I am trying to visualize some mechanical systems that have the origin in the phase plane ($q, \dot{q}$) as an unstable equilibrium. And to show that the equilibrium is unstable I want to show the existence of an asympotic trajectory to the origin ($q = 0, \dot{q} = 0$) when $t \rightarrow -\infty$.
My questions: Do you have any example of this? (maybe the unstable point of the pendulum, but I wasn't able to show an asymptotic trajectory to the origin).
In the case of mechanical systems with unstable equibrium, the system cannot be conservative, right? I was wondering if it can be possible to have an unstable equilibrium in a conservative system.
Examples are
an upright pendulum (on a rod, not on a string obviously), or balancing a rod on your nose,
a roulette wheel, if it were just the bowl with the hill in the middle.