The parametric oscillator : the simple pendulum without friction in a various gravitational range.
In fact, this is an alternativ movement of the point O that reproduces such conditions.
We have a new expression for the gravity :
The expression of the dynamic equation is now:
This equation is not integrable in the case of an ordinary law for g(t).
Moreover the origin point is not necessary stable...
We take g periodical (Hill's equation) with b circular (Mathieu's equation):
So the equation becomes:
The solutions can be written under the form below:
Stability of the solutions:
For m small, and w» wo.
We obtain this diagram of stability: