Kowalewski top and complex Lie algebras

Abstract

This paper identifies a natural Hamiltonian on a ten dimensional complex Lie algebra that unravels the mysteries encountered in Kowalewski’s famous paper on the motions of a rigid body around its fixed point under the influence of gravity. This system reveals that the enigmatic conditions of Kowalewski, namely, two principal moments of inertia equal to each other and twice the value of the remaining moment of inertia, and the centre of gravity in the plane spanned by the directions corresponding to the equal moments of inertia, are both necessary and sufficient for the existence of an isospectral representation $\frac{dL(\lambda )}{dt}=[M(\lambda ),L(\lambda )]$ with a spectral parameter $\lambda$. This representation then yields a crucial spectral invariant that naturally accounts for all the integrals of motion, known as Kowalewski type integrals in the literature of the top. This result is fundamentally dependent on a preliminary discovery that the equality of two principal moments of inertia and the placement of the centre of mass in the plane spanned by the corresponding directions is intimately tied to the existence of another integral of motion on whose zero level surface the above spectral representation resides. The link between mechanical tops and Hamiltonian systems on Lie algebras is provided by an earlier result in which it is shown that the equations of mechanical tops with a linear potential, (heavy tops, in particular) can be represented on certain coadjoint orbits in the semi-direct product ${\mathfrak {g}}={\mathfrak {p}}\rtimes {\mathfrak {k}}$ induced by a closed subgroup K of the underlying group G. The passage to complex Lie algebras is motivated by Kowalewski’s mysterious use of complex variables. It is shown that the complex variables in her paper are naturally identified with complex quaternions and the representation of $\mathfrak {so}(4,{{\mathbb {C}}})$ as the product $\mathfrak {sl}(2,{{\mathbb {C}}})\times \mathfrak {sl}(2,{{\mathbb {C}}})$. The paper also shows that all the equations of Kowalewski type can be solved by a uniform integration procedure over the Jacobian of a hyperelliptic curve, as in the original paper of Kowalewski.