Abstract
“Glued geodesic flows” and, in particular, “generalized billiard flows” on Riemannian manifolds with boundary, and geodesic flows on piecewise smooth Riemannian manifolds are studied. We develop the approaches of Lazutkin (1993) and Tabachnikov (1993) for proving the Poncelet type closure theorems via applying the classical Liouville theorem to the billiard flow (respectively to the billiard map). We prove that the condition on the refraction/reflection law to respect the Huygens principle is not only sufficient, but also necessary for “local Liouville integrability” of the glued geodesic flow, more precisely for pairwise commutation of the “glued flows” corresponding to a maximal collection of local first integrals in involution homogeneous in momenta. A similar criterion for “local Liouville integrability” of the succession/billiard map is obtained.
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References
E. Y. Amiran, Smooth Convex Planar Domains for Which the Billiard Ball Map Is Integrable Are Ellipses, Preprint, Math. Depart. Western Washington Univ., Bellingam (1991).
A. Besse, Manifolds All of Whose Geodesics Are Closed, Springer, Berlin (1978).
S. V. Bolotin, “Integrable billiards of Birkhoff,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 33–36 (1990).
A. Cayley, “Developments on the porism of the in-and-circumscribed polygon,” Philos. Mag., 7, 339–345 (1854).
S. J. Chang, B. Crespi, and K. J. Shi, “Elliptical billiard systems and the full Poncelets theorem in n dimensions,” J. Math. Phys., 34, 2242–2256 (1993).
S. J. Chang and R. Friedberg, “Elliptical billiards and Poncelet’s theorem,” J. Math. Phys., 29, 1537–1550 (1988).
M. Chasles, “Géométrie pure. Théorèmes sur les sections coniques confocales,” Ann. Math. Pures Appl., 18, 269–276 (1827/1828).
V. V. Fokicheva, “A topological classification of billiards in locally planar domains bounded by arcs of confocal quadrics,” Sb. Math., 206, No. 10, 1463–1507 (2015).
V. V. Fokicheva and A. T. Fomenko, “Topology and singularities of the billiards,” in: Proc. Int. Conf. “Voronezh Winter Math. School of S. G. Krein,” 2014, Voronezh Univ. Publ., Voronezh (2014), pp. 372–385.
B. Halpern, “Strange billiard tables,” Trans. Am. Math. Soc., 237, 297–305 (1977).
V. Lazutkin, KAM Theory and Semiclassical Approximations to Eigenfunctions, Springer, Berlin, 1993.
J.-V. Poncelet, Traité des propriétés projectives des figures, Mett, Paris (1822).
V. Rom-Kedar and D. Turaev, “Billiards: a singular perturbation limit of smooth Hamiltonian flows,” Chaos, 22, No. 2, 026102 (2012).
I. V. Sypchenko and D. S. Timonina, “Closed geodesics on piecewise smooth surfaces of revolution with constant curvature,” Sb. Math., 206, No. 5, 738–769 (2015).
S. Tabachnikov, “Poncelet’s theorem and dual billiards,” Enseign. Math., 39, 189–194 (1993).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 20, No. 3, pp. 113–152, 2015.
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Kudryavtseva, E.A. Liouville Integrable Generalized Billiard Flows and Poncelet Type Theorems. J Math Sci 225, 611–638 (2017). https://doi.org/10.1007/s10958-017-3482-5
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DOI: https://doi.org/10.1007/s10958-017-3482-5