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Phase velocity vector

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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)


Comments

References

[a1] V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Springer (1983) (Translated from Russian)
How to Cite This Entry:
Phase velocity vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Phase_velocity_vector&oldid=14363
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article