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.