Student Seminar: Classical and Quantum Integrable Systems

One can study the structure of the curves of intersection by fixing the ellipsoid E > 0
and changing the radius J of the sphere.
Note that alternatively the Euler equations can be rewritten as the equations for
the angular velocity Ω:
dΩ 1
dt + I 3 − I 2
Ω 2 Ω 3 = 0 ,
I 1
dΩ 2
dt + I 1 − I 3
Ω 3 Ω 1 = 0 ,
I 2
dΩ 3
dt + I 2 − I 1
Ω 1 Ω 2 = 0.
I 3
We could express Ω 1 and Ω 3 from the conservation laws
Ω 2 1
(
)
1 =
(2EI 3 − J 2 ) − I 2 (I 3 − I 2 )Ω 2 2 ,
I 1 (I 3 − I 1 )
Ω 2 1
(
)
3 =
(J 2 − 2EI 1 ) − I 2 (I 2 − I 1 )Ω 2 2 .
I 3 (I 3 − I 1 )
Then plugging this into the Euler equation for Ω 2 we obtain
dΩ 2
dt
=
√
1 ( )(
)
√ (2EI 3 − J 2 ) − I 2 (I 3 − I 2 )Ω 2 2 (J 2 − 2EI 1 ) − I 2 (I 2 − I 1 )Ω 2 2 .
I 2 I1 I 3
We assume that I 3 > I 2 > I 1 and further that M 2 > 2EI 2 . Then making the
substitutions
√
√
(I 3 − I 2 )(J
τ = t
− 2EI 1 )
I 2 (I 3 − I 2 )
, s = Ω 2
I 1 I 2 I 3 2EI 3 − J 2
and introducing the positive parameter k 2 < 1 by
we obtain
k 2 = (I 2 − I 1 )(2EI 3 − J 2 )
(I 3 − I 2 )(J 2 − 2EI 1 )
τ =
∫ s
0
ds
√
(1 − s2 )(1 − k 2 s 2 ) .
The initial time τ = 0 is chosen such that for s = 0 one has Ω 2 = 0. Inverting the
last integral one gets the Jacobi elliptic function 4
Using two other elliptic functions
s = sn τ .
cn 2 τ + sn 2 τ = 1 , dn 2 τ + k 2 sn 2 τ = 1
4 Elliptic functions were first applied to this problem in Rueb, Specimen inaugural, Utrecht, 1834.
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