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<html>
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<title>Solar system</title>
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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numeric_odeint.tutorial.solar_system"></a><a class="link" href="solar_system.html" title="Solar system">Solar system</a>
</h3></div></div></div>
<div class="toc"><dl>
<dt><span class="section"><a href="solar_system.html#boost_numeric_odeint.tutorial.solar_system.gravitation_and_energy_conservation">Gravitation
and energy conservation</a></span></dt>
<dt><span class="section"><a href="solar_system.html#boost_numeric_odeint.tutorial.solar_system.define_the_system_function">Define
the system function</a></span></dt>
</dl></div>
<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="boost_numeric_odeint.tutorial.solar_system.gravitation_and_energy_conservation"></a><a class="link" href="solar_system.html#boost_numeric_odeint.tutorial.solar_system.gravitation_and_energy_conservation" title="Gravitation and energy conservation">Gravitation
and energy conservation</a>
</h4></div></div></div>
<p>
The next example in this tutorial is a simulation of the outer solar system,
consisting of the sun, Jupiter, Saturn, Uranus, Neptune and Pluto.
</p>
<p>
<span class="inlinemediaobject"><img src="../../solar_system.jpg" alt="solar_system"></span>
</p>
<p>
Each planet and of course the sun will be represented by mass points. The
interaction force between each object is the gravitational force which
can be written as
</p>
<p>
<span class="emphasis"><em>F<sub>&#8203;ij</sub> = -&#947; m<sub>&#8203;i</sub> m<sub>&#8203;j</sub> ( q<sub>&#8203;i</sub> - q<sub>&#8203;j</sub> ) / | q<sub>&#8203;i</sub> - q<sub>&#8203;j</sub> | <sup>3</sup></em></span>
</p>
<p>
where <span class="emphasis"><em>&#947;</em></span> is the gravitational constant, <span class="emphasis"><em>m<sub>&#8203;i</sub></em></span>
and <span class="emphasis"><em>m<sub>&#8203;j</sub></em></span> are the masses and <span class="emphasis"><em>q<sub>&#8203;i</sub></em></span>
and <span class="emphasis"><em>q<sub>&#8203;j</sub></em></span> are the locations of the two objects. The equations
of motion are then
</p>
<p>
<span class="emphasis"><em>dq<sub>&#8203;i</sub> / dt = p<sub>&#8203;i</sub></em></span>
</p>
<p>
<span class="emphasis"><em>dp<sub>&#8203;i</sub> / dt = 1 / m<sub>&#8203;i</sub> &#931;<sub>&#8203;ji</sub> F<sub>&#8203;ij</sub></em></span>
</p>
<p>
where <span class="emphasis"><em>p<sub>&#8203;i</sub></em></span> is the momenta of object <span class="emphasis"><em>i</em></span>.
The equations of motion can also be derived from the Hamiltonian
</p>
<p>
<span class="emphasis"><em>H = &#931;<sub>&#8203;i</sub> p<sub>&#8203;i</sub><sup>2</sup> / ( 2 m<sub>&#8203;i</sub> ) + &#931;<sub>&#8203;j</sub> V( q<sub>&#8203;i</sub> , q<sub>&#8203;j</sub> )</em></span>
</p>
<p>
with the interaction potential <span class="emphasis"><em>V(q<sub>&#8203;i</sub>,q<sub>&#8203;j</sub>)</em></span>. The Hamiltonian
equations give the equations of motion
</p>
<p>
<span class="emphasis"><em>dq<sub>&#8203;i</sub> / dt = dH / dp<sub>&#8203;i</sub></em></span>
</p>
<p>
<span class="emphasis"><em>dp<sub>&#8203;i</sub> / dt = -dH / dq<sub>&#8203;i</sub></em></span>
</p>
<p>
In time independent Hamiltonian system the energy and the phase space volume
are conserved and special integration methods have to be applied in order
to ensure these conservation laws. The odeint library provides classes
for separable Hamiltonian systems, which can be written in the form <span class="emphasis"><em>H
= &#931;
p<sub>&#8203;i</sub><sup>2</sup> / (2m<sub>&#8203;i</sub>) + H<sub>&#8203;q</sub>(q)</em></span>, where <span class="emphasis"><em>H<sub>&#8203;q</sub>(q)</em></span> only
depends on the coordinates. Although this functional form might look a
bit arbitrary, it covers nearly all classical mechanical systems with inertia
and without dissipation, or where the equations of motion can be written
in the form <span class="emphasis"><em>dq<sub>&#8203;i</sub> / dt = p<sub>&#8203;i</sub></em></span> / m<sub>&#8203;i</sub> , <span class="emphasis"><em>dp<sub>&#8203;i</sub> / dt =
f( q<sub>&#8203;i</sub> )</em></span>.
