The free oscillator : the simple pendulum without any friction ( non-dissipativ system).

The picture below show the different components of this system.

The fundamental principle of dynamic permit to write the equation:

For a small angle, this equation can be simplified:

This is a periodical motion, which solution is

T is the period and w the pulsation.

Two values are sufficient to describe the whole instantaneous movement : the angle (q ) and the angular speed (dq /dt).

Furthermore it is not necessary to integrate because of the representation that can be used to describe the movement. You can easily express the angular speed (dq /dt) in function of the angle (q ). The geometrical representation is called "portrait des phases".

In this case you can use E(q ,dq /dt):

This is proportional to the energy of the pendulum.

So the curves solution of the equation are isoenergetical curves in the phase-space ("espace des phases"). Cf. the picture below.

• the circles centered around of q =0± 2pn (nÎ N) correspond to isochronic oscillations. The period is independant of the magnitude and of the mass (linear approximation).
• But, in the non-linear domain, the period of the oscillations increase when the magnitude and so the energy increase... The circles become ovals. And when the energy is higher than 2g, the oscillations are replaced by rotations in one of the two ways possible (there are two equal possibilities).