Particle Orbits in Central Potential

Orbits of particles in a $r^{-1}$ potential are illustrated. The position of the central singularity is at $x=y=0$ . The particle orbits are calculated according to the equations of motion

$\ddot r = \frac{L^2}{r^3} -\frac{1}{r^2}$

$\dot \phi = \frac{L}{r}$

where $x=r\cos\phi$ and $y=r\sin\phi$ , the constant $L$ is a constant of integration corresponding to the angular momentum.

The black bar indicates the position from where the equations of motion are integrated. The numerical method used for the integration is a fourth order Runge-Kutta scheme. The initial position for all five particles is $(x,y) = (1, 0.25)$ . In addition all particles have the same initial velocity $\dot y=0$ , while the velocity $\dot x$ in $x-$ direction is different. The different values for the angular momentum $L=x\dot y-y\dot x$ result from this different value of $\dot x$ . The graphic shows how distinct initial values for a particle result in different orbits in a central potential.