What's the difference between time-dependent flow (isotopy) and time-independent flow? 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)?
 A: I'll use Amitai Yuval's notation.
Notice that if $X_t=X$ does not depend on $t$, then the equation
$$\frac{d}{dt}\varphi_t(p)=X(\varphi_t(p))$$
means that $\varphi$ is the flow of $X$. In particular, $\varphi_t\circ\varphi_s=\varphi_{t+s}$ for all $t,s\in\mathbb{R}$ (which is a natural property of the flow).
Conversely, assume $\varphi_t\circ\varphi_s=\varphi_{t+s}$ $\forall t,s\in\mathbb{R}$. In particular $\varphi_t^{-1}=\varphi_{-t}$. By definition of $X_t$ we have
\begin{align*}
X_t(p)&=\left.\frac{d}{ds}\right|_{s=t}\varphi_s(\varphi_t^{-1}(p))\\
&=\left.\frac{d}{ds}\right|_{s=t}\varphi_{s-t}(p)\\
&=\left.\frac{d}{du}\right|_{u=0}\varphi_u(p),\,\,\,\text{where }u=s-t.
\end{align*}
This shows that $X_t$ does not depend on $t$. So we have just proved that

$X_t$ is independent of $t\Leftrightarrow$ $\varphi_t\circ\varphi_s=\varphi_{t+s}$ for all $t,s\in\mathbb{R}$.

A: 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.
