Editing Euler's three-body problem (section)

==Constants of motion==
The problem of two fixed centers conserves [[energy]]; in other words, the total energy <math>E</math> is a [[constant of motion]].  The [[potential energy]] is given by

:<math>
V(\mathbf{r}) = - \frac{\mu_1}{r_1} - \frac{\mu_2}{r_2}
</math>

where <math>\mathbf{r}</math> represents the particle's position, and <math>r_1</math> and <math>r_2</math> are the distances between the particle and the centers of force; <math>\mu_1</math> and <math>\mu_2</math> are constants that measure the strength of the first and second forces, respectively.  The total energy equals sum of this potential energy with the particle's [[kinetic energy]]

:<math>
E = \frac{\mathbf{p}^2}{2 m} + V(\mathbf{r})
</math>

where <math>m</math> and <math>\mathbf{p}</math> are the particle's mass and [[linear momentum]], respectively.

The particle's [[linear momentum|linear]] and [[angular momentum]] are not conserved in Euler's problem, since the two centers of force act like external forces upon the particle, which may yield a net force and torque on the particle.  Nevertheless, Euler's problem has a second constant of motion

:<math>
C = r_{1}^{2}\,r_{2}^{2}\,\frac{d\theta_{1}}{dt} \frac{d\theta_{2}}{dt} + 2\,a \left( \mu_{1} \cos \theta_{1} - \mu_{2} \cos \theta_{2} \right),
</math>

where <math>2\,a</math> is the separation of the two centers of force, <math>\theta_1</math> and <math>\theta_2</math> are the angles of the lines connecting the particle to the centers of force, with respect to the line connecting the centers.  This second constant of motion was identified by [[E. T. Whittaker]] in his work on analytical mechanics,<ref name="whittaker_constant" >Whittaker [[Analytical Dynamics of Particles and Rigid Bodies]], p. 283.</ref> and generalized to <math>n</math> dimensions by [[Charles Coulson|Coulson]] and Joseph in 1967.<ref name="coulson_joseph" >{{cite journal | author = [[Charles Coulson|Coulson CA]], Joseph A | year = 1967 | title = A Constant of Motion for the Two-Centre Kepler Problem | journal = International Journal of Quantum Chemistry | volume = 1 | issue = 4 | pages = 337&ndash;447 | doi = 10.1002/qua.560010405|bibcode = 1967IJQC....1..337C }}</ref>  In the Coulson–Joseph form, the constant of motion is written

:<math>
B = \mathbf{L}^2 +  a^2 p_n^2 + 2\,a\,x_n \left(\frac{\mu_1 }{r_1} - \frac{\mu_2}{r_2} \right),
</math>

where <math>p_n</math> denotes the momentum component along the <math>x_n</math> axis on which the attracting centers are located.{{refn|group=note|The latter expression differs from the constant C above by the additional term <math>2\,c^2 E</math>}} This constant of motion corresponds to the total [[angular momentum]] squared <math>\mathbf{L}^2</math> in the limit when the two centers of force converge to a single point (<math>a\rightarrow 0</math>), and proportional to the [[Laplace–Runge–Lenz vector]] <math>\mathbf{A}</math> in the limit when one of the centers goes to infinity (<math>a\rightarrow\infty</math> while <math>|x_n - a|</math> remains finite).