Student Seminar: Classical and Quantum Integrable Systems

is the bounded kinetic energy). According to the known theorem in topology, a
two-dimensional compact orientable manifold admitting globally defined vector field
is isomorphic to a torus. This is our Liouville torus 3 . According to the Liouville
theorem motion on the torus will be characterized by two frequencies ω 1 and ω 2 . If
their ratio is not a rational number then the body never returns to its original state
of motion.
Consider a rigid body rotation around a fixed point O and denote by K a coordinate
system rotating with the body around O: in K the body is at rest. Every
vector in K is carried to k by an operator B. By definition of the angular momentum
we have
M = q × m ˙q = m q × (ω × q) .
Denote by J and by Ω the angular momentum and angular velocity in the moving
frame K. We have
J = m Q × (Ω × Q) .
This defines a linear map A: K → K such that AΩ = J. This operator is symmetric:
(AX, Y ) = (m Q × (X × Q), Y ) = m(Q × X, Q × Y )
because the r.h.s. is symmetric function of X, Y . The operator A is called the inertia
tensor. We see that taking X = Y = Ω we get
E = T = 1 2 (AΩ, Ω) = 1 2 (J, Ω) = m 2 (Q × Ω, Q × Ω) = m 2 ˙Q 2 = m 2 ˙q2 .
being a symmetric operator A is diagonalizable and it defines three mutually orthogonal
characteristic directions. In the basis where A is diagonal the inertia operator
and the kinetic energy take a very simple form
J i = I i Ω i ,
T = 1 3∑
I i Ω 2 i .
2
The axes of this particular coordinate system are called the principle inertia axes.
Problem. Rewrite expression for energy via the quantities of the stationary frame k.
We have
i=1
E = 1 2 (AΩ, Ω) = 1 2 (J, Ω) = 1 2 (M, ω) = m 2 (q × (q × ω), ω) = m (q × ω, q × ω)
2
= 1 (ω
2 m 2 q 2 − (ωq) 2) = 1 )
2 ω iω j m
(x 2 i δ ij − x i x j .
} {{ }
inertia tensor
3 We cannot use the Liouville theorem to derive this result, because the integrals M i do not
commute with each other and, therefore, the Frobenious theorem cannot be applied to deduce that
the level set is a smooth manifold. Nevertheless we can identify the Liouville torus by different
means.
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