</p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
A short physical note: While the two-body-problem is known to be integrable,
that means it can be solved with purely analytic techniques, already
the three-body-problem is not solveable. This was found in the end of
the 19th century by H. Poincare which lead to the whole new subject of
<a href="http://en.wikipedia.org/wiki/Chaos_theory" target="_top">Chaos Theory</a>.
</p></td></tr>
</table></div>
</div>
<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="boost_numeric_odeint.tutorial.solar_system.define_the_system_function"></a><a class="link" href="solar_system.html#boost_numeric_odeint.tutorial.solar_system.define_the_system_function" title="Define the system function">Define
the system function</a>
</h4></div></div></div>
<p>
To implement this system we define a 3D point type which will represent
the space as well as the velocity. Therefore, we use the operators from
<a href="http://www.boost.org/doc/libs/release/doc/html/operators.html" target="_top">Boost.Operators</a>:
</p>
<p>
</p>
<pre class="programlisting"><span class="comment">/*the point type */</span>
<span class="keyword">template</span><span class="special">&lt;</span> <span class="keyword">class</span> <span class="identifier">T</span> <span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Dim</span> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">point</span> <span class="special">:</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">additive1</span><span class="special">&lt;</span> <span class="identifier">point</span><span class="special">&lt;</span> <span class="identifier">T</span> <span class="special">,</span> <span class="identifier">Dim</span> <span class="special">&gt;</span> <span class="special">,</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">additive2</span><span class="special">&lt;</span> <span class="identifier">point</span><span class="special">&lt;</span> <span class="identifier">T</span> <span class="special">,</span> <span class="identifier">Dim</span> <span class="special">&gt;</span> <span class="special">,</span> <span class="identifier">T</span> <span class="special">,</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiplicative2</span><span class="special">&lt;</span> <span class="identifier">point</span><span class="special">&lt;</span> <span class="identifier">T</span> <span class="special">,</span> <span class="identifier">Dim</span> <span class="special">&gt;</span> <span class="special">,</span> <span class="identifier">T</span>
<span class="special">&gt;</span> <span class="special">&gt;</span> <span class="special">&gt;</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="keyword">const</span> <span class="keyword">static</span> <span class="identifier">size_t</span> <span class="identifier">dim</span> <span class="special">=</span> <span class="identifier">Dim</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">value_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">point</span><span class="special">&lt;</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">dim</span> <span class="special">&gt;</span> <span class="identifier">point_type</span><span class="special">;</span>
<span class="comment">// ...</span>
<span class="comment">// constructors</span>
<span class="comment">// ...</span>
<span class="comment">// operators</span>
<span class="keyword">private</span><span class="special">:</span>
<span class="identifier">T</span> <span class="identifier">m_val</span><span class="special">[</span><span class="identifier">dim</span><span class="special">];</span>
<span class="special">};</span>
<span class="comment">//...</span>
<span class="comment">// more operators</span>
</pre>
<p>
</p>
<p>
The next step is to define a container type storing the values of <span class="emphasis"><em>q</em></span>
and <span class="emphasis"><em>p</em></span> and to define system functions. As container
type we use <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span></code>
</p>
<p>
</p>
<pre class="programlisting"><span class="comment">// we simulate 5 planets and the sun</span>
<span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">6</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">point</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">,</span> <span class="number">3</span> <span class="special">&gt;</span> <span class="identifier">point_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span> <span class="identifier">point_type</span> <span class="special">,</span> <span class="identifier">n</span> <span class="special">&gt;</span> <span class="identifier">container_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">n</span> <span class="special">&gt;</span> <span class="identifier">mass_type</span><span class="special">;</span>
</pre>
<p>
</p>
<p>
The <code class="computeroutput"><span class="identifier">container_type</span></code> is different
from the state type of the ODE. The state type of the ode is simply a
<code class="computeroutput"><span class="identifier">pair</span><span class="special">&lt;</span>
<span class="identifier">container_type</span> <span class="special">,</span>
<span class="identifier">container_type</span> <span class="special">&gt;</span></code>
since it needs the information about the coordinates and the momenta.
