Velimir Jurdjevic

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.

Control theory, initially conceived in the 1950’s as an engineering subject motivated by the needs of automatic control, has undergone an important mathematical transformation since then, in which its basic question, understood in a larger geometric context, led to a theory that provides distinctive and innovative insights, not only to the original problems of engineering, but also to the problems of differential geometry and mechanics.
This paper elaborates the contributions of control theory to geometry and mechanics by focusing on the class of problems which have played an important part in the evolution of integrable systems. In particular the paper identifies a large class of Hamiltonians obtained by the Maximum principle that admit isospectral representation on the Lie algebras $\mathfrak{g} = \mathfrak{p} ⊕ \mathfrak{k}$ of the form
$$\frac{dL_\lambda}{dt} = [\Omega_\lambda, L_\lambda] L_\lambda = L_{\mathfrak{p}} − \lambda L_{\mathfrak{k}} − (\lambda^2 − s)A, \quad L_{\mathfrak{p}} \in \mathfrak{p}, \quad L_{\mathfrak{k}} \in \mathfrak{k}.$$
The spectral invariants associated with $L_\lambda$ recover the integrability results of C.G.J. Jacobi concerning the geodesics on an ellipsoid as well as the results of C. Newmann for mechanical problem on the sphere with a quadratic potential. More significantly, this study reveals a large class of integrable systems in which these classical examples appear only as very special cases.