Editing Euler's three-body problem (section)

==Mathematical solutions==

===Original Euler problem===
In the original Euler problem, the two centers of force acting on the particle are assumed to be fixed in space; let these centers be located along the ''x''-axis at ±''a''.   The particle is likewise assumed to be confined to a fixed plane containing the two centers of force. The potential energy of the particle in the field of these centers is given by

:<math>
V(x, y) = \frac{-\mu_1}{\sqrt{\left( x - a \right)^2 + y^2}} - \frac{\mu_2}{\sqrt{\left( x + a \right)^2 + y^2}} .
</math>

where the proportionality constants μ<sub>1</sub> and μ<sub>2</sub> may be positive or negative.  The two centers of attraction can be considered as the foci of a set of ellipses.  If either center were absent, the particle would move on one of these ellipses, as a solution of the [[Kepler problem]].  Therefore, according to [[Bonnet's theorem]], the same ellipses are the solutions for the Euler problem.

Introducing [[elliptic coordinates]],

:<math>
\,x = \,a \cosh \xi \cos \eta,
</math>

:<math>
\,y = \,a \sinh \xi \sin \eta,
</math>

the potential energy can be written as

:<math>
\begin{align}
V(\xi, \eta) & = \frac{-\mu_{1}}{a\left( \cosh \xi - \cos \eta \right)} - \frac{\mu_{2}}{a\left( \cosh \xi + \cos \eta \right)} \\[8pt]
& = \frac{-\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right)}{a\left( \cosh^{2} \xi - \cos^{2} \eta \right)},
\end{align}
</math>

and the kinetic energy as

:<math>
T = \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right) \left( \dot{\xi}^{2} + \dot{\eta}^{2} \right).
</math>

This is a [[Liouville dynamical system]] if ξ and η are taken as φ<sub>1</sub> and φ<sub>2</sub>, respectively; thus, the function ''Y'' equals

:<math>
\,Y = \cosh^{2} \xi - \cos^{2} \eta
</math>

and the function ''W'' equals

:<math>
W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right).
</math>

Using the general solution for a [[Liouville dynamical system]],<ref name="liouville_1849">{{cite journal | author = Liouville J | year = 1849 | title = Mémoire sur l'intégration des équations différentielles du mouvement d'un nombre quelconque de points matériels | journal = Journal de Mathématiques Pures et Appliquées | volume = 14 | pages = 257&ndash;299 | url = http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16393&Deb=263&Fin=305&E=PDF| author-link = Joseph Liouville }}</ref> one obtains

:<math>
\frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma
</math>

:<math>
\frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma
</math>

Introducing a parameter ''u'' by the formula

:<math>
du = \frac{d\xi}{\sqrt{E \cosh^2 \xi + \left( \frac{\mu_1 + \mu_2}{a} \right) \cosh \xi - \gamma}} = 
\frac{d\eta}{\sqrt{-E \cos^2 \eta + \left( \frac{\mu_1 - \mu_2}{a} \right) \cos \eta + \gamma}},
</math>

gives the [[parametric solution]]

:<math>
u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = 
\int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}.
</math>

Since these are [[elliptic integral]]s, the coordinates ξ and η can be expressed as elliptic functions of ''u''.