How Can the Stability of a Kapitza Pendulum Be Demonstrated?

AI Thread Summary
The discussion focuses on the stability of a Kapitza pendulum, particularly when the condition $$A_0 \gg g$$ allows for the simplification of the equation of motion. Participants express confusion regarding the stability of the pendulum, especially with the parameter ##T_d##, noting that when ##T_d=\infty##, the inverted pendulum becomes unstable. There is acknowledgment that no specific conditions for ##T_d## were provided, leading to uncertainty about the problem's formulation. The stability of the Kapitza pendulum is ultimately linked to KAM theory, which is considered advanced and not typically covered in undergraduate studies. The conversation highlights the complexities and misunderstandings surrounding the mechanics of the Kapitza pendulum.
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Homework Statement
I am having confusion with regards to the proof of my problem. This involves a pendulum with harmonically driven pivot and the task is to show that the pendulum is stable in the inverted position when the amplitude of the driving acceleration is sufficiently high. For reference, consider the problem below.

> Consider a pendulum with harmonically driven pivot. The equation of motion is
$$\frac{d^2\theta}{dt^2}=-\frac{g+a_d(t)}{L}\sin\theta$$
where $$a_d(t)=A_0\sin(2\pi t/T_d)$$ is the time-varying acceleration of the pivot. Show that when the amplitude of the driving acceleration is sufficiently high $$A_0\gg g$$ the pendulum is stable in the inverted position i.e., if $$\theta(t=0)\approx180°$$.
Relevant Equations
$$\frac{d^2\theta}{dt^2}=-\frac{g+a_d(t)}{L}\sin\theta$$
where $$a_d(t)=A_0\sin(2\pi t/T_d)$$ is the time-varying acceleration of the pivot.
I understand that when $$A_0 \gg g$$, the g term in the equation of motion can be dropped. The equation of motion then becomes
$$\frac{d^2\theta}{dt^2}=-\frac{a_d(t)}{L}\sin\theta$$

But how can I show that the pendulum is stable for such case? I am totally clueless.
 
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Is there any condition on ##T_d##? Say ##T_d=\infty## the inverted pendulum is unstable.
 
looks like an ill - understood Kapitza's pendulum
 
mitochan said:
Is there any condition on ##T_d##? Say ##T_d=\infty## the inverted pendulum is unstable.
Hi, mitochan! There were no specified conditions for ##T_d## given in the problem.
 
wrobel said:
looks like an ill - understood Kapitza's pendulum
It is quite consistent with the mechanics of Kapitza's pendulum. I presume that something is wrong in the problem, right?
 
I do not know.

Anyway stability of the Kapitza pendulum is proved by means of KAM theory. All this field is far beyond undergraduate courses.
 
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