Let $a,b \in \mathbb{R^2}$ and consider $U: \mathbb{R^2}-\{a,b\} \to \mathbb{R}$ be a smooth function which satisfies $\limsup_{|q| \to \infty} |U(q)| =1.$ Consider the system of ODEs for $(p(t),q(t)) \in \mathbb{R^2} \times (\mathbb{R^2} - \{a,b\}):$ $$ \begin{cases} \dot{p} = \nabla U(q(t)) & \\ \dot{q} = p, & \end{cases} $$ with initial data prescribed at $t = 0.$
Show that for $0 < T < \infty,$ then $\sup_{t \in [0,T)} |p(t)| < \infty$ and $\sup_{t \in [0,T)} |q(t)| <\infty,$ where $(p(t),q(t))$ is a solution to the system above. Furthermore, show that if $[0,T)$ is the maximal interval of existence, then $\lim_{t\to T} q(t) = a \text{ or } b$.
I am not really sure where to start such a problem... I am posting here mostly just for people's intuition on how to start such problems (solutions are always helpful too). I can recognize that this system is essentially Newton's law, where the potential is $-U(x),$ the position is $q,$ and momentum is $p.$
As for showing the limit of the position approaches $a$ or $b,$ we have that the system can be written as the gradient of a scalar field, so therefore there can not be any limit cycles. Since there can't be any limit cycles and the solutions don't blow up (from the previous part), we must have that $q$ approaches $a$ or $b$. This is the argument I had in mind... but I have no clue how to formalize this mathematically.
Thanks for your comments in advance!