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| | Let me initial begin by introducing myself. My name is Boyd Butts although it is not the title on my birth certificate. Since she was 18 she's been working as a receptionist but her promotion by no means arrives. South Dakota is where me and my husband live. Doing ceramics is what adore performing.<br><br>Review my web page; [http://www.article-galaxy.com/profile.php?a=165109 www.article-galaxy.com] |
| In [[mathematics]], a '''separatrix''' refers to the boundary separating two modes of behaviour in a [[ordinary differential equation|differential equation]].
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| ==Example==
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| Consider the differential equation describing the motion of a simple [[pendulum]]:
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| :<math>{d^2\theta\over dt^2}+{g\over l} \sin\theta=0. </math>
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| 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
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| <math>H = \frac{\dot{\theta}^2}{2} - \frac{g}{l}\cos\theta.</math>
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| 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.
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| [[Image:Separatrix for a Simple Pendulum.png|thumb|The phase space for the simple pendulum]]
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| If <math>H<-\frac{g}{l}</math> then no curve exist (<math>\dot{\theta}</math> must be [[complex number|imaginary]]).
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| 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.
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| If <math>\frac{g}{l}<H</math> then the curve is open, and this corresponds to pendulum forever swinging through complete circles.
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| 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.
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| ==External links==
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| * [http://mathworld.wolfram.com/Separatrix.html Separatrix] from [[MathWorld]].
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| {{mathapplied-stub}}
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| [[Category:Dynamical systems]]
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Let me initial begin by introducing myself. My name is Boyd Butts although it is not the title on my birth certificate. Since she was 18 she's been working as a receptionist but her promotion by no means arrives. South Dakota is where me and my husband live. Doing ceramics is what adore performing.
Review my web page; www.article-galaxy.com