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

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.

DOI: https://doi.org/10.4213/faa258

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

Bibliographic databases:

UDC: 514.85+515.178+531.01

Citation: Yu. N. Fedorov, “An Ellipsoidal Billiard with a Quadratic Potential”, Funktsional. Anal. i Prilozhen., 35:3 (2001), 48–59; Funct. Anal. Appl., 35:3 (2001), 199–208

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa258
• https://doi.org/10.4213/faa258
• http://mi.mathnet.ru/eng/faa/v35/i3/p48

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Dragović V., Jovanović B., Radnović M., “On elliptical billiards in the Lobachevsky space and associated geodesic hierarchies”, J. Geom. Phys., 47:2-3 (2003), 221–234
2. Dragović V., Radnović M., “Cayley-type conditions for billiards within $k$ quadrics in $\mathbb R^d$”, J. Phys. A, 37:4 (2004), 1269–1276
3. Abenda S., Fedorov Y., “Integrable ellipsoidal billiards with separable polynomial potentials”, Equadiff 2003: International Conference on Differential Equations, 2005, 687–692
4. V. Dragović, M. Radnović, “Integrable billiards and quadrics”, Russian Math. Surveys, 65:2 (2010), 319–379
5. Abenda S., Grinevich P.G., “Periodic billiard orbits on n-dimensional ellipsoids with impacts on confocal quadrics and isoperiodic deformations”, J Geom Phys, 60:10 (2010), 1617–1633
6. Jovanovic B., “The Jacobi-Rosochatius Problem on an Ellipsoid: the Lax Representations and Billiards”, Arch. Ration. Mech. Anal., 210:1 (2013), 101–131
7. Radnovic M., “Topology of the Elliptical Billiard With the Hooke'S Potential”, Theor. Appl. Mech., 42:1 (2015), 1–9
8. Jovanovic B., Jovanovic V., “Geodesic and Billiard Flows on Quadrics in Pseudo-Euclidean Spaces: l-a Pairs and Chasles Theorem”, Int. Math. Res. Notices, 2015, no. 15, 6618–6638
9. Jovanovic B., Jovanovic V., “Virtual billiards in pseudo-Euclidean spaces: discrete Hamiltonian and contact integrability”, Discret. Contin. Dyn. Syst., 37:10 (2017), 5163–5190
10. Božidar Jovanović, Vladimir Jovanović, “Heisenberg Model in Pseudo-Euclidean Spaces II”, Regul. Chaotic Dyn., 23:4 (2018), 418–437
11. Vladimir Dragović, Milena Radnović, “Caustics of Poncelet Polygons and Classical Extremal Polynomials”, Regul. Chaotic Dyn., 24:1 (2019), 1–35
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