Euler's three-body problem

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In physics and astronomy, Euler's three-body problem, named after Leonhard Euler, is to solve for the motion of a test mass that is free to move in the presence of the gravitational field of a primary and secondary mass which are fixed in space. This problem is the simplest three-body problem that retains physical significance. Euler discussed it in memoirs published in 1760.

The problem is analytically solvable but requires the evaluation of elliptic integrals.^[1] Numerical methods may be used, such as Runge-Kutta, to solve the resulting ordinary differential equations approximately and to gain some feel for the physics.

1 Example of bicentric orbits
- 1.1 Solution
2 Constant of motion
3 See also
4 References

[edit] Example of bicentric orbits

In classical mechanics, Euler's three-body problem describes the motion of a particle in a plane under the influence of two fixed centers, each of which attract the particle with an inverse-square force such as Newtonian gravity or Coulomb's law. Examples of the bicenter problem include a planet moving around two slowly moving stars, or an electron moving in the electric field of two positively charged nuclei, such as the first ion of the hydrogen molecule H₂. The strength of the two attractions need not be equal; thus, the two stars may have different masses or the nuclei two different charges.

[edit] Solution

Let the fixed centers of attraction be located along the x-axis at ±a. The potential energy of the moving particle is given by

$V(x, y) = \frac{-\mu_{1}}{\sqrt{\left( x - a \right)^{2} + y^{2}}} - \frac{\mu_{2}}{\sqrt{\left( x + a \right)^{2} + y^{2}}} .$

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 bicenter problem.

Introducing elliptic coordinates,

$\,x = \,a \cosh \xi \cos \eta,$

$\,y = \,a \sinh \xi \sin \eta,$

the potential energy can be written as

$V(\xi, \eta) = \frac{-\mu_{1}}{a\left( \cosh \xi - \cos \eta \right)} - \frac{\mu_{2}}{a\left( \cosh \xi + \cos \eta \right)} = \frac{-\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right)}{a\left( \cosh^{2} \xi - \cos^{2} \eta \right)},$

and the kinetic energy as

$T = \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right) \left( \dot{\xi}^{2} + \dot{\eta}^{2} \right).$

This is a Liouville dynamical system if ξ and η are taken as φ₁ and φ₂, respectively; thus, the function Y equals

$\,Y = \cosh^{2} \xi - \cos^{2} \eta$

and the function W equals

$W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right)$

Using the general solution for a Liouville dynamical system,^[2] one obtains

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

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

Introducing a parameter u by the formula

$du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}},$

gives the parametric solution

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

Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u.

[edit] Constant of motion

The bicentric problem conserves energy, i.e., the total energy E is a constant of motion. However, the problem has a second constant of motion, namely,

$r_{1}^{2} r_{2}^{2} \left( \frac{d\theta_{1}}{dt} \right) \left( \frac{d\theta_{2}}{dt} \right) - 2c \left[ \mu_{1} \cos \theta_{1} + \mu_{2} \cos \theta_{2} \right],$

from which the problem can be solved using the method of the last multiplier.

[edit] See also

[edit] References

^ Whittaker, ET (1937). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies, 4th ed., New York: Dover Publications, pp. 97–99. ASIN B0006AQI82.
^ Liouville (1849). "Unknown title". Journal de Mathematique 14: 257–?.

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Euler's three-body problem

From Wikipedia, the free encyclopedia

Contents

[edit] Example of bicentric orbits

[edit] Solution

[edit] Constant of motion

[edit] See also

[edit] References

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