What's the difference between time-dependent flow (isotopy) and time-independent flow?

Question

Regarding the fact that both time-independent and time-dependent vector fields correspond with family of diffeomorphisms, i.e. $\{\phi_t | t\in\Re, \phi_t: M\to M\}$, what's the difference between these two families, i.e, time-independent and time-dependent flows (isotopy)?

Answer 1

Let $\varphi_t:M\to M,\;t\in \mathbb{R}$, be a smooth family of diffeomorphisms. "Smooth" means here that the corresponding map $$\varphi:M\times\mathbb{R}\to M,\quad(p,t)\mapsto\varphi_t(p),$$ is smooth. For every $t$, define the vector field $X_t$ by $$X_t(p)=\left.\frac{d}{ds}\right|_{s=t}\varphi_s\left(\varphi_t^{-1}(p)\right).$$Then the family $(\varphi_t)$ is generated by the time-dependent vector field $X_t$, in the sense that for every $q$ and $t$ we have $$\frac{d}{dt}\varphi_t(q)=X_t(\varphi_t(q)).$$In other words, the family $(\varphi_t)$ is the solution to the ODE given by the time-dependent vector field $X_t$. Now, in very specific cases $X_t$ may turn out to be independent of $t$, but it does not change anything, essentially.