Student Seminar: Classical and Quantum Integrable Systems

we obtain the solution
Ω 1 =
Ω 2 =
Ω 3 =
√
2EI 3 − J 2
I 1 (I 3 − I 1 ) cn τ ,
√
2EI 3 − J 2
I 2 (I 3 − I 1 ) sn τ ,
√
J 2 − 2EI 1
I 3 (I 3 − I 1 ) dn τ .
Period of all these three elliptic functions is given by 4K, where K is the complete
elliptic integral of the first kind:
K =
∫ 1
0
ds
√
(1 − s2 )(1 − k 2 s 2 ) .
Period in time t is therefore given by
√
I 1 I 2 I 3
T = 4K
(I 3 − I 2 )(J 2 − 2EI 1 ) .
After this time both Ω and J will return to their original values. Thus, Ω or J
perform a strictly periodic motion. What is remarkable, is that the top itself does
not return in its original position in the stationary coordinate system k.
We have obtained that the angular momentum J moves periodically with the
period T . On the other hand, we know that the Liouville torus has the dimension
two! This means that the actual motion of the body should be parameterized by two
frequencies ω 1,2 . Let us express the angular velocity Ω via the Euler angles and their
derivatives. Let x 1 , x 2 , x 3 be the axes of the moving frame k. Components of ˙θ on
x 1 are
˙θ 1 = ˙θ cos ψ , ˙θ2 = − ˙θ sin ψ , ˙θ3 = 0 .
The velocity ˙φ is directed along Z. Its projections are
˙φ 1 = ˙φ sin θ sin ψ , ˙φ2 = ˙φ sin θ cos ψ , ˙φ3 = ˙φ cos θ .
Finally, the velocity ˙ψ is directed along x 3 . Thus, we can write the components of
the angular velocity in the moving frame as
Ω 1 = ˙φ sin θ sin ψ + ˙θ cos ψ
Ω 2 = ˙φ sin θ cos ψ − ˙θ sin ψ
Ω 3 = ˙φ cos θ + ˙ψ .
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