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 workbegins 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 showsthat these integrals of motion can be naturally extracted from a canonical Poisson system on the dual of \(so(4,\mathbb{C})\) generated byan affine quadratic Hamiltonian \(H\) (Kirchhoff – Kowalewski type).
The paper shows that the passage to complex variablesis 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.
