Student Seminar: Classical and Quantum Integrable Systems

to show that ˙⃗ R = 0. The last formula can be proved by noting that the vector
⃗r × (⃗r × ⃗v) = α⃗r + β⃗v
is orthogonal to ⃗r. Thus, multiplying both sides by ⃗r we get
0 = αr 2 + β(⃗v⃗r) .
On the other hand, multiplying both sides by ⃗v we get
(⃗v, ⃗r × (⃗r × ⃗v)) = α(⃗v⃗r) + βv 2
which gives
(⃗v, ⃗r × (⃗r × ⃗v)) = −(⃗r × ⃗v, ⃗r × ⃗v = −r 2 v 2 sin φ = −r 2 v 2 (1 − cos 2 φ)
= −r 2 v 2 + (⃗v⃗r) 2 = α(⃗v⃗r) + βv 2 .
These two equations allows one to find
α = (⃗v⃗r) , β = −r 2 .
2.3 Rigid body
2.3.1 Moving coordinate system
Let K and k will be two oriented Euclidean spaces. A motion of K relative to k is
a mapping smoothly depending on t:
D t : K → k ,
which preserves the metric and orientation. Every motion can be uniquely written
as the composition of a rotation (D t which maps the origin of K into the origin
of k, i.e. D t is linear mapping) and a translation C t : k → k. Let call K and k
moving and stationary coordinate systems respectively. Let q(t) and Q(t) will be the
radius-vector of a point in a stationary and moving coordinate systems respectively.
Then
q(t) = D t Q(t) = B t Q(t) + r(t) .
} {{ } }{{}
rotation translation
Differentiating we get an addition formula for velocities
˙q =
ḂQ +B
}{{}
˙Q + ṙ .
transferred rotation
Suppose a point does not move w.r.t. to the moving frame, i.e.
r = ṙ = 0. Then
˙q = ḂQ = ḂB−1 q = Aq ,
˙Q = 0 and also that
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