Special Issue: On the 175th Anniversary of S.V. Kovalevskaya (Issue Editors: Vladimir Dragović, Andrey Mironov, and Sergei Tabachnikov)

Sonya Kowalewski's Legacy to Mechanics and Complex Lie Algebras

Velimir Jurdjevic

Department of Mathematics, University of Toronto, M5S 3G3 Toronto, Ontario, Canada

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.

Keywords: Riemannian and semi-Riemannian manifolds, connections, parallel transport, isometries, Lie groups actions, Pontryagin Maximum Principle, extremal curves, integrable systems

MSC: 53C17, 53C22, 53B21, 53C25, 30C80, 26D05, 49J15, 58E40

Received: 01.06.2025
Accepted: 12.09.2025

Language: English

DOI: 10.1134/S1560354725050028