</p>
<p>
Next we define the system's equations. As we will use a stepper that accounts
for the Hamiltonian (energy-preserving) character of the system, we have
to define the rhs different from the usual case where it is just a single
function. The stepper will make use of the separable character, which means
the system will be defined by two objects representing <span class="emphasis"><em>f(p) =
-dH/dq</em></span> and <span class="emphasis"><em>g(q) = dH/dp</em></span>:
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">gravitational_constant</span> <span class="special">=</span> <span class="number">2.95912208286e-4</span><span class="special">;</span>
<span class="keyword">struct</span> <span class="identifier">solar_system_coor</span>
<span class="special">{</span>
<span class="keyword">const</span> <span class="identifier">mass_type</span> <span class="special">&amp;</span><span class="identifier">m_masses</span><span class="special">;</span>
<span class="identifier">solar_system_coor</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">mass_type</span> <span class="special">&amp;</span><span class="identifier">masses</span> <span class="special">)</span> <span class="special">:</span> <span class="identifier">m_masses</span><span class="special">(</span> <span class="identifier">masses</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&amp;</span><span class="identifier">p</span> <span class="special">,</span> <span class="identifier">container_type</span> <span class="special">&amp;</span><span class="identifier">dqdt</span> <span class="special">)</span> <span class="keyword">const</span>
<span class="special">{</span>
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special">&lt;</span><span class="identifier">n</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span>
<span class="identifier">dqdt</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span> <span class="identifier">m_masses</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
<span class="special">}</span>
<span class="special">};</span>
</pre>
<p>
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">solar_system_momentum</span>
<span class="special">{</span>
<span class="keyword">const</span> <span class="identifier">mass_type</span> <span class="special">&amp;</span><span class="identifier">m_masses</span><span class="special">;</span>
<span class="identifier">solar_system_momentum</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">mass_type</span> <span class="special">&amp;</span><span class="identifier">masses</span> <span class="special">)</span> <span class="special">:</span> <span class="identifier">m_masses</span><span class="special">(</span> <span class="identifier">masses</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&amp;</span><span class="identifier">q</span> <span class="special">,</span> <span class="identifier">container_type</span> <span class="special">&amp;</span><span class="identifier">dpdt</span> <span class="special">)</span> <span class="keyword">const</span>
<span class="special">{</span>
<span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="identifier">q</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span>
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special">&lt;</span><span class="identifier">n</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span>
<span class="special">{</span>
<span class="identifier">dpdt</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">j</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">j</span><span class="special">&lt;</span><span class="identifier">i</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">j</span> <span class="special">)</span>
<span class="special">{</span>
<span class="identifier">point_type</span> <span class="identifier">diff</span> <span class="special">=</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">j</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
<span class="keyword">double</span> <span class="identifier">d</span> <span class="special">=</span> <span class="identifier">abs</span><span class="special">(</span> <span class="identifier">diff</span> <span class="special">);</span>
<span class="identifier">diff</span> <span class="special">*=</span> <span class="special">(</span> <span class="identifier">gravitational_constant</span> <span class="special">*</span> <span class="identifier">m_masses</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">m_masses</span><span class="special">[</span><span class="identifier">j</span><span class="special">]</span> <span class="special">/</span> <span class="identifier">d</span> <span class="special">/</span> <span class="identifier">d</span> <span class="special">/</span> <span class="identifier">d</span> <span class="special">);</span>
<span class="identifier">dpdt</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">+=</span> <span class="identifier">diff</span><span class="special">;</span>
<span class="identifier">dpdt</span><span class="special">[</span><span class="identifier">j</span><span class="special">]</span> <span class="special">-=</span> <span class="identifier">diff</span><span class="special">;</span>
<span class="special">}</span>
<span class="special">}</span>
<span class="special">}</span>
<span class="special">};</span>
</pre>
<p>
</p>
<p>
In general a three body-system is chaotic, hence we can not expect that
arbitrary initial conditions of the system will lead to a dynamic which
is comparable with the solar system. That is we have to define proper initial
conditions, which are taken from the book of Hairer, Wannier, Lubich.
