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[[File:Henon heiles potential.svg|thumb|Contour plot of the Hénon-Heiles potential]]
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The '''Hénon-Heiles equation''' is used to model [[star]]s. It is expressed as<ref>{{Citation
|first=        Michel
|last=          Hénon
|title=          Chaotic Behaviour of Deterministic Systems
|chapter=    Numerical exploration of Hamiltonian Systems
|pages=      53–170
|editor-first = G.
|editor-last=  Iooss
|publisher= Elsevier Science Ltd
|year=        1983
|isbn=        044486542X
}}</ref> 
:<math>
V(x,y) = \frac{1}{2}(x^2+y^2+2 x^2y - \frac{2}{3}y^3)
</math>
 
While at [[Princeton University|Princeton]] in 1962, [[Michel Hénon]] and [[Carl Heiles]] worked on the non-linear motion of a star around a galactic center where the motion is restricted to a plane. They published a paper<ref>{{cite journal
|author=    Hénon, M.; Heiles, C.
|title=        The applicability of the third integral of motion: Some numerical experiments
|journal=  The Astrophysical Journal
|volume=  69
|pages=    73–79
|year=      1964
|bibcode= 1964AJ.....69...73H
|doi = 10.1086/109234 }}</ref>
that describes their work in 1964.
 
The Hénon-Heiles System (HHS) is defined by the following four equations:
:<math>
dx/dt=u
</math>
:<math>
dy/dt=v
</math>
:<math>
du/dt=-Ax-2xy
</math>
:<math>
dv/dt=-By+\epsilon y^2-x^2
</math>
 
where <math> A,B, \epsilon \in \R, A > 0</math> and <math>B > 0 </math>. Since HHS is specified in <math>\R^2</math>, we need a Hamiltonian of degrees of freedom two to model it.
 
It can be solved for some cases using Painlevé Analysis.
The Hamiltonian for the HHS is
 
:<math> H=\frac1 2 (u^2+v^2+Ax^2+By^2 )+x^2 y-\frac 1 3 \epsilon y^3</math>
 
== References ==
{{Reflist}}
== External links ==
* http://mathworld.wolfram.com/Henon-HeilesEquation.html
 
{{DEFAULTSORT:Henon-Heiles Equation}}
[[Category:Stellar astronomy]]

Revision as of 00:43, 3 March 2014

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