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The problem of the motion of a solid in an ideal fluid. Integration of the Clebsch’s case. (English) Zbl 0982.35085

This paper deals with a geometric approach to the integration of Clebsch’s case of equations describing the motion of a solid body in an ideal fluid. This problem is defined by a nonlinear system of 6 differential equations admitting 4 polynomial first integrals. The author shows that the intersection of surface levels of these integrals can be completed to an Abelian surface, i.e., a 2-dimensional algebraic torus. Moreover, he proves that the problem can be linearized, i.e., it can be written in terms of Abelian integrals on a Prym variety of a genus 3 curve obtained naturally.

MSC:

35Q35 PDEs in connection with fluid mechanics
14H40 Jacobians, Prym varieties
70E40 Integrable cases of motion in rigid body dynamics
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70G55 Algebraic geometry methods for problems in mechanics
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