Difference between revisions of "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

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

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