In my previous question, I asked about the Galilean invariance of the Hamiltonian. I've got already two answers, probably good but I have difficulties interpreting them. Both answers write the Hamiltonian with the coordinates of the moving observer. I cannot interpret this. The Hamiltonian is a function defined on the phase space. Since phase space doesn't contain the time, I see no sense of such thing as "Hamiltonian from the viewpoint of a moving observer", or such expression as $H(x+vt,p+mv)$. From a physical viewpoint, of course, I understand this but I don't see the clear mathematical model behind it. I can express the Galilean invariance of the Hamiltonian function only so that I require that for each solution $(x(t),p(t))$ of the Hamiltonian equations $(x(t)+vt,p'(t))$ is also a solution with some function $t\mapsto p'(t)$. This is a different thing than "transforming" the Hamiltonian. So, my question is, how can be modeled the Galilean invariance of the Hamiltonian mathematically correctly?