Physical meaning of theorem

Question

This is the image of theorem from V.I Arnold's Mathematical method of mechanics. I understood the example given in text. But I want to know what is physical meaning of example? Can anybody help?

Answer 1

FWIW, the example shows that a symplectic vector field $X$ is not necessarily a globally defined Hamiltonian vector field. If $X$ is the time-evolution vector field, it means physically that there is not a globally defined notion of energy.