Symplectomorphism, intuitive interpretation.

Question

What is an intuitive way to think about symplectomorphisms? A symplectomorphism between two symplectic spaces is a map $(M_1, \omega_1)\xrightarrow{\phi} (M_2,\omega_2)$ such that for the pullback $\phi^*\omega_2=\omega_1$. But how to think about $\phi$? What would it typically represent in classical mechanics?

Answer 1

You might have heard them called "canonical transformations" or "contact transformation",they are changes of coordinates that don't change the form of Hamilton's equations of motion.