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Some recent generalizations of the classical rigid body systems. (English) Zbl 1378.70007
From the text: Some recent generalizations of the classical rigid body systems are reviewed. The cases presented include dynamics of a heavy rigid body fixed at a point in three-dimensional space, the Kirchhoff equations of motion of a rigid body in an ideal incompressible fluid as well as their higher-dimensional generalizations.
The paper is organized as follows. The basic facts about three-dimensional motion of a rigid body are presented in Sect. 2. In the same Section, the basic steps of the algebro-geometric integration procedure for the Hess-Appel’rot case of motion of three-dimensional rigid body are given. A recent approach to the Kowalevski integration procedure is given in Sect. 3. The basic facts of higher-dimensional rigid body dynamics are presented in Sect. 4. The same Section provides the definition of the isoholomorphic systems, such as the Lagrange bitop and \(n\)-dimensional Hess-Appel’rot systems. The importance of the isoholomorphic systems has been underlined by S. Grushevsky and I. Krichever [Duke Math. J. 152, No. 2, 317–371 (2010; Zbl 1217.14022)]. In Sect. 5 we review the classical Grioli precessions and present its quite recent higher-dimensional generalizations. The four-dimensional generalizations of the Kirchhoff and Chaplygin cases of motion of a rigid body in an ideal fluid are given in Sect. 6.
Contents:
1. Introduction.
2. The Hess-Appel’rot case of rigid body motion:
Basic notions of heavy rigid body fixed at a point; Integrable cases; Definition of the Hess-Appel’rot system; A Lax representation for the classical Hess-Appel’rot system: an algebro-geometric integration procedure. Zhukovski’s geometric interpretation.
3. Kowalevski top, discriminantly separable polynomials, and two valued groups:
Discriminantly separable polynomials; Two-valued groups; 2-valued group structure on \(\mathrm{CP}^1\) and the Kowalevski fundamental equation; Fundamental steps in the Kowalevski integration procedure; Systems of the Kowalevski type: definition; An example of systems of the Kowalevski type; Another example of an integrable system of the Kowalevski type; Another class of systems of the Kowalevski type; A deformation of the Kowalevski top.
4. The Lagrange bitop and the \(n\)-dimensional Hess-Appel’rot systems:
Higher-dimensional generalizations of rigid body dynamics; The heavy rigid body equations on \(e(n)\); The heavy rigid body equations on \(s = so(n) \times_{\text{ad}} so(n)\); Four-dimensional rigid body motion; The Lagrange bitop: definition and a Lax representation; Classical integration; Properties of the spectral curve; Four-dimensional Hess-Appel’rot systems; The \(n\)-dimensional Hess-Appel’rot systems; Classical integration of the four-dimensional Hess-Appel’rot system.
5. Four-dimensional Grioli-type precessions:
The classical Grioli case; Four-dimensional Grioli case. 6. Motion of a rigid body in an ideal fluid: the Kirchhoff equations:
Integrable cases; Three-dimensional Chaplygin’s second case; Classical integration procedure; Lax representation for the Chaplygin case; Four-dimensional Kirchhoff and Chaplygin cases.
References.

MSC:
70E40 Integrable cases of motion in rigid body dynamics
70E17 Motion of a rigid body with a fixed point
70E45 Higher-dimensional generalizations in rigid body dynamics
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
14H70 Relationships between algebraic curves and integrable systems
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