Hello! Right know I'm trying to make an orbit integrator for solving a logarithmic potential with the form:(adsbygoogle = window.adsbygoogle || []).push({});

\begin{equation}

\Phi= \frac{v_0^2}{2} ln(x^2+ \frac{y^2}{u^2} + r_0^2)

\end{equation}

where,v_{0}, anduare constantsr_{0}

My approach is to use,

\begin{equation}

\ddot{q} = -\bigtriangledown \Phi

\end{equation}

Then the system equations,

\begin{equation}

\ddot{x} = -v_o^2 \frac{x}{x^2+ \frac{y^2}{u^2} + r_0^2}

\end{equation}

\begin{equation}

\ddot{y} = -\frac{v_o^2}{u^2} \frac{y}{x^2+ \frac{y^2}{u^2} + r_0^2}

\end{equation}

My guess is that in order to solve for x and y using Runge Kutta or leapfrog, I need to decouple the system, but I don't know how to do so.

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# Oribit integrator for a logarithmic potential

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