The pendulum in the free oscillation case Image from F.Dabireau

Equations

If we write the momentum equation in O with :
- a the rope length
- I the inertial momentum for the pendulum in O, in the direction perpendicular to the movement

we obtain: That is equal to: (1)

With (in the case of a non-weighting rope): The equation (1) can be rewritten in a system: Solutions

The solution of this system in the phase plan (u,v which means angle,speed) is an ellipse or an periodic curve, as we can see in the following picture: Image F.Dabireau

Physically, the pendulum is never stopped by any friction. So it will always keep its energy. If there is oscillations, it gives an ellipse. If the initial condition on the pendulum makes him do a complete rotation, it will continue to turn. it gives a periodic diagram with an increasing angle and a periodic speed.

In the case of small oscillation, we can write the conservation of the energy and we see that it leads to an ellipse.   So, if we the energy is constant, it leads to the following equation: Results of the program  We can see that the results are similar to the theory. The numerical scheme used seems to be stable. The temporal evolution is periodic.