Capillary floating and the billiard ball problem

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 p/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Math Fluid Mech 11:464-465, 2009) pointed out a connection with an independent problem on billiard caustics citing the unpublished work (Gutkin in Proceedings of the Workshop on Dynamics and Related Questions, PennState University, 1993) of the present author.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 ? in this set we exhibit a family of convex, non-circular cylinders that float in every orientation with contact angle ?. Our discussion contains other material of independent interest for the billiard ball problem. © 2011 Springer Basel AG.