On a General Solution of the Three-Body Problem

[Guest]

We are compelled to tackle this historically difficult and important elegant problem with much gumption and gusto... Any general solution to that problem requires, we believe, some knowledge of differential equations, topology, number theory, harmonic analysis, dynamical systems theory/chaos theory, general mathematical analysis...

Relevant Reference Links:

'Three-body problem',

https://en.wikipedia.org/wiki/Three-body_problem#cite_note-PrincetonCompanion-1;

'The compelling mathematical challenge of the three-body problem' by Tim Stephens and Prof. R. Montgomery,

https://phys.org/news/2019-08-compelling-mathematical-three-body-problem.html.

Dave.

[Guest]

Relevant Reference Link:

'An Introduction to Dynamical Systems and Chaos' by Prof. G.C. Layek,

https://www.researchgate.net/publication/296950820_An_Introduction_to_Dynamical_Systems_and_Chaos.

[Guest]

Relevant Reference Link:

'An Introduction to Dynamical Systems and Chaos' by Prof. G.C. Layek,

https://www.researchgate.net/publication/296950820_An_Introduction_to_Dynamical_Systems_and_Chaos.

Please also consider the relevant textbook, 'Dynamical Systems and Chaos' by Prof. H Broer and Prof. F. Takens,

https://epdf.pub/dynamical-systems-and-chaos.html.

[Guest]

"The conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus." -- David Hilbert.

Hmm. Since there has been considerable work done... to solve our difficult, elegant, and important three-body problem, we expect a general solution to our problem within 48 months in honor of the late great David Hilbert.

"We must know — we will know!" -- David Hilbert.

Relevant Reference Links:

'n-body problem',

https://en.wikipedia.org/wiki/N-body_problem;

'David Hilbert',

https://en.wikipedia.org/wiki/David_Hilbert.

Dave.

[Guest]

Relevant Reference Link:

'An Introduction to the Classical Three-Body Problem: From Periodic Solutions to Instabilities and Chaos' by Prof. G. S. Krishnaswami and Prof. H. Senapati,

https://www.ias.ac.in/article/fulltext/reso/024/01/0087-0114.

[Guest]

Relevant Reference Link:

'Three-Body Problem',

https://math.stackexchange.com/search?q=three-body+problem.

[Guest]

Current Status of the General Solution of the Three-Body Problem:

"...In place of the two masses, put three, and you have the three-body problem. Like its predecessor, its orbits are solutions to a system of differential equations. Unlike its

predecessor, however, it is difficult to impossible to find explicit formulas for orbits. To this day, despite modern computers and centuries of work by some of

the best physicists and mathematicians, we only have explicit formulas for five families of orbits, three found by Leonhard Euler (in 1767) and two by Joseph-Louis Lagrange

(in 1772). In 1890 Henri Poincaré discovered chaotic dynamics within the three-body problem, a finding that implies we can never know all the solutions to the problem at a

level of detail remotely approaching Newton's complete solution to the two-body problem. Yet through a process called numerical integration, done efficiently on a computer,

we can nonetheless generate finite segments of approximate orbits, a process essential to the planning of space missions. By extending the run-time of the computer, we can

make the approximations as accurate as we want..." Professor Richard Montgomery.

[Guest]

"Simple seeks the simplest (best) solution."

Formulation of a General Solution for the Three-Body Problem:

Initially, we have three masses, [tex]m_{1 }, m_{2 }, m_{3 }[/tex], each at a distinct spatial point, [tex]p_{1 } = (x_{1 }, y_{1 }, z_{1 }), p_{2 } = (x_{2 }, y_{2 }, z_{2 }), p_{3 } = (x_{3 }, y_{3 }, z_{3 })[/tex], respectively. And where [tex](x_{n }, y_{n}, z_{n }) = (x_{n }(t), y_{n}(t), z_{n }(t))[/tex] for [tex]n\in[/tex] {1, 2, 3}.

According to Newton's Law of Universal Gravitation, we generate a system of nine second-order differential equations:

Step 1:

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

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

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

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

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

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

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

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

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

Step 2: ...

[Guest]

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

Formulation of a General Solution for the Three-Body Problem:

Initially, we have three masses, [tex]m_{1 }, m_{2 }, m_{3 }[/tex], each at a distinct spatial point, [tex]p_{1 } = (x_{1 }, y_{1 }, z_{1 }), p_{2 } = (x_{2 }, y_{2 }, z_{2 }), p_{3 } = (x_{3 }, y_{3 }, z_{3 })[/tex], respectively. And where [tex](x_{n }, y_{n}, z_{n }) = (x_{n }(t), y_{n}(t), z_{n }(t))[/tex] for [tex]n\in[/tex] {1, 2, 3}.

