Editing Inverted pendulum (section)

===Kapitza's pendulum===
{{main|Kapitza's pendulum}}

An inverted pendulum in which the pivot is oscillated rapidly up and down can be stable in the inverted position.  This is called [[Kapitza's pendulum]], after Russian physicist [[Pyotr Kapitza]] who first analysed it.  The equation of motion for a pendulum connected to a massless, oscillating base is derived the same way as with the pendulum on the cart. The position of the point mass is now given by:
:<math>\left( -\ell \sin \theta , y + \ell \cos \theta    \right)</math>
and the velocity is found by taking the first derivative of the position:
:<math>v^2=\dot y^2-2 \ell \dot y \dot \theta \sin \theta  + \ell^2\dot \theta ^2.</math>[[Image:Inverted pendulum oscillatory base.svg|thumb|right|500px|Plots for the inverted pendulum on an oscillatory base. The first plot shows the response of the pendulum on a slow oscillation, the second the response on a fast oscillation]]

The [[Lagrangian mechanics|Lagrangian]] for this system can be written as:
:<math>
 L = \frac{1 }{2} m \left ( \dot y^2-2 \ell \dot y \dot \theta \sin \theta  + \ell^2\dot \theta ^2   \right) - m g \left( y + \ell \cos \theta  \right )
</math>
and the equation of motion follows from:
:<math>
{\mathrm{d} \over \mathrm{d}t}{\partial{L}\over \partial{\dot \theta}} - {\partial{L}\over \partial \theta} = 0
</math>
resulting in:
:<math>
\ell \ddot \theta - \ddot y \sin \theta = g \sin \theta.
</math>
If ''y'' represents a [[simple harmonic motion]], <math>y = A \sin \omega t</math>, the following [[differential equation]] is:
:<math>
\ddot \theta - {g \over \ell} \sin \theta = -{A \over \ell} \omega^2 \sin \omega t \sin \theta.
</math>

This equation does not have elementary closed-form solutions, but can be explored in a variety of ways.  It is closely approximated by the [[Mathieu equation]], for instance, when the amplitude of oscillations are small. Analyses show that the pendulum stays upright for fast oscillations. The first plot shows that when <math>y</math> is a slow oscillation, the pendulum quickly falls over when disturbed from the upright position. The angle <math>\theta</math> exceeds 90° after a short time, which means the pendulum has fallen on the ground. If  <math>y</math> is a fast oscillation the pendulum can be kept stable around the vertical position.  The second plot shows that when disturbed from the vertical position, the pendulum now starts an oscillation around the vertical position (<math>\theta = 0</math>). The deviation from the vertical position stays small, and the pendulum doesn't fall over.
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