TY - JOUR
T1 - Remarks on Joachimsthal Integral and Poritsky Property
AU - Arnold, Maxim
AU - Tabachnikov, Serge
N1 - Funding Information:
We are grateful to A. Akopyan, M. Bialy, A. Petrunin, and especially to A. Glutsyuk for useful discussions. Many thanks to the referees for their suggestions. ST was supported by NSF grant DMS-2005444.
Publisher Copyright:
© 2021, Institute for Mathematical Sciences (IMS), Stony Brook University, NY.
PY - 2021/9
Y1 - 2021/9
N2 - 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.
AB - 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.
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U2 - 10.1007/s40598-021-00180-0
DO - 10.1007/s40598-021-00180-0
M3 - Article
AN - SCOPUS:85107527781
SN - 2199-6792
VL - 7
SP - 483
EP - 491
JO - Arnold Mathematical Journal
JF - Arnold Mathematical Journal
IS - 3
ER -