}[/tex];

9. [tex]\frac{1}{2} * m_{3 }*(\frac{dz_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{z_{3 }-z_{1 }}{r_{31}^{3}} + m_{2 }* \frac{z_{3 }-z_{2 }}{r_{32}^{3}}) * |z_{3}( t + \triangle t) - z_{3}(t)|= E_{z_{3}}[/tex].

Next, we list the important position functions corresponding to our spatial components:

10. [tex]x_{1 } = x_{1 }(t) = \frac{1}{2} * a_{11 }*t^{2} + b_{11 }*t + c_{11 }[/tex];

11. [tex]y_{1 } = y_{1 }(t) = \frac{1}{2} * a_{12 }*t^{2} + b_{12 }*t + c_{12 }[/tex];

12. [tex]z_{1 } = z_{1 }(t) = \frac{1}{2} * a_{13 }*t^{2} + b_{13 }*t + c_{13 }[/tex];

13. [tex]x_{2} = x_{2 }(t) = \frac{1}{2} * a_{21 }*t^{2} + b_{21 }*t + c_{21 }[/tex];

14. [tex]y_{2 } = y_{2 }(t) = \frac{1}{2} * a_{22 }*t^{2} + b_{22 }*t + c_{22 }[/tex];

15. [tex]z_{2 } = z_{2 }(t) = \frac{1}{2} * a_{23 }*t^{2} + b_{23 }*t + c_{23 }[/tex];

16. [tex]x_{3 } = x_{3 }(t) = \frac{1}{2} * a_{31 }*t^{2} + b_{31 }*t + c_{31 }[/tex];

17. [tex]y_{3 } = y_{3 }(t) = \frac{1}{2} * a_{32 }*t^{2} + b_{32 }*t + c_{32 }[/tex];

18. [tex]z_{3 } = z_{3 }(t) = \frac{1}{2} * a_{33 }*t^{2} + b_{33 }*t + c_{33 }[/tex]

where [tex]a_{ij }[/tex] is the acceleration variable; [tex]b_{ij }[/tex] is the speed variable; [tex]c_{ij }[/tex] is the initial position variable (coordinate).

Therefore, there are 27 (1a - 9a plus 1-18) system equations with 36 unknowns (independent variables) which helps to explain the rich diversity of possible orbits for the three-body problem... Of course, if we select the energy associated with each spatial component, then we have 27 system equations with 27 unknowns (independent variables). Yes! The time variable, t, is also independent. However, we do not count it as unknown...

And if the position functions, 10 - 18, are generally correct, then we have solved the three-body problem generally.

David Cole,

https://www.researchgate.net/profile/David_Cole29.

Go Blue! :D

P.S. We apologize for any errors posted here, and for the 'never-ending' updates too.
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Thu Sep 19, 2019 8:36 pm

Guest wrote:FYI: E is the total energy for our three-body system, and it must remain constant while the energy associated with each spatial component may vary.

[tex]E = E_{x_{1 }} + E_{y_{1 }} + E_{z_{1 }} + E_{x_{2 }} + E_{y_{2 }} + E_{z_{2 }} + E_{x_{3 }} + E_{y_{3}} + E_{z_{1 }}[/tex].


FYI: 'Modelling the Three-Body Problem in Classical Mechanics using Python:
An overview of the fundamentals of gravitation, the odeint solver in Scipy and 3D plotting in Matplotlib',

https://towardsdatascience.com/modelling-the-three-body-problem-in-classical-mechanics-using-python-9dc270ad7767.

We believe theory and practice (experiments/applications) must complement/confirm each other... We strongly encourage the reader to verify our general system equations for the three-body problem or the n-body problem via calculations/simulations/experiments... Good luck! :)
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Fri Sep 20, 2019 9:44 pm

Guest wrote:
Guest wrote:FYI: E is the total energy for our three-body system, and it must remain constant while the energy associated with each spatial component may vary:

[tex]E_{x_{1 }} + E_{y_{1 }} + E_{z_{1 }} + E_{x_{2 }} + E_{y_{2 }} + E_{z_{2 }} + E_{x_{3 }} + E_{y_{3}} + E_{z_{3 }} = E[/tex].


FYI: 'Modelling the Three-Body Problem in Classical Mechanics using Python:
An overview of the fundamentals of gravitation, the odeint solver in Scipy and 3D plotting in Matplotlib',

https://towardsdatascience.com/modelling-the-three-body-problem-in-classical-mechanics-using-python-9dc270ad7767.

We believe theory and practice (experiments/applications) must complement/confirm each other... We strongly encourage the reader to verify our general system equations for the three-body problem or the n-body problem via calculations/simulations/experiments... Good luck! :)
Guest
 

Re: Dave's Solution to the Three-Body Problem

Postby Guest » Tue Nov 05, 2019 2:41 pm

Guest wrote:An Update:

"Simple seeks simplest (best) solution." -- Dave.

In our closed system, the total and constant Energy (E) = total Kinetic Energy (K) + total Gravitational Energy (U) of our three-body system.

We recall our system equations with regards to acceleration (a) due to gravitational energy:

1a. [tex]\frac{d^{2}x_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{x_{1 }-x_{2 }}{r_{12}^{3}} + m_{3 }* \frac{x_{1 }-x_{3 }}{r_{13}^{3}})[/tex];

2a. [tex]\frac{d^{2}y_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{y_{1 }-y_{2 }}{r_{12}^{3}} + m_{3 }* \frac{y_{1 }-y_{3 }}{r_{13}^{3}})[/tex];

3a. [tex]\frac{d^{2}z_{1 }}{dt^{2}} = -G* ( m_{2 }* \frac{z_{1 }-z_{2 }}{r_{12}^{3}} + m_{3 }* \frac{z_{1 }-z_{3 }}{r_{13}^{3}})[/tex];

4a. [tex]\frac{d^{2}x_{2 }}{dt^{2}} = -G* (m_{1 }* \frac{x_{2 }-x_{1 }}{r_{21}^{3}} + m_{3 }* \frac{x_{2 }-x_{3 }}{r_{23}^{3}})[/tex];

5a. [tex]\frac{d^{2}y_{2 }}{dt^{2}} = -G* ( m_{1 }* \frac{y_{2 }-y_{1 }}{r_{21}^{3}} + m_{3 }* \frac{y_{2 }-y_{3 }}{r_{23}^{3}})[/tex];

6a. [tex]\frac{d^{2}z_{2 }}{dt^{2}} = -G* ( m_{1 }* \frac{z_{2 }-z_{1 }}{r_{21}^{3}} + m_{3 }* \frac{z_{2 }-z_{3 }}{r_{23}^{3}})[/tex];

7a. [tex]\frac{d^{2}x_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{x_{3 }-x_{1 }}{r_{31}^{3}} + m_{2 }* \frac{x_{3 }-x_{2 }}{r_{32}^{3}})[/tex];

8a. [tex]\frac{d^{2}y_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{y_{3 }-y_{1 }}{r_{31}^{3}} + m_{2 }* \frac{y_{3 }-y_{2 }}{r_{32}^{3}})[/tex];

9a. [tex]\frac{d^{2}z_{3 }}{dt^{2}} = -G* ( m_{1 }* \frac{z_{3 }-z_{1 }}{r_{31}^{3}} + m_{2 }* \frac{z_{3 }-z_{2 }}{r_{32}^{3}})[/tex]

with [tex]r_{ij} = \sqrt{(x_{i }-x_{j })^{2} + (y_{i }-y_{j })^{2} + (z_{i }-z_{j })^{2} }[/tex] where G is the gravitational constant.

Now, we generate our final system equations corresponding to the energy conservation law for our closed system:

1. [tex]\frac{1}{2} * m_{1 }*(\frac{dx_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{x_{1 }-x_{2 }}{r_{12}^{3}} + m_{3 }* \frac{x_{1 }-x_{3 }}{r_{13}^{3}}) * |x_{1}( t + \triangle t) - x_{1}(t)| =E_{x_{1 }}[/tex];

2. [tex]\frac{1}{2} * m_{1 }*(\frac{dy_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{y_{1 }-y_{2 }}{r_{12}^{3}} + m_{3 }* \frac{y_{1 }-y_{3 }}{r_{13}^{3}}) * |y_{1}( t + \triangle t) - y_{1}(t)|= E_{y_{1}}[/tex];

3. [tex]\frac{1}{2} * m_{1 }*(\frac{dz_{1 }}{dt})^{2} - G* m_{1 }*( m_{2 }* \frac{z_{1 }-z_{2 }}{r_{12}^{3}} + m_{3 }* \frac{z_{1 }-z_{3 }}{r_{13}^{3}}) * |z_{1}( t + \triangle t) - z_{1}(t)|= E_{z_{1}}[/tex];

4. [tex]\frac{1}{2} * m_{2 }*(\frac{dx_{2 }}{dt})^{2} - G* m_{2 }*( m_{1 }* \frac{x_{2 }-x_{1 }}{r_{21}^{3}} + m_{3 }* \frac{x_{2 }-x_{3 }}{r_{23}^{3}}) * |x_{2}( t + \triangle t) - x_{2}(t)|= E_{x_{2}}[/tex];

5. [tex]\frac{1}{2} * m_{2}*(\frac{dy_{2 }}{dt})^{2} - G* m_{2 }*( m_{2 }* \frac{y_{2 }-y_{1 }}{r_{21}^{3}} + m_{3 }* \frac{y_{2 }-y_{3 }}{r_{23}^{3}}) * |y_{2}( t + \triangle t) - y_{2}(t)|= E_{y_{2}}[/tex];

6. [tex]\frac{1}{2} * m_{2 }*(\frac{dz_{2 }}{dt})^{2} - G* m_{2 }*( m_{1 }* \frac{z_{2 }-z_{1 }}{r_{21}^{3}} + m_{3 }* \frac{z_{2 }-z_{3 }}{r_{23}^{3}}) * |z_{2}( t + \triangle t) - z_{2}(t)|= E_{z_{2}}[/tex];

7. [tex]\frac{1}{2} * m_{3 }*(\frac{dx_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{x_{3 }-x_{1 }}{r_{31}^{3}} + m_{2 }* \frac{x_{3 }-x_{2 }}{r_{32}^{3}}) * |x_{3}( t + \triangle t) - x_{3}(t)|= E_{x_{3}}[/tex];

8. [tex]\frac{1}{2} * m_{3 }*(\frac{dy_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{y_{3 }-y_{1 }}{r_{31}^{3}} + m_{2 }* \frac{y_{3 }-y_{2 }}{r_{32}^{3}}) * |y_{3}( t + \triangle t) - y_{3}(t)|= E_{y_{3}}[/tex];

9. [tex]\frac{1}{2} * m_{3 }*(\frac{dz_{3 }}{dt})^{2} - G* m_{3 }*( m_{1 }* \frac{z_{3 }-z_{1 }}{r_{31}^{3}} + m_{2 }* \frac{z_{3 }-z_{2 }}{r_{32}^{3}}) * |z_{3}( t + \triangle t) - z_{3}(t)|= E_{z_{3}}[/tex].

Next, we list the important position functions corresponding to our spatial components:

10. [tex]x_{1 } = x_{1 }(t) = \frac{1}{2} * a_{11 }*t^{2} + b_{11 }*t + c_{11 }[/tex];

11. [tex]y_{1 } = y_{1 }(t) = \frac{1}{2} * a_{12 }*t^{2} + b_{12 }*t + c_{12 }[/tex];

12. [tex]z_{1 } = z_{1 }(t) = \frac{1}{2} * a_{13 }*t^{2} + b_{13 }*t + c_{13 }[/tex];

13. [tex]x_{2} = x_{2 }(t) = \frac{1}{2} * a_{21 }*t^{2} + b_{21 }*t + c_{21 }[/tex];

14. [tex]y_{2 } = y_{2 }(t) = \frac{1}{2} * a_{22 }*t^{2} + b_{22 }*t + c_{22 }[/tex];

15. [tex]z_{2 } = z_{2 }(t) = \frac{1}{2} * a_{23 }*t^{2} + b_{23 }*t + c_{23 }[/tex];

16. [tex]x_{3 } = x_{3 }(t) = \frac{1}{2} * a_{31 }*t^{2} + b_{31 }*t + c_{31 }[/tex];

17. [tex]y_{3 } = y_{3 }(t) = \frac{1}{2} * a_{32 }*t^{2} + b_{32 }*t + c_{32 }[/tex];

18. [tex]z_{3 } = z_{3 }(t) = \frac{1}{2} * a_{33 }*t^{2} + b_{33 }*t + c_{33 }[/tex]

where [tex]a_{ij }[/tex] is the acceleration variable; [tex]b_{ij }[/tex] is the speed variable; [tex]c_{ij }[/tex] is the initial position variable (coordinate).

Therefore, there are 27 (1a - 9a plus 1-18) system equations with 36 unknowns (independent variables) which helps to explain the rich diversity of possible orbits for the three-body problem... Of course, if we select the energy associated with each spatial component, then we have 27 system equations with 27 unknowns (independent variables). Yes! The time variable, t, is also independent. However, we do not count it as unknown...

And if the position functions, 10 - 18, are generally correct, then we have solved the three-body problem generally.

David Cole,

https://www.researchgate.net/profile/David_Cole29.

Go Blue! :D

P.S. We apologize for any errors posted here, and for the 'never-ending' updates too.


Hmm. For equations 10 - 18, we strongly suggest [tex]a_{ij } = a_{ij }(t)[/tex] and [tex]b_{ij } = b_{ij }(t)[/tex]. Right?
Guest
 

Re: Dave's General Solution to the Three-Body Problem

Postby Guest » Thu Nov 07, 2019 6:20 pm

Relevant Reference Links:

'Three-Body Problem',

https://wiki.tfes.org/Three_Body_Problem;

'Newton vs the machine: solving the chaotic three-body problem using deep neural networks',

https://arxiv.org/abs/1910.07291.
Attachments
nBody.jpg
nBody.jpg (17.76 KiB) Viewed 14 times
Guest
 


Return to Number Theory



Who is online

Users browsing this forum: No registered users and 2 guests