Namespaces
Variants
Actions

Difference between revisions of "Phase velocity vector"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
The vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072620/p0726201.png" /> originating at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072620/p0726202.png" /> of the [[Phase space|phase space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072620/p0726203.png" /> of the [[Autonomous system|autonomous system]]
+
{{TEX|done}}
 +
The vector $f(x)$ originating at a point $x$ of the [[Phase space|phase space]] $G$ of the [[Autonomous system|autonomous system]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072620/p0726204.png" /></td> </tr></table>
+
$$\dot x=f(x),\quad f\in C^1(G),\quad G\subset\mathbf R^n.$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072620/p0726205.png" /> be the [[Phase trajectory|phase trajectory]] of the system passing through a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072620/p0726206.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072620/p0726207.png" />, then the phase velocity vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072620/p0726208.png" /> is tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072620/p0726209.png" /> and represents the instantaneous rate of the motion along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072620/p07262010.png" /> of a representative point of the system at the moment of passing through the position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072620/p07262011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072620/p07262012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072620/p07262013.png" /> is an [[Equilibrium position|equilibrium position]].
+
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
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article