Student Seminar: Classical and Quantum Integrable Systems

2.3.3 Euler’s top
Consider the motion of a rigid body around a fixed point O. Let J and Ω will be
the vector of angular momentum and the angular momentum in the body, i.e. in the
moving coordinate system K. We have AΩ = J, where A is the inertia tensor. The
angular momentum M = B t J of the body in space is preserved. Thus, we have
0 = Ṁ = ḂJ + B ˙ J = ḂB−1 M + B ˙ J = ω × M + B ˙ J = B
(
Ω × J + J ˙
)
.
From here we find
dJ
dt = J × Ω = J × A−1 J .
These are the famous Euler equations which describe the motion of the angular momentum
insider the rigid body. If one takes the coordinate adjusted to the principle
axes then one gets the following system of equations
dJ 1
= a 1 J 2 J 3 ,
dt
dJ 2
= a 2 J 3 J 1 ,
dt
dJ 3
= a 3 J 1 J 2 .
dt
Here
a 1 = I 2 − I 3
, a 2 = I 3 − I 1
, a 3 = I 1 − I 2
.
I 2 I 3 I 1 I 3 I 1 I 2
In this way the Euler equations can be viewed as equations for the components of
the angular momentum insider the body.
Consider the energy
H = 1 2 (J, A−1 J) = 1 2
3∑
i=1
J 2 i
I i
.
It is easy to verify explicitly that it is conserved due to eoms:
3∑ J
(
i a1
Ḣ = J i = J 1 J 2 J 3 + a 2
+ a )
3
= 0 .
I i I 1 I 2 I 3
i=1
Verify the conservation of the length of the angular momentum
3∑
(
)
J˙
2 = J i J˙
i = J 1 J 2 J 3 a 1 + a 2 + a 3 = 0 .
i=1
This is of course agrees with the fact that M is conserved and that M 2 = J 2 . Thus,
we have proved that the Euler equations have two quadratic integrals: the energy
and M 2 = J 2 . Thus, J lies on the intersection of an ellipsoid and a sphere:
2E = J 2 1
I 1
+ J 2 2
I 2
+ J 2 3
I 3
, J 2 = J 2 1 + J 2 2 + J 2 3 .
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