Goryachev-Chaplygin top

A rigid body rotating about a fixed point, for which:

a) the principal moments of inertia $ \lambda = ( \lambda _ {1} , \lambda _ {2} , \lambda _ {3} ) $, with regard to the fixed point, satisfy the relation $ \lambda _ {1} = \lambda _ {2} = 4 \lambda _ {3} $;

b) the centre of mass belongs to the equatorial plane through the fixed point;

c) the principal angular momentum is perpendicular to the direction of gravity, i.e., $ \langle {m, \gamma } \rangle = 0 $. First introduced by D. Goryachev [a4] in 1900, the system was later integrated by S.A. Chaplygin [a3] in terms of hyper-elliptic integrals (cf. also Hyper-elliptic integral). The system merely satisfying a) and b) is not algebraically integrable, but on the locus, defined by c), it is; namely, it has an extra invariant of homogeneous degree $ 3 $:

$$ Q _ {4} = ( m ^ {2} _ {1} + m _ {2} ^ {2} ) m _ {3} + 2m _ {1} \gamma _ {3} . $$

C. Bechlivanidis and P. van Moerbeke [a1] have shown that the problem has asymptotic solutions which are meromorphic in $ \sqrt t $; the system linearizes on a double cover of a hyper-elliptic Jacobian (i.e., of the Jacobi variety of a hyper-elliptic curve; cf. also Plane real algebraic curve), ramified exactly along the two hyper-elliptic curves, where the phase variables blow up; see also [a5]. An elementary algebraic mapping transforms the Goryachev–Chaplygin equations into equations closely related to the $ 3 $- body Toda lattice. A Lax pair is given in [a2]:

$$ - { \frac{i}{2} } ( { {h _ {- 1 } } tilde } h ^ {- 1 } + { {L _ {0} } tilde } + { {L _ {1} } tilde } h ) ^ \bullet = $$

$$ = [ { {L _ {- 1 } } tilde } h ^ {- 1 } + { {L _ {0} } tilde } + { {L _ {1} } tilde } h, { {B _ {0} } tilde } - { {L _ {1} } tilde } h ] , $$

where $ { {L _ {0} } tilde } $ and $ { {L _ {1} } tilde } $ are given by the $ ( 3 \times 3 ) $ right-lower corner of $ L _ {0} $ and $ L _ {1} $ and where

$$ { {L _ {- 1 } } tilde } = { \frac{1}{2} } \left ( \begin{array}{ccc} 0 &- y _ {3} & 0 \\ y _ {3} & 0 &y _ {1} - x _ {1} ^ {2} \\ 0 &- y _ {2} + x _ {2} ^ {2} & 0 \\ \end{array} \right ) , $$

$$ { {B _ {0} } tilde } = \left ( \begin{array}{ccc} { \frac{3}{2} } x _ {3} & 0 &- x _ {1} \\ 0 &{ \frac{3}{2} } x _ {3} & 0 \\ - x _ {2} & 0 &- x _ {3} \\ \end{array} \right ) . $$

See also Kowalewski top.

References

[a1]	C. Bechlivanidis, P. van Moerbeke, "The Goryachev–Chaplygin top and the Toda lattice" Comm. Math. Phys. , 110 (1987) pp. 317–324
[a2]	A.I. Bobenko, V.B. Kuznetsov, "Lax representation and new formulae for the Goryachev–Chaplygin top" J. Phys. A , 21 (1988) pp. 1999–2006
[a3]	S.A. Chaplygin, "A new case of rotation of a rigid body, supported at one point" , Collected works , I , Gostekhizdat (1948) pp. 118–124 (In Russian)
[a4]	D. Goryachev, "On the motion of a rigid material body about a fixed point in the case " Mat. Sb. , 21 (1900) (In Russian)
[a5]	L. Piovan, "Cyclic coverings of Abelian varieties and the Goryachev–Chaplygin top" Math. Ann. , 294 (1992) pp. 755–764