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<html>
<head>
<title>
THREE_BODY_SIMULATION - Planar Three Body Problem Simulation
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
THREE_BODY_SIMULATION <br> Planar Three Body Problem Simulation
</h1>
<hr>
<p>
<b>THREE_BODY_SIMULATION</b>
is a C++ program which
simulates the solution of the planar three body problem.
</p>
<p>
Three bodies, regarded as point masses, are constrained to lie in a plane.
The masses of each body are given, as are the positions and velocities
at a starting time T = 0. The bodies move in accordance with the
gravitational force between them.
</p>
<p>
The force exerted on the 0-th body by the 1st body can be written:
<pre>
F = - m0 m1 ( p0 - p1 ) / |p0 - p1|^3
</pre>
assuming that units have been normalized to that the gravitational
coefficient is 1. Newton's laws of motion can be written:
<pre>
m0 p0'' = - m0 m1 ( p0 - p1 ) / |p0 - p1|^3
- m0 m2 ( p0 - p2 ) / |p0 - p2|^3
m1 p1'' = - m1 m0 ( p1 - p0 ) / |p1 - p0|^3
- m1 m2 ( p1 - p2 ) / |p1 - p2|^3
m2 p2'' = - m2 m0 ( p2 - p0 ) / |p2 - p0|^3
- m2 m1 ( p2 - p1 ) / |p2 - p1|^3
</pre>
</p>
<p>
Letting
<pre>
y1 = p0(x)
y2 = p0(y)
y3 = p0'(x)
y4 = p0'(y)
</pre>
and using similar definitions for p1 and p2, the 3 second order vector
equations can be rewritten as 12 first order equations. In particular,
the first four are:
<pre>
y1' = y3
y2' = y4
y3' = - m1 ( y1 - y5 ) / |(y1,y2) - (y5,y6) |^3
- m2 ( y1 - y9 ) / |(y1,y2) - (y9,y10)|^3
y4' = - m1 ( y2 - y6 ) / |(y1,y2) - (y5,y6) |^3
- m2 ( y2 - y10 ) / |(y1,y2) - (y9,y10)|^3
</pre>
and so on.
This first order system can be integrated by a standard ODE solver.
</p>
<p>
Note that when any two bodies come close together, the solution changes
very rapidly, and very small steps must be taken by the ODE solver.
For this system, the first near collision occurs around T=15.8299, and
the results produced by the ODE solver will not be very accurate after
that point.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>THREE_BODY_SIMULATION</b> is available in
<a href = "../../c_src/three_body_simulation/three_body_simulation.html">a C version</a> and
<a href = "../../cpp_src/three_body_simulation/three_body_simulation.html">a C++ version</a> and
<a href = "../../f77_src/three_body_simulation/three_body_simulation.html">a FORTRAN77 version</a> and
<a href = "../../f_src/three_body_simulation/three_body_simulation.html">a FORTRAN90 version</a> and
<a href = "../../m_src/three_body_simulation/three_body_simulation.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/brownian_motion_simulation/brownian_motion_simulation.html">
BROWNIAN_MOTION_SIMULATION</a>,
a C++ program which
simulates Brownian motion in an M-dimensional region.
</p>
<p>
<a href = "../../cpp_src/duel_simulation/duel_simulation.html">
DUEL_SIMULATION</a>,
a C++ program which
simulates N repetitions of a duel between two players, each of
whom has a known firing accuracy.
</p>
<p>
<a href = "../../cpp_src/fair_dice_simulation/fair_dice_simulation.html">
FAIR_DICE_SIMULATION</a>,
a C++ program which
simulates N tosses of 2 dice, making a histogram of the results.
</p>
<p>
<a href = "../../cpp_src/high_card_simulation/high_card_simulation.html">
HIGH_CARD_SIMULATION</a>,
a C++ program which
simulates a situation in which you see the cards in a deck one by one,
and must select the one you think is the highest and stop;
the program uses GNUPLOT for graphics.
</p>
<p>
<a href = "../../cpp_src/ising_2d_simulation/ising_2d_simulation.html">
ISING_2D_SIMULATION</a>,
a C++ program which
carries out a Monte Carlo simulation of an Ising model.
a 2D array of positive and negative charges,
each of which is likely to "flip" to be in agreement with neighbors.
</p>
<p>
<a href = "../../cpp_src/poisson_simulation/poisson_simulation.html">
POISSON_SIMULATION</a>,
a C++ library which
simulates a Poisson process in which events randomly occur with an
average waiting time of Lambda.
</p>
<p>
<a href = "../../cpp_src/reactor_simulation/reactor_simulation.html">
REACTOR_SIMULATION</a>,
a C++ program which
a simple Monte Carlo simulation of the shielding effect of a slab
of a certain thickness in front of a neutron source. This program was
provided as an example with the book "Numerical Methods and Software."
</p>
<p>
<a href = "../../cpp_src/rkf45/rkf45.html">
RKF45</a>,
a C++ library which
implements the Runge-Kutta-Fehlberg (RKF) solver
for the approximate solution of an ordinary differential equation (ODE) system.
</p>
<h3 align = "center">
Author:
</h3>
<p>
Original MATLAB version by Dominik Gruntz, Joerg Waldvogel;
C++ version by John Burkardt.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Dominik Gruntz, Joerg Waldvogel,<br>
"Orbits in the Planar Three-Body Problem",<br>
Walter Gander, Jiri Hrebicek,<br>
Solving Problems in Scientific Computing using Maple and Matlab,<br>
Springer, 1997,<br>
ISBN: 3-540-61793-0,<br>
LC: Q183.9.G36.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "simple_rkf45.cpp">simple_rkf45.cpp</a>, the source code.
</li>
<li>
<a href = "simple_rkf45.sh">simple_rkf45.sh</a>,
BASH commands to compile the source code.
</li>
<li>
<a href = "simple_rkf45_output.txt">simple_rkf45_output.txt</a>,
printed output from the run.
</li>
<li>
<a href = "simple_rkf45_t.txt">simple_rkf45_t.txt</a>,
the values of time T at every 0.1 seconds.
</li>
<li>
<a href = "simple_rkf45_y.txt">simple_rkf45_y.txt</a>,
the values of the 12 state variables at every time.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<b>SIMPLE_RKF45</b> simulates the problem by calling the ODE integrator RKF45.
This approach loses accuracy when the bodies come close to colliding, which is
likely to happen often.
<ul>
<li>
<b>MAIN</b> is the main program for SIMPLE_RKF45.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>SIMPLE_RKF45_RUN</b> runs the simple three body ODE system.
</li>
<li>
<b>SIMPLE_F</b> returns the right hand side of the three body ODE system.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last modified on 13 October 2012.
</i>
<!-- John Burkardt -->
</body>
</html>