Abstract
In a study of capillary floating, Finn (J Math Fluid Mech 11:443–458, 2009) described a procedure for determining cross-sections of non-circular, infinite convex cylinders that float horizontally on a liquid surface in every orientation with contact angle π/2. Finn’s procedure yielded incomplete results for other contact angles; he raised the question as to whether an analogous construction would be feasible in that case. In the note, Finn (J Math Fluid Mech 11:464–465, 2009) pointed out a connection with an independent problem on billiard caustics citing the unpublished work (Gutkin in Proceedings of the Workshop on Dynamics and Related Questions, PennState University, 1993) of the present author. Here we present a solution of the billiard problem in full detail, thus settling Finn’s question in a surprising way. In particular, we show that such floating cylinders exist if and only if the contact angle lies in a certain, explicitly described countably dense set. Moreover, for each element γ in this set we exhibit a family of convex, non-circular cylinders that float in every orientation with contact angle γ. Our discussion contains other material of independent interest for the billiard ball problem.
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Arnold V.I.: Singularities of Caustics and Wave Fronts. Kluwer Academic Publishers, Dordrecht (1990)
Auerbach H.: Sur un problème de M. Ulam concernant l’equilibre des corps flottants. Studia Math. 7, 121–142 (1938)
Bangert, V.: Mather sets for twist maps and geodesics on tori. Dynamics reported, vol. 1, pp. 1–56. Wiley, Chichester (1988)
Bangert V., Gutkin E.: Insecurity for compact surfaces of positive genus. Geom. Dedicata 146, 165–191 (2010)
Baryshnikov Y., Zharnitsky V.: Sub-Riemannian geometry and periodic orbits in classical billiards. Math. Res. Lett. 13, 587–598 (2006)
Bennequin D.: Caustique mystique (d’après Arnold et al.). Séminaire Bourbaki, 1984/85. Astérisque 133–134, 19–56 (1986)
Birkhoff G.D.: Dynamical systems with two degrees of freedom. Trans. Am. Math. Soc. 18, 199–300 (1917)
Boltyanski V.G., Yaglom I.M.: Convex Figures. Rinehart and Winston, New York (1960)
Borisov A., Filaseta M., Lam T.Y., Trifonov O.: Classes of polynomials having only one non-cyclotomic irreducible factor. Acta Arith. 90, 121–153 (1999)
Bracho J., Montejano L., Oliveros D.: Zindler curves and the floating body problem. Period. Math. Hungar. 49, 9–23 (2004)
Bruce J.W., Giblin P.J.: Curves and Singularities. A Geometrical Introduction to Singularity Theory. Cambridge University Press, Cambridge (1984)
Cyr, V.: A number theoretic question arising in the geometry of plane curves and in billiard dynamics (2011) arXiv:1103.5072, Proc. AMS, in press
Douady, R.: Applications du théorème des tores invariants. Thése de troisième cycle, Paris VI (1982)
Finn R.: Floating bodies subject to capillary attractions. J. Math. Fluid Mech. 11, 443–458 (2009)
Finn R.: Remarks on “Floating bodies in neutral equilibrium”. J. Math. Fluid Mech. 11, 466–467 (2009)
Finn R.: Remarks on “Floating Bodies subject to capillary attraction”. J. Math. Fluid Mech. 11, 464–465 (2009)
Finn R., Sloss M.: Floating Bodies in neutral equilibrium. J. Math. Fluid Mech. 11, 459–463 (2009)
Finn R., Vogel T.: Floating criteria in three dimensions. Analysis 29, 387–402 (2009)
Gutkin E.: Billiard flows on almost integrable polyhedral surfaces. Erg. Theory Dyn. Syst. 4, 560–584 (1984)
Gutkin, E.: Billiard tables of constant width and dynamical characterizations of the circle. pp. 21–24. In: Proceedings of the Workshop on Dynamics and Related Questions, PennState University (1993)
Gutkin, E.: A few remarks on the billiard ball problem, pp. 157–165. Contemp. Math. vol. 173. A. M. S., Providence, RI (1994)
Gutkin E.: Two applications of calculus to triangular billiards. Am. Math. Mon. 104, 618–622 (1997)
Gutkin E.: Billiard dynamics: a survey with the emphasis on open problems. Reg. Chaot. Dyn. 8, 1–13 (2003)
Gutkin E.: Blocking of billiard orbits and security for polygons and flat surfaces. GAFA: Geom. Funct. Anal. 15, 83–105 (2005)
Gutkin E.: Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces. Reg. Chaot. Dyn. 15, 482–503 (2010)
Gutkin, E.: Capillary floating and the billiard ball problem (2010) arXiv:1012.2448, preprint
Gutkin E., Katok A.: Caustics for inner and outer billiards. Comm. Math. Phys. 173, 101–133 (1995)
Gutkin E., Knill O.: Billiards that share a triangular caustic, pp. 199–213. World Sci. Publ., River Edge, NJ (1996)
Gutkin E., Rams M.: Growth rates for geometric complexities and counting functions in polygonal billiards. Erg. Theory Dyn. Sys. 29, 1163–1183 (2009)
Innami N.: Convex curves whose points are vertices of billiard triangles. Kodai Math. J. 11, 17–24 (1988)
Katok, A.: Billiard table as a playground for a mathematician, pp. 216–242. London Math. Soc. Lecture Notes, vol. 321. Cambridge University Press, Cambridge (2005)
de Laplace P.S.: Traité de Mécanique Céleste, Vol. 4, Supplements au Livre X. Gauthier-Villars, Paris (1806)
Lazutkin V.F.: Existence of caustics for the billiard problem in a convex domain. Izv. Akad. Nauk SSSR 37, 186–216 (1973)
Mather, J.N., Forni, G.: Action minimizing orbits in Hamiltonian systems. In: Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991), pp. 92–186. Lecture Notes in Math. vol. 1589. Springer, Berlin (1994)
Raphaël E., di Meglio J.-M., Berger M., Calabi E.: Convex particles at interfaces. J. Phys. I France 2, 571–579 (1992)
The Scottish Book. Mathematics from the Scottish Café. In: Selected papers presented at the Scottish Book Conference held at North Texas State University. Birkhäuser, Boston (1981)
Tabachnikov S.: Billiards. Soc. Math. de France, Paris (1995)
Tabachnikov S.: Geometry and billiards. AMS, Providence (2005)
Tabachnikov S.: Tire track geometry: variations on a theme. Israel J. Math. 151, 1–28 (2006)
Varkonyi P.L.: Floating body problems in two dimensions. Stud. Appl. Math. 122, 195–218 (2009)
Wegner F.: Floating bodies of equilibrium. Stud. Appl. Math. 111, 167–183 (2003)
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Gutkin, E. Capillary Floating and the Billiard Ball Problem. J. Math. Fluid Mech. 14, 363–382 (2012). https://doi.org/10.1007/s00021-011-0071-0
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DOI: https://doi.org/10.1007/s00021-011-0071-0