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Oribit integrator for a logarithmic potential

  1. Jan 25, 2016 #1
    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.
     
  2. jcsd
  3. Jan 25, 2016 #2
    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.
     
  4. Jan 28, 2016 #3
    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).
     
  5. Jan 28, 2016 #4
    Sure it is possible. It is very straightforward. Just apply the method to all four equations at every step of the code.
     
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