Difference between revisions of "Phase velocity vector"
From Encyclopedia of Mathematics
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| − | The vector | + | {{TEX|done}} |
| + | The vector $f(x)$ originating at a point $x$ of the [[Phase space|phase space]] $G$ of the [[Autonomous system|autonomous system]] | ||
| − | + | $$\dot x=f(x),\quad f\in C^1(G),\quad G\subset\mathbf R^n.$$ | |
| − | Let | + | Let $\Gamma$ be the [[Phase trajectory|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|equilibrium position]]. |
====References==== | ====References==== | ||
Latest revision as of 15:27, 22 September 2014
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=33358
Phase velocity vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Phase_velocity_vector&oldid=33358
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article