Phase velocity vector

The vector $f(x)$ originating at a point $x$ of the phase space $G$ of the autonomous system

$$\dot x=f(x),\quad f\in C^1(G),\quad G\subset\mathbf R^n.$$

Let $\Gamma$ be the phase trajectory of the system passing through a point $\xi\in G$; if $f(\xi)\neq0$, then the phase velocity vector $f(\xi)$ is tangent to $\Gamma$ and represents the instantaneous rate of the motion along $\Gamma$ of a representative point of the system at the moment of passing through the position $\xi\in\Gamma$. If $f(\xi)=0$, then $\xi\in G$ is an equilibrium position.

References

[1]	L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)

References

[a1]	V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Springer (1983) (Translated from Russian)