On a General Solution of the Three-Body Problem

[Guest]

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

Relevant Reference Link:

'The Three-Body Problem', by Prof. Z. Musielak and Prof. B. Quarles,

https://www.researchgate.net/publication/262975895_The_three-body_problem?_sg=oJWFWnULr3KXRhEAsS3dah7CIwi5LRtW9VO6mGDBDK0iUObqfb3nADJMIZYy6NK7b9rQt4t8SMRqd9Y.

[Guest]

FYI: 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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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_{1 }}[/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];

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

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

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

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

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

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

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

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

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

Therefore, there are 27 (1a - 9a plus 1-18) system equations with 36 unknowns (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 (variables).

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

Dave Cole,

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

[Guest]

FYI: 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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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];

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

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

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

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

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

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

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

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

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

Therefore, there are 27 (1a - 9a plus 1-18) system equations with 36 unknowns (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 (variables).

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

Dave Cole,

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

[Guest]

Remarks: [tex]a_{ij }[/tex] is an acceleration variable; [tex]b_{ij }[/tex] is a speed variable; and [tex]c_{ij }[/tex] is an initial position variable.

Whatever, energy (e.g. [tex]E_{x_{1 }}[/tex]) selected for each spatial component (e.g. [tex]x_{1 }[/tex]) must be consistent systemwise. And our system equations are generally difficult to solve...

[Guest]

FYI: 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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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].5m_{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];

10. [tex]x_{1 } = x_{1 }(t) = .5a_{11 }*t^{2} + b_{11 }*t + c_{11 }[/tex];

11. [tex]y_{1 } = y_{1 }(t) = .5a_{12 }*t^{2} + b_{12 }*t + c_{12 }[/tex];

12. [tex]z_{1 } = z_{1 }(t) = .5a_{13 }*t^{2} + b_{13 }*t + c_{13 }[/tex];

13. [tex]x_{2} = x_{2 }(t) = .5a_{21 }*t^{2} + b_{21 }*t + c_{21 }[/tex];

14. [tex]y_{2 } = y_{2 }(t) = .5a_{22 }*t^{2} + b_{22 }*t + c_{22 }[/tex];

15. [tex]z_{2 } = z_{2 }(t) = .5a_{23 }*t^{2} + b_{23 }*t + c_{23 }[/tex];

16. [tex]x_{3 } = x_{3 }(t) = .5a_{31 }*t^{2} + b_{31 }*t + c_{31 }[/tex];

17. [tex]y_{3 } = y_{3 }(t) = .5a_{32 }*t^{2} + b_{32 }*t + c_{32 }[/tex];

18. [tex]z_{3 } = z_{3 }(t) = .5a_{33 }*t^{2} + b_{33 }*t + c_{33 }[/tex].

Therefore, there are 27 (1a - 9a plus 1-18) system equations with 36 unknowns (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 (variables).

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

Dave Cole,

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

[Guest]

Remarks: For the n-body problem where [tex]n \ge 3[/tex], we expect 9n equations with 12n unknowns (variables)... Right?

The investigation of the n-body problem is very important for learning physics at levels, for research, and for many applications (e.g. systems-modeling and prediction) in applied and theoretical physics as well as for advancing pure and applied mathematics... Please refer to relevant reference links in the previous posts for details.

[Guest]

Remarks: For the n-body problem where [tex]n \ge 3[/tex], we expect 9n equations with 12n unknowns (variables)... Right?

The investigation of the n-body problem is very important for learning physics at different levels, for research, and for many applications (e.g. systems-modeling and prediction) in applied and theoretical physics as well as for advancing pure and applied mathematics... Please refer to relevant reference links in the previous posts for details.

[Guest]

Remark: The Imperial Rule and the Numbers Game:

Orbits may be stable or become unstable, and energy associated with each spatial component may change, but the total energy of our closed system must remain constant (imperial rule)! Right?

[Guest]

On the Solutions of Stable Orbits for the Three-Body Problem (TBP) and Lyapunov's Second Method (L2M) for Stability:

We want to know how L2M helps us solve TBP and much more...

Relevant Reference Link:

'Important Question about the Lyapunov's Second Method for Stability',

https://www.researchgate.net/project/Important-Question-about-the-Lyapunovs-Second-Method-for-Stability.

[Guest]

An Update:

"Simple seeks simplest 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}})*\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 (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 (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!

[Guest]

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}})*\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!

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]

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].

[Guest]

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]

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]

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

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

[Guest]

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

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

[Guest]

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]

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)|[/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)|[/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)|[/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)|[/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)|[/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)|[/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)|[/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)|[/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)|[/tex].

[Guest]

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]

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!

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