**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.