|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{refimprove|date=September 2012}}
| | The author is known by the title of Numbers Wunder. To collect coins is what her family members and her enjoy. Minnesota is exactly where he's been living for many years. I used to be unemployed but now I am a librarian and the wage has been truly satisfying.<br><br>My homepage [http://www.gaysphere.net/blog/167593 over the counter std test] |
| In [[mathematics]], a '''separatrix''' refers to the boundary separating two modes of behaviour in a [[ordinary differential equation|differential equation]].
| |
| | |
| ==Example==
| |
| | |
| Consider the differential equation describing the motion of a simple [[pendulum]]:
| |
| | |
| :<math>{d^2\theta\over dt^2}+{g\over l} \sin\theta=0. </math>
| |
| | |
| where <math>l</math> denotes the length of the pendulum, <math>g</math> the [[gravitational acceleration]] and <math>\theta</math> the angle between the pendulum and vertically downwards. In this system there is a conserved quantity H (the [[Hamiltonian mechanics|Hamiltonian]]), which is given by
| |
| | |
| <math>H = \frac{\dot{\theta}^2}{2} - \frac{g}{l}\cos\theta.</math>
| |
| | |
| With this defined, one can plot a curve of constant H in the [[phase space]] of system. The phase space is a graph with <math>\theta</math> along the horizontal axis and <math>\dot{\theta}</math> on the vertical axis - see the thumbnail to the right. The type of resulting curve depends upon the value of H.
| |
| | |
| [[Image:Separatrix for a Simple Pendulum.png|thumb|The phase space for the simple pendulum]]
| |
| | |
| If <math>H<-\frac{g}{l}</math> then no curve exist (<math>\dot{\theta}</math> must be [[complex number|imaginary]]).
| |
| | |
| If <math>-\frac{g}{l}<H<\frac{g}{l}</math> then the curve will be a simple closed curve (see [[curve]]) which is nearly circular for small H and becomes "eye" shape when H approaches the upper bound. These curves correspond to the pendulum swinging periodically from side to side.
| |
| | |
| If <math>\frac{g}{l}<H</math> then the curve is open, and this corresponds to pendulum forever swinging through complete circles.
| |
| | |
| In this system the separatix is the curve that corresponds to <math>H=\frac{g}{l}</math>. It separates (hence the name) the phase space into two distinct areas. The region inside the separatrix has all those phase space curves which correspond to oscillating motion back and forth of the pendulum (whose equation of motion is given above), whereas the region outside the separatrix has all the phase space curves which correspond to the motion with the pendulum continuously turning through vertical planar circles.
| |
| | |
| ==External links==
| |
| * [http://mathworld.wolfram.com/Separatrix.html Separatrix] from [[MathWorld]].
| |
| | |
| | |
| {{mathapplied-stub}}
| |
| [[Category:Dynamical systems]]
| |
The author is known by the title of Numbers Wunder. To collect coins is what her family members and her enjoy. Minnesota is exactly where he's been living for many years. I used to be unemployed but now I am a librarian and the wage has been truly satisfying.
My homepage over the counter std test