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<html>
<head>
<title>
TWO_BODY_SIMULATION - Planar Two Body Problem Simulation
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
TWO_BODY_SIMULATION <br> Planar Two Body Problem Simulation
</h1>
<hr>
<p>
<b>TWO_BODY_SIMULATION</b>
is a MATLAB library which
simulates the solution of the planar two body problem.
</p>
<p>
Two 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. One body is assume to be much more massive than the other.
Therefore, the common motion of the two bodies about their center of mass
can be approximated by assuming that the large body remains fixed.
</p>
<p>
Under these assumptions, Newton's equations for (x(t),y(t)), the positition
of the lighter body with respect to the heavy body at (0,0), can be written as:
<pre>
r(t) = sqrt ( x(t)^2 + y(t)^2 )
x''(t) = - x(t) / r^3
y''(t) = - y(t) / r^3
</pre>
</p>
<p>
These two second order equations can be rewritten as a system of four
first order equations using the variable u = [ x(t), x'(t), y(t), y'(t) ],
resulting in the equations:
<pre>
r = sqrt ( u(1)^2 + u(3)^2 )
u'(1) = u(2)
u'(2) = - u(1) / r^3
u'(3) = u(4)
u'(4) = - u(3) / r^3
</pre>
</p>
<p>
By specifying some initial condition for u, the system can then be
integrated in time using a standard ODE solver.
</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>TWO_BODY_SIMULATION</b> is available in
<a href = "../../f_src/two_body_simulation/two_body_simulation.html">a FORTRAN90 version</a> and
<a href = "../../m_src/two_body_simulation/two_body_simulation.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/brownian_motion_simulation/brownian_motion_simulation.html">
BROWNIAN_MOTION_SIMULATION</a>,
a MATLAB program which
simulates Brownian motion in an M-dimensional region.
</p>
<p>
<a href = "../../m_src/dice_simulation/dice_simulation.html">
DICE_SIMULATION</a>,
a MATLAB program which
simulates N tosses of M dice, making a histogram of the results.
</p>
<p>
<a href = "../../m_src/duel_simulation/duel_simulation.html">
DUEL_SIMULATION</a>,
a MATLAB program which
simulates N repetitions of a duel between two players, each of
whom has a known firing accuracy.
</p>
<p>
<a href = "../../m_src/gamblers_ruin_simulation/gamblers_ruin_simulation.html">
GAMBLERS_RUIN_SIMULATION</a>,
a MATLAB program which
simulates the game of gambler's ruin.
</p>
<p>
<a href = "../../m_src/high_card_simulation/high_card_simulation.html">
HIGH_CARD_SIMULATION</a>,
a MATLAB 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.
</p>
<p>
<a href = "../../m_src/ising_2d_simulation/ising_2d_simulation.html">
ISING_2D_SIMULATION</a>,
a MATLAB 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 = "../../m_src/lorenz_simulation/lorenz_simulation.html">
LORENZ_SIMULATION</a>,
a MATLAB program which
solves the Lorenz equations and displays the solution, for various
starting conditions.
</p>
<p>
<a href = "../../m_src/poisson_simulation/poisson_simulation.html">
POISSON_SIMULATION</a>,
a MATLAB library which
simulates a Poisson process in which events randomly occur with an
average waiting time of Lambda.
</p>
<p>
<a href = "../../m_src/random_walk_1d_simulation/random_walk_1d_simulation.html">
RANDOM_WALK_1D_SIMULATION</a>,
a MATLAB program which
simulates a random walk in a 1-dimensional region.
</p>
<p>
<a href = "../../m_src/random_walk_2d_avoid_simulation/random_walk_2d_avoid_simulation.html">
RANDOM_WALK_2D_AVOID_SIMULATION</a>,
a MATLAB program which
simulates a self-avoiding random walk in a 2-dimensional region.
</p>
<p>
<a href = "../../m_src/random_walk_2d_simulation/random_walk_2d_simulation.html">
RANDOM_WALK_2D_SIMULATION</a>,
a MATLAB program which
simulates a random walk in a 2-dimensional region.
</p>
<p>
<a href = "../../m_src/random_walk_3d_simulation/random_walk_3d_simulation.html">
RANDOM_WALK_3D_SIMULATION</a>,
a MATLAB program which
simulates a random walk in a 3-dimensional region.
</p>
<p>
<a href = "../../m_src/reactor_simulation/reactor_simulation.html">
REACTOR_SIMULATION</a>,
a MATLAB 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 = "../../m_src/rkf45/rkf45.html">
RKF45</a>,
a MATLAB library which
implements the Runge-Kutta-Fehlberg ODE solver.
</p>
<p>
<a href = "../../m_src/sir_simulation/sir_simulation.html">
SIR_SIMULATION</a>,
a MATLAB program which
simulates the spread of a disease through a hospital room of M by N beds,
using the SIR (Susceptible/Infected/Recovered) model.
</p>
<p>
<a href = "../../m_src/three_body_simulation/three_body_simulation.html">
THREE_BODY_SIMULATION</a>,
a MATLAB program which
simulates the behavior of three planets, constrained to lie in a plane,
and moving under the influence of gravity,
by Walter Gander and Jiri Hrebicek.
</p>
<p>
<a href = "../../m_src/traffic_simulation/traffic_simulation.html">
TRAFFIC_SIMULATION</a>,
a MATLAB program which
simulates the cars waiting to get through a traffic light.
</p>
<p>
<a href = "../../m_src/truel_simulation/truel_simulation.html">
TRUEL_SIMULATION</a>,
a MATLAB program which
simulates N repetitions of a duel between three players, each of
whom has a known firing accuracy.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Charles VanLoan,<br>
Introduction to Scientific Computing,<br>
Prentice Hall, 1997,<br>
ISBN: 0-13-125444-8,<br>
LC: QA76.9.M35V375.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "kepler.m">kepler.m</a>,
evaluates the right hand side of the ODE system.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>INITIAL_ORBIT</b> simulates the problem over approximately two orbits.
<ul>
<li>
<a href = "initial_orbit.m">initial_orbit.m</a>,
the source code.
</li>
<li>
<a href = "initial_orbit.png">initial_orbit.png</a>,
an image of the trajectory.
</li>
</ul>
</p>
<p>
<b>ORBITAL_DECAY</b> computes about twenty successive orbits, showing
how the orbit gradually decays to a more elliptical form.
<ul>
<li>
<a href = "orbital_decay.m">orbital_decay.m</a>,
the source code.
</li>
<li>
<a href = "orbital_decay.png">orbital_decay.png</a>,
an image of the trajectory.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last modified on 05 November 2012.
</i>
<!-- John Burkardt -->
</body>
</html>