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<html>
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<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Symplectic System</title>
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<link rel="stylesheet" href="../../boostbook.css" type="text/css">
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<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
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<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
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<link rel="up" href="../concepts.html" title="Concepts">
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<link rel="prev" href="system.html" title="System">
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<link rel="next" href="simple_symplectic_system.html" title="Simple Symplectic System">
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
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<div class="spirit-nav">
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<a accesskey="p" href="system.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="simple_symplectic_system.html"><img src="../../images/next.png" alt="Next"></a>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_numeric_odeint.concepts.symplectic_system"></a><a class="link" href="symplectic_system.html" title="Symplectic System">Symplectic
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System</a>
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</h3></div></div></div>
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<h5>
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<a name="boost_numeric_odeint.concepts.symplectic_system.h0"></a>
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<span><a name="boost_numeric_odeint.concepts.symplectic_system.description"></a></span><a class="link" href="symplectic_system.html#boost_numeric_odeint.concepts.symplectic_system.description">Description</a>
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</h5>
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<p>
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This concept describes how to define a symplectic system written with generalized
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coordinate <code class="computeroutput"><span class="identifier">q</span></code> and generalized
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momentum <code class="computeroutput"><span class="identifier">p</span></code>:
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</p>
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<p>
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<span class="emphasis"><em>q'(t) = f(p) </em></span>
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</p>
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<p>
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<span class="emphasis"><em>p'(t) = g(q) </em></span>
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</p>
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<p>
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Such a situation is typically found for Hamiltonian systems with a separable
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Hamiltonian:
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</p>
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<p>
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<span class="emphasis"><em>H(p,q) = H<sub>kin</sub>(p) + V(q) </em></span>
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</p>
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<p>
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which gives the equations of motion:
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</p>
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<p>
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<span class="emphasis"><em>q'(t) = dH<sub>kin</sub> / dp = f(p) </em></span>
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</p>
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<p>
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<span class="emphasis"><em>p'(t) = dV / dq = g(q) </em></span>
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</p>
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<p>
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The algorithmic implementation of this situation is described by a pair of
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callable objects for <span class="emphasis"><em>f</em></span> and <span class="emphasis"><em>g</em></span> with
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a specific parameter signature. Such a system should be implemented as a
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std::pair of functions or a functors. Symplectic systems are used in symplectic
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steppers like <code class="computeroutput"><span class="identifier">symplectic_rkn_sb3a_mclachlan</span></code>.
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</p>
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<h5>
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<a name="boost_numeric_odeint.concepts.symplectic_system.h1"></a>
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<span><a name="boost_numeric_odeint.concepts.symplectic_system.notation"></a></span><a class="link" href="symplectic_system.html#boost_numeric_odeint.concepts.symplectic_system.notation">Notation</a>
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</h5>
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<div class="variablelist">
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<p class="title"><b></b></p>
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<dl>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">System</span></code></span></dt>
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<dd><p>
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A type that is a model of Symplectic_System
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">Coor</span></code></span></dt>
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<dd><p>
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The type of the coordinate <span class="emphasis"><em>q</em></span>
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">Momentum</span></code></span></dt>
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<dd><p>
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The type of the momentum <span class="emphasis"><em>p</em></span>
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">CoorDeriv</span></code></span></dt>
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<dd><p>
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The type of the derivative of coordinate <span class="emphasis"><em>q'</em></span>
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">MomentumDeriv</span></code></span></dt>
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<dd><p>
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The type of the derivative of momentum <span class="emphasis"><em>p'</em></span>
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">sys</span></code></span></dt>
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<dd><p>
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An object of the type <code class="computeroutput"><span class="identifier">System</span></code>
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">q</span></code></span></dt>
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<dd><p>
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Object of type Coor
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">p</span></code></span></dt>
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<dd><p>
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Object of type Momentum
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">dqdt</span></code></span></dt>
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<dd><p>
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Object of type CoorDeriv
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">dpdt</span></code></span></dt>
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<dd><p>
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Object of type MomentumDeriv
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</p></dd>
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</dl>
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</div>
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<h5>
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<a name="boost_numeric_odeint.concepts.symplectic_system.h2"></a>
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<span><a name="boost_numeric_odeint.concepts.symplectic_system.valid_expressions"></a></span><a class="link" href="symplectic_system.html#boost_numeric_odeint.concepts.symplectic_system.valid_expressions">Valid
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expressions</a>
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</h5>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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<p>
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Name
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</p>
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</th>
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<th>
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<p>
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Expression
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</p>
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</th>
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<th>
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<p>
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Type
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</p>
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</th>
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<th>
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<p>
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Semantics
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Check for pair
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</p>
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</td>
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<td>
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<p>
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<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">is_pair</span><span class="special"><</span>
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<span class="identifier">System</span> <span class="special">>::</span><span class="identifier">type</span></code>
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</p>
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</td>
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<td>
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<p>
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<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">true_</span></code>
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</p>
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</td>
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<td>
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<p>
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Check if System is a pair
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Calculate <span class="emphasis"><em>dq/dt = f(p)</em></span>
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</p>
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</td>
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<td>
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<p>
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<code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">first</span><span class="special">(</span>
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<span class="identifier">p</span> <span class="special">,</span>
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<span class="identifier">dqdt</span> <span class="special">)</span></code>
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</p>
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</td>
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<td>
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<p>
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<code class="computeroutput"><span class="keyword">void</span></code>
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</p>
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</td>
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<td>
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<p>
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Calculates <span class="emphasis"><em>f(p)</em></span>, the result is stored into
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<code class="computeroutput"><span class="identifier">dqdt</span></code>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Calculate <span class="emphasis"><em>dp/dt = g(q)</em></span>
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</p>
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</td>
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<td>
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<p>
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<code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">second</span><span class="special">(</span>
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<span class="identifier">q</span> <span class="special">,</span>
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<span class="identifier">dpdt</span> <span class="special">)</span></code>
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</p>
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</td>
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<td>
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<p>
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<code class="computeroutput"><span class="keyword">void</span></code>
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</p>
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</td>
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<td>
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<p>
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Calculates <span class="emphasis"><em>g(q)</em></span>, the result is stored into
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<code class="computeroutput"><span class="identifier">dpdt</span></code>
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
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and Mario Mulansky<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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</p>
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</div></td>
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</tr></table>
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<hr>
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<div class="spirit-nav">
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<a accesskey="p" href="system.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="simple_symplectic_system.html"><img src="../../images/next.png" alt="Next"></a>
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