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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.
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.

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
Full Text: DOI
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