Collinear Solution of the Three-Body Problem
Author
Luciano Quaresma
Title
Collinear Solution of the Three-Body Problem
Description
Show the collinear motion of the three-body problem
Category
Educational Materials
Keywords
Three-Body Problem
URL
http://www.notebookarchive.org/2021-05-4olzajb/
DOI
https://notebookarchive.org/2021-05-4olzajb
Date Added
2021-05-10
Date Last Modified
2021-05-10
File Size
62.19 kilobytes
Supplements
Rights
CC BY 4.0
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Collinear Solution of the Three-Body Problem
Collinear Solution of the Three-Body Problem
By Luciano Quaresma and Manuel Rodrigues
The Three-Body Problem deals with the motion on three mutually interacting bodies. In this Notebook, this interaction is described by Newton’s Second Law and his Law of Universal Gravitation. Each body has six degrees of freedom, three for positions and three for velocities, so the Dynamic System used to describe them has a total of 18 equations. These are coupled nonlinear differential equations that doesn’t have a exact solution, so the numerical approach is used. To do this, first it is convenient restrict the motion to the x-y plane, which leads to a system with 12 equations, and to represent the terms of the equations as
In[]:=
g
i_,j_
m
i
m
j
d
i_,j_
x
i
x
j
y
i
y
j
dx
i_
v
x
i
dy
i_
v
y
i
dv
x
i_
g
i,j
m
i
x
i
x
j
d
i,j
dv
y
i_
g
i,j
m
i
y
i
y
j
d
i,j
Furthermore, it is useful to define the value of some parameters of the equations
In[]:=
G:=1:=2:=0.5:=0.5
m
1
m
2
m
3
It is important to note that specific solutions have some requirements on the masses used, such as the solution in this notebook.
To distinguish the trajectories of each body, it is also defined some style parameters to be used later.
To distinguish the trajectories of each body, it is also defined some style parameters to be used later.
In[]:=
c
1
c
2
c
3
The dynamic system is solved with the NDSolve command and the solution is represented by s as a function of its 12 initial conditions.
In[]:=
s[x10_,x20_,x30_,y10_,y20_,y30_,vx10_,vx20_,vx30_,vy10_,vy20_,vy30_]:=NDSolve'[t],'[t],'[t],'[t],'[t],'[t],'[t],'[t],'[t],'[t],'[t],'[t],[0]x10,[0]x20,[0]x30,[0]y10,[0]y20,[0]y30,[0]vx10,[0]vx20,[0]vx30,[0]vy10,[0]vy20,[0]vy30,{,,,,,,,,,,,},{t,0,100}
x
1
dx
1
x
2
dx
2
x
3
dx
3
y
1
dy
1
y
2
dy
2
y
3
dy
3
v
x
1
dv
x
1
v
x
2
dv
x
2
v
x
3
dv
x
3
v
y
1
dv
y
1
v
y
2
dv
y
2
v
y
3
dv
y
3
x
1
x
2
x
3
y
1
y
2
y
3
v
x
1
v
x
2
v
x
3
v
y
1
v
y
2
v
y
3
x
1
x
2
x
3
y
1
y
2
y
3
v
x
1
v
x
2
v
x
3
v
y
1
v
y
2
v
y
3
To show the final results, some terms are defined as a function of an instant T at which the graph is plotted.
In[]:=
p[i_,T_]:=ParametricPlot[Evaluate[{[t],[t]}/.s[-1,1.254953728,2.745046272,0,0,0,0,0,0,-0.3660350371,0.4593570344,1.004783114]],{t,0,T},PlotStyle{Black,}]
x
i
y
i
c
i
In[]:=
e
i_
x
i
y
i
In[]:=
b
i_
e
i
m
i
Finally the solution is plotted and the solution is chosen from the initial conditions and the parameters defined above. The representation for the so called Collinear Solution at T=10 is given by
In[]:=
Collinear[T_]:={p[1,T],p[2,T],p[3,T],[T],[T],[T],l[T]}
b
1
b
2
b
3
In[]:=
Show[Collinear[10],AxesLabel{"x","y"},PlotPoints1000,BaseStyle{FontSize22},ImageSize{1050,1050},PlotRange{{-3,3},{-3,3}},AxesOrigin{0,0},AxesStyleArrowheads[{0.0,0.02}]]
Out[]=
From the image it is clear why this solution is called Collinear as the bodies always stays on the same line. Many of the solutions for the Three-Body Problem show chaotic behavior, so even the slightest change on the initial conditions leads to the collapse of the apparent stability of this and other solutions.
Cite this as: Luciano Quaresma, "Collinear Solution of the Three-Body Problem" from the Notebook Archive (2021), https://notebookarchive.org/2021-05-4olzajb
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