Integrability of Nonlinear Systems

36 B. Grammaticos and A. Ramani
At that time one of the small miracles that are often associated with integrability,
occured [13]. Kovalevskaya, who was Fuchs’s student, set out to study the
integrability of a physical problem using singularity-analysis techniques. If she
had just confirmed the integrability of the already known integrable cases her
work would have been, at best, interesting and soon forgotten. What happened
was that Kovalevskaya discovered a new, highly nontrivial, case. She set out to
study the motion of a heavy top, spinning around a fixed point. The equations of
motion with respect to a moving Cartesian coordinate system based on the principal
axes of inertia with origin at its fixed point, known as Euler’s equations,
are:
A dp
dt =(B − C)qr + Mg(γy 0 − βz 0 )
B dq
dt =(C − A)pr + Mg(αz 0 − γx 0 )
C dr
dt =(A − B)pq + Mg(βx 0 − αy 0 )
(2.6)
dα
= βr − γq
dt
dβ
= γp − αr
dt
dγ
= αq − βp,
dt
where (p, q, r) are the components of angular velocity, (α, β, γ) the directions
cosines of the direction of gravity, (A, B, C) the moments of inertia, (x 0 ,y 0 ,z 0 )
the position of the centre of mass of the system, M the mass of the top, and g the
acceleration due to gravity. The complete integrability of the system requires the
knowledge of four integrals of motion. Three such integrals are straightforward,
the geometric constraint,
α 2 + β 2 + γ 2 =1, (2.7)
the total energy,
Ap 2 + Bq 2 + Cr 2 − 2Mg(αx 0 + βy 0 + γz 0 )=K 1 , (2.8)
and the projection of the angular momentum on the direction of gravity,
Aαp + Bβq + Cγr = K 2 . (2.9)
A fourth integral was known only in three cases:
Spherical: A = B = C with integral px 0 + qy 0 + rz 0 = K,
Euler: x 0 = y 0 = z 0 with integral A 2 p 2 + B 2 q 2 + C 2 r 2 = K, and
Lagrange: A = B and x 0 = y 0 = 0 with integral Cr = K.
In each of these cases the solutions of the equations of motion were given in
terms of elliptic functions and were thus meromorphic in time t. Kovalevskaya