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Rationality conditions for the ratio of elliptic integrals and the great Poncelet theorem. (Russian, English) Zbl 1083.33016
Vestn. Mosk. Univ., Ser. I 2003, No. 4, 6-13 (2003); translation in Mosc. Univ. Math. Bull. 58, No. 4, 1-7 (2003).
The author establishes sufficient rationality conditions for the ratio of elliptic integrals of a certain type. They are presented in the form of Diophantine equations satisfied by the coefficients of the characteristic third or fourth order polynomial. These conditions are derived from the known Cayley theorem and the Poncelet theorem. The possibilities of extension of the great Poncelet theorem for the spaces of constant curvature are discussed.

33E05 Elliptic functions and integrals
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
53A05 Surfaces in Euclidean and related spaces
70F99 Dynamics of a system of particles, including celestial mechanics
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics