On a General Solution of the Three-Body Problem

Re: On a General Solution of the Three-Body Problem

Postby Guest » Fri Sep 13, 2019 11:28 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.
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Re: On a General Solution of the Three-Body Problem

Postby Guest » Thu Sep 19, 2019 8:34 pm

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! :)
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