According to Newton's Law of Universal Gravitation, we generate a system of nine second-order differential equations:

Step 1:

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

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

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

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

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

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

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

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

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

Step 2: We shall review all significant work done thus far to solve our three-body problem.

Step 3: ...

[Guest]

Equations of Step 1 need to be corrected according to the true distance between points p_1, p_2, and p_3.-

For example, |p_1 - p_2| = sqrt ((x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2). Right?

[Guest]

Equations of Step 1 need to be corrected according to the true distance between points p_1, p_2, and p_3.

For example, |p_1 - p_2| = sqrt ((x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2). Right?

Hmm. You may be right! I'll recheck those equations. Thanks!

[Guest]

An Update:

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

Formulation of a General Solution for the Three-Body Problem:

Initially, we have three masses, [tex]m_{1 }, m_{2 }, m_{3 }[/tex], each at a distinct spatial point, [tex]p_{1 } = (x_{1 }, y_{1 }, z_{1 }), p_{2 } = (x_{2 }, y_{2 }, z_{2 }), p_{3 } = (x_{3 }, y_{3 }, z_{3 })[/tex], respectively. And where [tex](x_{n }, y_{n}, z_{n }) = (x_{n }(t), y_{n}(t), z_{n }(t))[/tex] for [tex]n\in[/tex] {1, 2, 3}.

According to Newton's Laws of Motion, we generate a system of nine second-order differential equations:

Step 1:

1. [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];

2. [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];

3. [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];

4. [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];

5. [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];

6. [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];

7. [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];

8. [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];

9. [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.


Step 2: We shall review all significant work done thus far to solve our three-body problem.

Step 3: ...

[Guest]

Remark on Step 1 Equations:

If [tex]\frac{d^{2}x_{n }}{dt^{2}}
= \frac{dv_{x_{n }}}{dt} = \frac{d^{2}y_{n }}{dt^{2}}
= \frac{dv_{y_{n }}}{dt} = \frac{d^{2}z_{n }}{dt^{2}}
= \frac{dv_{z_{n }}}{dt} = 0[/tex], then there is no relative distance or motion between the three masses ...

Time, t, is the only independent variable. And we must consider stable and unstable orbits and the impact of all parameters...

[Guest]

Remark on Step 1 Equations:

If [tex]\frac{d^{2}x_{n }}{dt^{2}}
= \frac{dv_{x_{n }}}{dt} = \frac{d^{2}y_{n }}{dt^{2}}
= \frac{dv_{y_{n }}}{dt} = \frac{d^{2}z_{n }}{dt^{2}}
= \frac{dv_{z_{n }}}{dt} = 0[/tex], then there is no relative distance or motion between the three masses ...

Time, t, is the only independent variable. And we must consider stable and unstable orbits and the impact of all parameters...

Our parameters include initial conditions at time, [tex]t_{0 }[/tex], and those initial conditions 'should be' consistent with our system equations of step one.

[Guest]

Hmm. We may need to make some reasonable assumptions about how the motion of each mass move componentwise under the force of gravitional attraction. Furthermore, we may also need to make some assumptions about how the three masses as a single system move through space under the force of gravitional attraction given the reasonable initial conditions of mass magnitude, position, velocity, and acceleration of each mass in accordance with our system equations...

[Guest]

Remark on Step 1 Equations:

If the initial condition, [tex]\frac{d^{2}x_{n }}{dt^{2}}
= \frac{dv_{x_{n }}}{dt} = \frac{d^{2}y_{n }}{dt^{2}}
= \frac{dv_{y_{n }}}{dt} = \frac{d^{2}z_{n }}{dt^{2}}
= \frac{dv_{z_{n }}}{dt} = 0[/tex], then there is no relative distance or motion between the three masses ...

Time, t, is the only independent variable. And we must consider stable and unstable orbits and the impact of all parameters...

Our parameters include initial conditions at time, [tex]t_{0 }[/tex], and those initial conditions 'should be' consistent (no singularities/blowups/etc.) with our system equations of step one. Therefore, that initial condition stated above can never occur as an intermediate or final state after our three masses are separated and set into motion ([tex]r_{ij } \ne 0[/tex])...

[Guest]

Remark: We believe a deep analysis of the three-body problem must begin at the system level down to individual orbits for each mass in our system. The system determines the possible orbits of each mass in the system.

[Guest]

FYI: Gravitational constant, G = 6.67408 × [tex]10^{-11} m^{3}/(kg * s^{2})[/tex].

[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.

[Guest]

FYI:

'Scientists discover more than 600 new periodic orbits of the famous three-body problem',

https://phys.org/news/2017-10-scientists-periodic-orbits-famous-three-body.html.

'The restricted three-body problem and holomorphic curves',

https://www.youtube.com/watch?v=WI0klrK8dNA.