</p>
<p>
As mentioned above, we need to use some special integrators in order to
conserve phase space volume. There is a well known family of such integrators,
the so-called Runge-Kutta-Nystroem solvers, which we apply here in terms
of a <code class="computeroutput"><span class="identifier">symplectic_rkn_sb3a_mclachlan</span></code>
stepper:
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">symplectic_rkn_sb3a_mclachlan</span><span class="special">&lt;</span> <span class="identifier">container_type</span> <span class="special">&gt;</span> <span class="identifier">stepper_type</span><span class="special">;</span>
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">100.0</span><span class="special">;</span>
<span class="identifier">integrate_const</span><span class="special">(</span>
<span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">,</span>
<span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">solar_system_coor</span><span class="special">(</span> <span class="identifier">masses</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">solar_system_momentum</span><span class="special">(</span> <span class="identifier">masses</span> <span class="special">)</span> <span class="special">)</span> <span class="special">,</span>
<span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">q</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">p</span> <span class="special">)</span> <span class="special">)</span> <span class="special">,</span>
<span class="number">0.0</span> <span class="special">,</span> <span class="number">200000.0</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="identifier">streaming_observer</span><span class="special">(</span> <span class="identifier">cout</span> <span class="special">)</span> <span class="special">);</span>
</pre>
<p>
</p>
<p>
These integration routine was used to produce the above sketch of the solar
system. Note, that there are two particularities in this example. First,
the state of the symplectic stepper is not <code class="computeroutput"><span class="identifier">container_type</span></code>
but a pair of <code class="computeroutput"><span class="identifier">container_type</span></code>.
Hence, we must pass such a pair to the integrate function. Since, we want
to pass them as references we can simply pack them into <a href="http://www.boost.org/doc/libs/release/doc/html/ref.html" target="_top">Boost.Ref</a>.
The second point is the observer, which is called with a state type, hence
a pair of <code class="computeroutput"><span class="identifier">container_type</span></code>.
The reference wrapper is also passed, but this is not a problem at all:
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">streaming_observer</span>
<span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&amp;</span> <span class="identifier">m_out</span><span class="special">;</span>
<span class="identifier">streaming_observer</span><span class="special">(</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span> <span class="special">&amp;</span><span class="identifier">out</span> <span class="special">)</span> <span class="special">:</span> <span class="identifier">m_out</span><span class="special">(</span> <span class="identifier">out</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
<span class="keyword">template</span><span class="special">&lt;</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">&gt;</span>
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">&amp;</span><span class="identifier">x</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
<span class="special">{</span>
<span class="identifier">container_type</span> <span class="special">&amp;</span><span class="identifier">q</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">first</span><span class="special">;</span>
<span class="identifier">m_out</span> <span class="special">&lt;&lt;</span> <span class="identifier">t</span><span class="special">;</span>
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special">&lt;</span><span class="identifier">q</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> <span class="identifier">m_out</span> <span class="special">&lt;&lt;</span> <span class="string">"\t"</span> <span class="special">&lt;&lt;</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
<span class="identifier">m_out</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="special">}</span>
<span class="special">};</span>
</pre>
<p>
</p>
<p>
The full example can be found here: <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/solar_system.cpp" target="_top">solar_system.cpp</a>
</p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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