Abstract
The classical Jacobi–Chasles theorem states that tangent lines drawn at all points of a geodesic curve on a quadric in n-dimensional Euclidean space are tangent, in addition to the given quadric, to n – 2 other confocal quadrics, which are the same for all points of the geodesic curve. This theorem immediately implies the integrability of a geodesic flow on an ellipsoid. In this paper, we prove a generalization of this result for a geodesic flow on the intersection of several confocal quadrics. Moreover, if we add the Hooke’s potential field centered at the origin to such a system, the integrability of the problem is preserved.
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REFERENCES
C. Jacobi, Lectures on Dynamics (ONTI, Leningrad, 1936) [in Russian].
M. Chasles, “Sur les lignes géodésiques et les lignes de courbure des surfaces du second degré,” J. Math. Pures Appl. 11, 5–20 (1846).
V. I. Arnold, “Several remarks on elliptic coordinates,” Zap. Nauchn. Sem. LOMI 133, 38–50 (1984).
V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1989; Springer-Verlag, New York, 1989).
V. V. Kozlov and D. V. Treshchev, Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts (Mosk. Gos. Univ., Moscow, 1991; Am. Math. Soc., Providence, R.I., 1991).
V. Dragović and M. Radnović, Integrable Billiards, Quadrics, and Multidimensional Poncelet Porisms (Regulyarnaya i Khaoticheskaya Dinamika, Moscow, 2010) [in Russian].
V. V. Kozlov, “Topological obstructions to the integrability of natural mechanical systems,” Dokl. Akad. Nauk SSSR 249 (6), 1299–1302 (1979).
S. Gitler and S. L. Medrano, “Intersections of quadrics, moment-angle manifolds and connected sums,” Geom. Topol. 17, 1497–1534 (2013).
V. V. Kozlov, “Some integrable extensions of Jacobi’s problem of geodesics on an ellipsoid,” J. Appl. Math. Mech. 59 (1), 1–7 (1995).
ACKNOWLEDGMENTS
The author is grateful to V.A. Kibkalo for setting up the problem and to A.T. Fomenko for a number of valuable remarks and comments.
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Translated by I. Ruzanova
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Belozerov, G.V. Integrability of a Geodesic Flow on the Intersection of Several Confocal Quadrics. Dokl. Math. 107, 1–3 (2023). https://doi.org/10.1134/S1064562423700382
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DOI: https://doi.org/10.1134/S1064562423700382