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JOURNALS // Funktsional'nyi Analiz i ego Prilozheniya // Archive

Funktsional. Anal. i Prilozhen., 2001 Volume 35, Issue 3, Pages 48–59 (Mi faa258)

This article is cited in 22 papers

An Ellipsoidal Billiard with a Quadratic Potential

Yu. N. Fedorov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: There exists an infinite hierarchy of integrable generalizations of the geodesic flow on an $n$-dimensional ellipsoid. These generalizations describe the motion of a point in the force fields of certain polynomial potentials. In the limit as one of semiaxes of the ellipsoid tends to zero, one obtains integrable mappings corresponding to billiards with polynomial potentials inside an $(n-1)$-dimensional ellipsoid.
In this paper, for the first time we give explicit expressions for the ellipsoidal billiard with a quadratic (Hooke) potential, its representation in Lax form, and a theta function solution. We also indicate the generating function of the restriction of the potential billiard map to a level set of an energy type integral. The method we use to obtain theta function solutions is different from those applied earlier and is based on the calculation of limit values of meromorphic functions on generalized Jacobians.

UDC: 514.85+515.178+531.01

Received: 29.09.2000

DOI: 10.4213/faa258


 English version:
Functional Analysis and Its Applications, 2001, 35:3, 199–208

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