On a General Solution of the Three-Body Problem

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

where $$a_{ij }$$ is the acceleration variable; $$b_{ij }$$ is the speed variable; $$c_{ij }$$ 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, and for the 'never-ending' updates too.
Guest

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

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',

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: On a General Solution of the Three-Body Problem

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

where $$a_{ij }$$ is the acceleration variable; $$b_{ij }$$ is the speed variable; $$c_{ij }$$ 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, and for the 'never-ending' updates too.

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

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

'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
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Guest

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

Optimal Controls and System Perturbation for the Three-Body Problem

How does one optimally perturb a three-body periodic & symmetrical orbit to generate a different configuration...?
Attachments
threebody.png (12.38 KiB) Viewed 162 times
Guest

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

Guest wrote:Optimal Controls and System Perturbation for the Three-Body Problem

How does one optimally perturb a three-body periodic & symmetrical orbit to generate a different configuration...?

Step 1: Define the orbit for one mass, $$m_{1}$$, for the four configurations (see the diagram, 'threebody.png') according to Newton's Law of Motion since all masses, $$m_{1}, m_{2}, m_{3}$$, will traverse the same orbit.

Find $$a_{11 }(t), a_{12 }(t), a_{13 }(t), b_{11 }(t), b_{12 }(t), b_{13 }(t), c_{11 }, c_{12 }, c_{13 }$$ for the following equations:

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

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

12. $$z_{1 }(t) = \frac{1}{2} * a_{13 }(t)*t^{2} + b_{13 }(t)*t + c_{13 }$$.

Step 2: Define a controlled space, U, that perturbs from one configuration to another configuration...

Good Luck!

Dave.
Attachments
threebody.png (12.38 KiB) Viewed 148 times
Guest

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

'Poincaré and the Three-Body Problem' by Prof. Alain Chenciner,

http://www.bourbaphy.fr/chenciner.pdf.
Guest

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

Hmm.

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

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

12. $$z_{1 }(t) = \frac{1}{2} * a_{13 }(t)*t^{2} + b_{13 }(t)*t + c_{13 }$$.

For equations 10 - 12, we shall consider tentatively:

$$a_{ij }(t)*t^{2} + b_{ij}(t)*t = F_{ij}( u_{ij}(t)*cos(\beta_{ij} t) , v_{ij}(t)*sin(\delta_{ij} t))$$

where the function, $$F_{ij}(t)$$, is generally a nonlinear function of t such that our solutions to the three-body problem are closed, symmetrical, and periodic orbits.
Guest

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

Guest wrote:Hmm.

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

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

12. $$z_{1 }(t) = \frac{1}{2} * a_{13 }(t)*t^{2} + b_{13 }(t)*t + c_{13 }$$.

For equations 10 - 12, we shall consider tentatively:

$$a_{ij }(t)*t^{2} + b_{ij}(t)*t = F_{ij}( u_{ij}(t)*cos(\beta_{ij} t) , v_{ij}(t)*sin(\delta_{ij} t))$$

where the function, $$F_{ij}(t)$$, is generally a nonlinear function of t such that our solutions to the three-body problem are closed, symmetrical, and periodic orbits.

Hmm. A function of a function may not be satisfactory here. Therefore, we reject the function, F. Instead, we shall consider tentatively:

$$a_{ij }(t)*t^{2} + b_{ij}(t)*t = u_{ij}(t) = \left \{ \right. u_{ijm}(t)*cos(\beta_{ijm} t)$$ from $$m = 0$$ to $$m = n_{ij}$$.

where the function, $$u_{ij}(t)$$, is generally a nonlinear piecewise function of t such that our solutions to the three-body problem are closed, symmetrical and periodic orbits...

Keywords: operator, continuity, phase difference, translation, rotation, change of coordinate system...

Moreover, the spatial components, x(t), y(t), and z(t), for each mass of our three-body system must also be synchronized and simplified appropriately...

Now, we may begin our difficult analysis and synthesis here while recalling and obeying our system equations based on Newton's Laws of Motion... Good Luck!
Attachments
threebody.png (12.38 KiB) Viewed 133 times
Guest

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

Our goal is to find a general solution (closed-form) to the n-body problem where $$n \ge 3$$.
Guest

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

FYI:

Each time-dependent spatial component, x(t), y(t), and z(t), for each mass of our n-body system, has a lower bound and an upper bound:

$$x_{min} \le x(t) \le x_{max}$$;

$$y_{min} \le y(t) \le y_{max}$$;

$$z_{min} \le z(t) \le z_{max}$$.

...
Guest

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

Guest wrote:FYI:

Each time-dependent spatial component, x(t), y(t), and z(t), for each mass of our n-body system, has a lower bound and an upper bound:

$$x_{min} \le x(t) \le x_{max}$$;

$$y_{min} \le y(t) \le y_{max}$$;

$$z_{min} \le z(t) \le z_{max}$$.

...

FYI: The periodic motion between the minimum and maximum points, inclusively, is quite fascinating.
Attachments
Smple Example of a Possible Mass Motion Profile.png (11.73 KiB) Viewed 100 times
Guest

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

FYI: The diagram below indicates motion profiles for our three-body problem at t=k. And each arrow indicates the velocity of motion for each mass at a point (bar) along a spatial component.
Attachments
Three-Body Motion Profiles at t = k.png (28.7 KiB) Viewed 93 times
Guest

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

Guest wrote:FYI: The diagram below indicates motion profiles for our three-body problem at t=k. And each arrow indicates the velocity of motion for each mass at a point (bar) along a spatial component.

FYI: Corrected Diagram is attached below. Changed x(k) to z(k) for the last motion profile.
Attachments
Three-Body Motion Profiles at t = k.png (11.69 KiB) Viewed 92 times
Guest

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

Keywords: Transition, Limit Cycles, and Stable Orbits...

Hmm. It's the initial positions of the three bodies of our three-body problem along with their respective mass, initial speed, initial direction, initial acceleration, and initial energy that determine the final closed, periodic, and symmetrical orbit (solution) according to Newton's Laws of Motion. Of course, there's a transitional period from the initial positions to a final stable orbit for the three-body system. So we are mindful of limit cycles too.

...
Guest

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

Guest wrote:Keywords: Transition, Limit Cycles, and Stable Orbits...

Hmm. It's the initial positions of the three bodies of our three-body problem along with their respective mass, initial speed, initial direction, initial acceleration, and initial energy that determine the final closed, periodic, and symmetrical orbit (solution) according to Newton's Laws of Motion. Of course, there's a transitional period from the initial positions to a final stable orbit for the three-body system. So we are mindful of limit cycles too.

...

Hmm. The initial energy is not required since it is calculated from the other initial parameters...
Guest

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

We are almost ready to begin a deeper analysis and synthesis of the three-body problem. We shall review our current system equations before we begin. And hopefully, everything is fine thus far.

Next, we shall start with some initial conditions and try to find a solution...

'David Hilbert's Sixteenth Problem requires a BIG SOLUTION!',

https://www.math10.com/forum/viewtopic.php?f=63&t=8429;

Dave.
Guest

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

FYI: 'Researchers crack Newton's elusive three-body problem?'

https://phys.org/news/2019-12-newton-elusive-three-body-problem.html.

Remark: The title of the story is misleading! But we are glad it reports some significant progress...
Guest

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

FYI: 'Chapter 7: Mechanisms in Programs and Nature
Section 4: Chaos Theory and Randomness from Initial Conditions',

https://www.wolframscience.com/nks/notes-7-4--three-body-problem/.
Guest

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