# Oribit integrator for a logarithmic potential

Hello! Right know I'm trying to make an orbit integrator for solving a logarithmic potential with the form:

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

where v0, u, and r0 are constants
My approach is to use,

\ddot{q} = -\bigtriangledown \Phi

Then the system equations,

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

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

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.

## Answers and Replies

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Dr. Courtney
Education Advisor
Gold Member
2020 Award
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.

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
Education Advisor
Gold Member
2020 Award
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.