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which is nearly circular for small H and becomes "eye" shaped when H approaches the upper bound. These curves correspond to the pendulum swinging periodically from side to side.
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741:. Trajectories to the left of the separatrix converge to the left stable equilibrium, and similarly for the right. The separatrix itself is the
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the angle between the pendulum and vertically downwards. In this system there is a conserved quantity H (the
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on the vertical axis – see the thumbnail to the right. The type of resulting curve depends upon the value of
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then the curve is open, and this corresponds to the pendulum forever swinging through complete circles.
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380:{\displaystyle H={\frac {{\dot {\theta }}^{2}}{2}}-{\frac {g}{\ell }}\cos \theta .}
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The separatrix is clearly visible by numerically solving for trajectories
233:{\displaystyle {d^{2}\theta \over dt^{2}}+{g \over \ell }\sin \theta =0.}
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Consider the differential equation describing the motion of a simple
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for the saddle point in the middle. Details are found in the page.
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575:{\displaystyle -{\frac {g}{\ell }}<H<{\frac {g}{\ell }}}
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782:, 3rd Ed., 2006, John Wiley and Sons, Hoboken, NJ, pg. 65.
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is the boundary separating two modes of behaviour in a
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With this defined, one can plot a curve of constant
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771:, 4th ed., 2012, Brooks/Cole, Boston, MA, pg. 469.
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490:{\displaystyle H<-{\frac {g}{\ell }}}
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615:{\displaystyle {\frac {g}{\ell }}<H}
582:then the curve will be a simple closed
457:The phase space for the simple pendulum
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655:{\displaystyle H={\frac {g}{\ell }}}
263:denotes the length of the pendulum,
55:adding citations to reliable sources
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629:is the curve that corresponds to
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439:{\displaystyle {\dot {\theta }}}
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718:{\displaystyle b=2,I_{ext}=3.5}
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18:Separatrix (dynamical systems)
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66:"Separatrix" mathematics
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285:gravitational acceleration
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300:{\displaystyle \theta }
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62:Find sources:
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40:This article
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49:Please help
44:verification
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395:phase space
309:Hamiltonian
131:mathematics
793:Separatrix
756:References
739:separatrix
627:separatrix
135:separatrix
77:newspapers
797:MathWorld
727:backwards
648:ℓ
602:ℓ
568:ℓ
549:ℓ
541:−
528:imaginary
511:˙
508:θ
483:ℓ
475:−
431:˙
428:θ
405:θ
372:θ
369:
361:ℓ
353:−
336:˙
333:θ
295:θ
251:ℓ
222:θ
219:
211:ℓ
182:θ
807:Category
729:in time.
526:must be
156:pendulum
145:Examples
733:In the
393:in the
91:scholar
243:where
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795:from
674:When
584:curve
98:JSTOR
84:books
607:<
560:<
554:<
472:<
287:and
283:the
133:, a
70:news
713:3.5
589:If
533:If
530:).
461:If
366:cos
216:sin
129:In
53:by
809::
450:.
228:0.
158::
141:.
799:.
710:=
705:t
702:x
699:e
695:I
691:,
688:2
685:=
682:b
645:g
640:=
637:H
610:H
599:g
565:g
557:H
546:g
480:g
469:H
448:H
391:H
375:.
358:g
348:2
343:2
324:=
321:H
271:g
225:=
208:g
203:+
195:2
191:t
187:d
177:2
173:d
120:)
114:(
109:)
105:(
95:·
88:·
81:·
74:·
47:.
20:)
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