I am studying Chapter 3, section 2.14 of the book Stability Theory by Lyapunov Direct Method by Laloy.
Given the equations of motion
$\dot{q} = \frac{\partial H}{\partial p}(q, p)$ and $\dot{p} = -\frac{\partial H}{\partial q}(q, p)$, where $H(q, p) = T(q, p) + \pi(q)$.
$T(q, p)$ is the kinetic energy and is positive definite and is $C^1$. $\pi(q)$ is the potential energy, is $C^1$, and $\pi(0) = 0$ and $\frac{\partial \pi}{\partial q}(0) = 0$.
Suppose that the origin q = p = 0 is unstable, that is, there exists an $\eta >0$ such that for all open set $0\in \phi\subset B_\eta$, there exists q$\in \partial \phi$ such that $\pi(q)\leq 0$.
The author says that the set A = {$q \in R^n | \pi(q) \leq 0$ or $||q|| \geq \eta$} is closed and contains the origin. But I din't understand how to show this.
Also, the author says: let $A_0$ be a connected component of A which contains the origin, and suppose that $A_0\in B_\eta$. Then $A_0$ is compact and disjoint from A.
I am having problem in visualizing the set $A_0$. More specifically, I didn't understand how is $A_0$ a connected component of A. Is it a curve starting at the origin, formed by the q's where $\pi(q)\leq 0$?