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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.