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In Arnold's Classical Mechanics book, he says "We assume that every solution to Hamilton's equations can be extended to the whole time axis", and adds that 'For this it is sufficient, for example, that the level sets of $H$ be compact'. How so?

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He is assuming that there are no local issues with the solution of the differential equation, which is reasonable. This means that you can always extend the solution a little bit in time around any point.

Then the only way that a solution will fail to extend in time is if the limit at some point is not in the phase space, so that the solution runs away to infinity. An example is for a Hamiltonian of the form

$$p^2 - x^4$$

If you start off at a certain position, by conservation of energy, the speed of the particle when it reaches position x goes as ${x^2}$ and the time required to reach infinity is proportional to the integral

$$ \int {1\over x^2} dx$$

which is convergent. There are similar blow ups in finite time in many systems where the energy isn't bounded below, including Newtonian gravitating point particles. He is pointing out that if the energy surface is compact, you can't run away.

The formal statement is that the local extension property allows you to continue from any point a little ways, and the compactness property means that the limit of the trajectory is in the space, so that you have a standard connectedness proof of global existence.

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