In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows:[1]
T=
1 | |
2 |
\left\{u1(q1)+u2(q2)+ … +us(qs)\right\} \left\{v1(q1)
q |
2 | |
1 |
+v2(q2)
q |
2 | |
2 |
+ … +vs(qs)
q |
2 | |
s |
\right\}
V=
w1(q1)+w2(q2)+ … +ws(qs) | |
u1(q1)+u2(q2)+ … +us(qs) |
The solution of this system consists of a set of separably integrable equations
\sqrt{2 | |
where E = T + V is the conserved energy and the
\gammas
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, namely the hydrogen molecular ion or 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.
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)=
-\mu1 | |
\sqrt{\left(x-a\right)2+y2 |
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\cosη,
y=a\sinh\xi\sinη,
the potential energy can be written as
V(\xi,η)=
-\mu1 | |
a\left(\cosh\xi-\cosη\right) |
-
\mu2 | |
a\left(\cosh\xi+\cosη\right) |
=
-\mu1\left(\cosh\xi+\cosη\right)-\mu2\left(\cosh\xi-\cosη\right) | |
a\left(\cosh2\xi-\cos2η\right) |
,
and the kinetic energy as
T=
ma2 | |
2 |
\left(\cosh2\xi-\cos2η\right)\left(
\xi |
2+
η |
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=\cosh2\xi-\cos2η
and the function W equals
W=-\mu1\left(\cosh\xi+\cosη\right)-\mu2\left(\cosh\xi-\cosη\right)
Using the general solution for a Liouville dynamical system below, one obtains
ma2 | |
2 |
\left(\cosh2\xi-\cos2η\right)2
\xi |
2=E\cosh2\xi+\left(
\mu1+\mu2 | |
a |
\right)\cosh\xi-\gamma
ma2 | |
2 |
\left(\cosh2\xi-\cos2η\right)2
η |
2=-E\cos2η+\left(
\mu1-\mu2 | |
a |
\right)\cosη+\gamma
Introducing a parameter u by the formula
du=
d\xi | |||||
|
gives the parametric solution
u=\int
d\xi | |||||
|
Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u.
The bicentric problem has a constant of motion, namely,
2 | |
r | |
1 |
2 | |
r | |
2 |
d\theta1 | |
dt |
d\theta2 | |
dt |
+2c\left(\mu1\cos\theta1-\mu2\cos\theta2\right),
from which the problem can be solved using the method of the last multiplier.
To eliminate the v functions, the variables are changed to an equivalent set
\varphir=\intdqr\sqrt{vr(qr)},
giving the relation
v1(q1)
q |
2 | |
1 |
+v2(q2)
q |
2 | |
2 |
+ … +vs(qs)
q |
2 | ||
= | ||
s |
\varphi |
2 | |
1 |
+
\varphi |
2 | |
2 |
+ … +
\varphi |
2 | |
s |
=F,
which defines a new variable F. Using the new variables, the u and w functions can be expressed by equivalent functions χ and ω. Denoting the sum of the χ functions by Y,
Y=\chi1(\varphi1)+\chi2(\varphi2)+ … +\chis(\varphis),
the kinetic energy can be written as
T=
1 | |
2 |
YF.
Similarly, denoting the sum of the ω functions by W
W=\omega1(\varphi1)+\omega2(\varphi2)+ … +\omegas(\varphis),
the potential energy V can be written as
V=
W | |
Y |
.
The Lagrange equation for the rth variable
\varphir
d | |
dt |
\left(
\partialT | ||||
|
\right)=
d | |
dt |
\left(Y
\varphi |
r\right)=
1 | |
2 |
F
\partialY | - | |
\partial\varphir |
\partialV | |
\partial\varphir |
.
Multiplying both sides by
2Y
\varphi |
r
2Y
\varphi |
r
d | |
dt |
\left(Y
\varphi |
r\right)=2T
\varphi |
r
\partialY | |
\partial\varphir |
-2Y
\varphi |
r
\partialV | |
\partial\varphir |
=2
\varphi |
r
\partial | |
\partial\varphir |
\left[(E-V)Y\right],
which may be written as
d | |
dt |
\left(Y2
\varphi |
2 | |
r |
\right)=2E
\varphi |
r
\partialY | |
\partial\varphir |
-2
\varphi |
r
\partialW | |
\partial\varphir |
=2E
\varphi |
r
d\chir | |
d\varphir |
-2
\varphi |
r
d\omegar | |
d\varphir |
,
where E = T + V is the (conserved) total energy. It follows that
d | |
dt |
\left(Y2
\varphi |
2 | |
r |
\right)=2
d | |
dt |
\left(E\chir-\omegar\right),
which may be integrated once to yield
1 | |
2 |
Y2
\varphi |
2 | |
r |
=E\chir-\omegar+\gammar,
where the
\gammar
s | |
\sum | |
r=1 |
\gammar=0.
Inverting, taking the square root and separating the variables yields a set of separably integrable equations:
\sqrt{2 | |
. E. T. Whittaker . 1937 . A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies. 4th . Dover Publications . New York . ISBN .