Oribit integrator for a logarithmic potential

[Serna]

Hello! Right know I'm trying to make an orbit integrator for solving a logarithmic potential with the form:
\begin{equation}
\Phi= \frac{v_0^2}{2} ln(x^2+ \frac{y^2}{u^2} + r_0^2)
\end{equation}
where v₀, u, and r₀ are constants
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.

 

[Dr. Courtney]

If by decouple the system, you mean separate the variables, it probably is not possible. But Runge-Kutta can be used to integrate the equations of motion as is. The Hamiltonian formulation (four equations with first derivatives) is usually easier.

 

[Serna]

Hello, thanks for your response!

By doing the Hamiltonian approach I still get equations (3) and (4) above, and the other two are are apparently of no use.
The problem is that I don't know how (if possible) to adapt the Runge-Kutta using two dependent variables (x,y) and the independent one (t).

 

[Dr. Courtney]

The problem is that I don't know how (if possible) to adapt the Runge-Kutta using two dependent variables (x,y) and the independent one (t).

Sure it is possible. It is very straightforward. Just apply the method to all four equations at every step of the code.