[pstricks] Searching for a numeric solution for coupled differential equation systems of order 2

[Alexander Grahn]

On Fri, May 25, 2012 at 01:52:18PM +0300, Manfred Braun wrote:
>On 25.05.2012, at 12:15, Alexander Grahn wrote:
>
>> On Thu, May 24, 2012 at 10:21:06PM +0200, Juergen Gilg wrote:
>>> Dear PSTricks list,
>>> 
>>> is there anybody out there, knowing about how to generate a PS code for 
>>> "Runge-Kutta 4" to solve a "coupled differential equation system of 
>>> order 2"?
>>> 
>>> There is the need for such a code to animate a double-pendulum in real 
>>> time, following the coupled differential equations with the variables 
>>> (\varphi_1, \varphi_2) and their derivatives like:
>>> 
>>> (1) 
>>> l_1\ddot{\varphi}_1+\frac{m_2}{m_1+m_2}l_2\ddot{\varphi}_2\cos(\varphi_1-\varphi_2)-\frac{m_2}{m_1+m_2}l_2\dot{\varphi}_2^2\sin(\varphi_1-\varphi_2)+g\sin\varphi_1=0
>>> 
>>> (2) 
>>> l_2\ddot{\varphi}_2+l_1\ddot{\varphi}_1\cos(\varphi_1-\varphi_2)-l_1\dot{\varphi}_1^2\sin(\varphi_1-\varphi_2)+g\sin\varphi_2=0
>> 
>> I'd first transform this set of DEs of second order into a set of four
>> DEs of first order,
>
>Yes, this is the right way.
>> 
>> However, it may still not be amenable to Runge-Kutta, because the right
>> hand sides of the resulting four equation set of DEs, put into normal
>> form, still contain derivatives:
>
>No! Take the following steps:
>
>1. Solve the two equations for \ddot{\varphi_1} and \ddot{\varphi_2}:
>    \ddot{\varphi}_1 = f_1(\varphi_1, \dot{\varphi}_1, \varphi_2, \dot{\varphi}_2)
>    \ddot{\varphi}_2 = f_2(\varphi_1, \dot{\varphi}_1, \varphi_2, \dot{\varphi}_2)
>
>2. Define  x_1 = \varphi_1,  x_2 = \dot{\varphi}_1,  x_3 = \varphi_2,  x_4 = \dot{\varphi}_2
>
>3. The new variables satisfy the system of four 1st-order equations
>    \dot{x}_1 = x_2
>    \dot{x}_2 = f_1(x_1, x_2, x_3, x_4)
>    \dot{x}_3 = x_4
>    \dot{x}_4 = f_2(x_1, x_2, x_3, x_4)
>
>This can be solved by Runge-Kutta or any other procedure.
>
>Manfred

Thanks, Manfred! In the meantime I tried to write the set of equations
down. Here is the result, without errors I hope:

The equations can be solved further to eliminate derivatives on the right
hand side. The resulting set of first order ODEs can be integrated by
\odesolve from the animate package documentation, provided the ODEs are
not stiff. Initial conditions for $\varphi_1$, $\dot\varphi_1$,
$\varphi_2$, $\dot\varphi_2$ are required:

\documentclass{article}
\usepackage{amsmath}
\parindent0pt
\begin{document}

\section{Original set of DEs of second order}

\begin{gather*}
l_1\ddot{\varphi}_1+\frac{m_2}{m_1+m_2}l_2\ddot{\varphi}_2\cos(\varphi_1-\varphi_2)-\frac{m_2}{m_1+m_2}l_2\dot{\varphi}_2^2\sin(\varphi_1-\varphi_2)+g\sin\varphi_1=0\\
l_2\ddot{\varphi}_2+l_1\ddot{\varphi}_1\cos(\varphi_1-\varphi_2)-l_1\dot{\varphi}_1^2\sin(\varphi_1-\varphi_2)+g\sin\varphi_2=0
\end{gather*}

\section{Set of first order ODEs in normal form, amenable to Runge-Kutta}

\begin{align*}
\dot\varphi_{11}& = \varphi_{12}\\
\dot\varphi_{12}& = \frac{\frac{a_1-b_1}{l_1}-\frac{c_1}{l_1}\frac{a_2-b_2}{l_2}}{1-\frac{c_1 c_2}{l_1 l_2}}\\
\dot\varphi_{21}& = \varphi_{22}\\                           
\dot\varphi_{22}& = \frac{\frac{a_2-b_2}{l_2}-\frac{c_2}{l_2}\frac{a_1-b_1}{l_1}}{1-\frac{c_1 c_2}{l_1 l_2}}\\
\end{align*}
with
\begin{align*}
a_1& = \frac{m_2}{m_1+m_2} l_2 \varphi_{22}^2 \sin(\varphi_{11}-\varphi_{21})\\
b_1& = g \sin\varphi_{11}\\
c_1& = \frac{m_2}{m_1+m_2} l_2 \cos(\varphi_{11}-\varphi_{21})\\
a_2& = l_1 \varphi_{12} \sin(\varphi_{11}-\varphi_{21})\\
b_2& = g \sin\varphi_{21}\\
c_2& = l_1 \cos(\varphi_{11}-\varphi_{21})
\end{align*}
\end{document}

Alexander