# Daily Archives: December 30, 2007

## The Painleve Equations of Motion

In the general theory of relativity, the orbits are given by geodesics. A geodesic is a path that extremizes the path length. The path length is defined as the integral of $ds$ over the path, where $ds^2$ is the metric. For the case of Painleve coordinates on the Schwarzschild metric, $ds^2$ is given by:
. Let’s let our path start at time t=0 and end at time t=1. For the path to be a geodesic, we must extremize the following integral (I’ll quickly sneak in a minus sign to make the path be timelike instead of spacelike):

To make life easier for us, we will make the assumption that the orbital motion is in the $\theta = \pi/2$ plane so there’s no $\theta$ dependence. That turns the angular part of the square root into $r^2\;(d\phi/dt)^2$. Furthermore, since the simulation is going to use Cartesian, (x,y) coordinates, we might as well replace $r^2\;(d\phi/dt)^2$ with $(x\;dy/dt-y\;dx/dt)^2$, and $dr/dt$ with $x\;dx/dt + y\;dy/dt$, their Cartesian equivalents. And put M=1, we can always fix it later by dimensional analysis.