inverted nonlinear pendulum, show unstable.
Hey there. I have a pendulum quandary.
Suppose we have a nonlinear pendulum. How can we show that the inverted
position is unstable?. And what is the exponential behavior of the angle in the
neighborhood of this unstable equilibrium position?.
I know the inverted position is at $\displaystyle {\theta}={\pi}$.
If even one initial condition causes the solution to tend away from equilibrium then it is unstable. But I do not know how to show that, much less explain the exponential behavior. I am thinking we can write the angle in terms of sine and cosine.
linearized stability analysis shows that if $\displaystyle f'(x_{E})<0$, then it is unstable. The displacement from equilibrium will grow exponentially for most initial conditions.
It would appear I need to take the derivative of some function using pi and show it is < 0. $\displaystyle sin(\pi)=0, \;\ cos(\pi)=1$
Does anyone have any thoughts?. Thank you.
