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I am looking for an algorithm to solve numerically the following system of differential equations. It is the geodesic equation on for example a 2-dimensional manifold. So, let $\gamma(t)=(\gamma_1(t),\gamma_2(t))$ be the geodesic passing through point $\gamma(0)=(x_0,y_0)$ with initial velocity $\gamma'(0)=(v_1,v_2)$. I know Runge-Kutta method for solving a differential equation of first order but no idea about a system of second order differential equations.

Eqaution of geodesic: $$ \gamma''_1(t)+ \Gamma_{111}(\gamma(t)) \hspace{0.2cm}\gamma_1'(t)\gamma'_1(t)+\Gamma_{121}(\gamma(t)) \hspace{0.2cm}\gamma_2'(t)\gamma'_1(t)+\Gamma_{112}(\gamma(t)) \hspace{0.2cm}\gamma_1'(t)\gamma'_2(t)+ \Gamma_{122}(\gamma(t)) \hspace{0.2cm}\gamma_2'(t)\gamma'_2(t)=0\\ \gamma''_2(t)+ \Gamma_{211}(\gamma(t)) \hspace{0.2cm}\gamma_1'(t)\gamma'_1(t)+\Gamma_{221}(\gamma(t)) \hspace{0.2cm}\gamma_2'(t)\gamma'_1(t)+\Gamma_{212}(\gamma(t)) \hspace{0.2cm}\gamma_1'(t)\gamma'_2(t)+ \Gamma_{222}(\gamma(t)) \hspace{0.2cm}\gamma_2'(t)\gamma'_2(t)=0$$

The set of functions $\Gamma_{ijk}$ are Christoffel symbols and are real valued functions of 2-variables.

I got a suggestion about reducing the order of equations by defining extra functions $f=\gamma'_1$ and $g=\gamma'_2$ and writing down the associated relations and forming a new system. Yet I have my question unanswered because I don't know the algorithm even I found a similar question here. But the mentioned question tries to give an algorithm to find two functions $z,y$.

Edited question:

Can any one generalize mentioned question's algorithm for my problem?

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  • $\begingroup$ One way is to write your equations using functions of $(x',y',x,y,t)$ rather than $(x,y,t)$. Then you can manage with only first-order equations. It’s not the only way, so maybe you’ll get a better answer. $\endgroup$
    – David K
    Sep 14, 2020 at 17:43
  • $\begingroup$ @DavidK Dear David, you say that I can define other two functions let say, $f=\gamma'_1$ and $g=\gamma'_2$, then rewrite the system. Now, I have a system of first order differential equations? $\endgroup$
    – Baghban
    Sep 14, 2020 at 18:00
  • $\begingroup$ That is the general idea. For example page 5 of engr.colostate.edu/~thompson/hPage/CourseMat/Tutorials/… (but note that we would write $t$ where they write $x$). $\endgroup$
    – David K
    Sep 14, 2020 at 18:34
  • $\begingroup$ On this network, in scientific computing, see scicomp.stackexchange.com/questions/19020/… or scicomp.stackexchange.com/questions/34257/solving-coupled-odes for a discussion of this situation. Also check the "Related" side bar. $\endgroup$
    – Lutz Lehmann
    Sep 15, 2020 at 9:46

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