Sonya Kowalewski’s Legacy to Mechanics and Complex Lie Algebras

Abstract

This paper provides an original rendition of the heavy top that unravels the mysteries behind S. Kowalewski’s seminal work on the motions of a rigid body around a fixed point under the influence of gravity. The point of departure for understanding Kowalewski’s work begins with Kirchhoff’s model for the equilibrium configurations of an elastic rod in \({\mathbb{R}}^{3}\) subject to fixed bending and twisting moments at its ends [17]. This initial orientation to the elastic problem shows, first, that the Kowalewski type integrals discovered by I. V. Komarov and V. B. Kuznetsov [24, 25] appear naturally on the Lie algebras associated with the orthonormal frame bundles of the sphere \(S^{3}\) and the hyperboloid \(H^{3}\) [17] and, secondly, it shows that these integrals of motion can be naturally extracted from a canonical Poisson system on the dual of \(so(4,\mathbb{C})\) generated by an affine quadratic Hamiltonian \(H\) (Kirchhoff – Kowalewski type).

The paper shows that the passage to complex variables is synonymous with the representation of \(so(4,\mathbb{C})\) as \(sl(2,\mathbb{C})\times sl(2,\mathbb{C})\) and the embedding of \(H\) into \(sp(4,\mathbb{C})\), an important intermediate step towards uncovering the origins of Kowalewski’s integral. There is a quintessential Kowalewski type integral of motion on \(sp(4,\mathbb{C})\) that appears as a spectral invariant for the Poisson system associated with a Hamiltonian \(\mathcal{H}\) (a natural extension of \(H\)) that satisfies Kowalewski’s conditions.

The text then demonstrates the relevance of this integral of motion for other studies in the existing literature [7, 35]. The text also includes a self-contained treatment of the integration of the Kowalewski type equations based on Kowalewski’s ingenuous separation of variables, the hyperelliptic curve and the solutions on its Jacobian variety.