Activities

• Select Graph to view the evolution in the equivalent one-dimensional problem.
• Select different initial condition and parameters and observe that, in general, orbits are open, i.e., non periodic. This is what one would expect due to Bertrand's theorem (which states that in a central forces field all bounded orbits are closed, the force is Newtonian, F∝-r^-2, or harmonic, F∝-r.
• However, nothing prevents the existence of isolated periodic orbits. Try, for instance,
  x = -0.854775390624998, y = 1.155, vx = 0, vy = -0.1, k/m = 1.
• How small must be a change in the above initial conditions to get an orbit which is quickly recognized as non periodic?
• With a bit of patience, you may ba able to find (good approximations to) other periodic orbits.