Guest wrote: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 wrote:"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 wrote: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 wrote: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 wrote:Guest wrote: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])...
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