Oribit integrator for a logarithmic potential

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1. Jan 25, 2016

Serna

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.

2. Jan 25, 2016

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.

3. Jan 28, 2016