Existence Of Phase Flow Provided Potential Energy is Positive

Question

I am reading through Arnold's "Mathematical Methods Of Classical Mechanics". In the section 4D on p. 21 concerning Phase Flow there is a question that reads as follows:

Show that if Potential Energy is postive, then there is a phase flow. Hint: Use the law of conservation of energy to show that a solution can be extended without bound.

I am stumped. Can somebody give me any other hints?

For context, Arnold defines a phase flow as a one-parameter group of diffeomorphisms of the phase plane to itself (the parameter in question being time so that the position of a point $M$ in the phase plane can be traced for all time $t$).

Answer 1

Hint: Given the energy integral $\frac{1}{2}\dot{x}^2+U(x)=E$, if the potential energy is bounded from below, the speed is bounded from above, so that a solution cannot reach spatial infinity in finite time, i.e. the flow can in principle be extended to the whole time-axis $\mathbb{R}$, which is one of the conditions of a phase flow listed on p. 20 (and the last line of OP).