Remarks on Joachimsthal Integral and Poritsky Property

Maxim Arnold, Serge Tabachnikov

Research output: Contribution to journalArticlepeer-review

Abstract

The billiard in an ellipse has a conserved quantity, the Joachimsthal integral. We show that the existence of such an integral characterizes conics. We extend this result to the spherical and hyperbolic geometries and to higher dimensions. We connect the existence of Joachimsthal integral with the Poritsky property, a property of billiard curves, called so after H. Poritsky whose important paper Poritsky (Ann Math 51:446–470, 1950) was one of the early studies of the billiard problem.

Original languageEnglish (US)
Pages (from-to)483-491
Number of pages9
JournalArnold Mathematical Journal
Volume7
Issue number3
DOIs
StatePublished - Sep 2021

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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