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Geodesic scattering on hyperboloids
Multi angle
Authors : Veselov, Alexander (Author of the conference)
CIRM (Publisher )
53C22 - Geodesics [See also 58E10]
CIRM (Publisher )
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Abstract :
Geodesic flow on ellipsoids is one of the most celebrated classical integrable systems considered by Jacobi in 1837.
Moser revisited this problem in 1978 revealing the link with the modern theory of solitons.
Surprisingly a similar question for hyperboloids did not get much attention, although the dynamics in this case is very different.
I will explain how to use the remarkable results of Moser and Knoerrer on the relations between Jacobi problem and integrable Neumann system on sphere to describe explicitly the geodesic scattering on hyperboloids. It will be shown also that Knoerrer's reparametrisation is closely related to the projectively equivalent metric on a quadric discovered in 1998 by Tabachnikov and, independently, by Matveev and Topalov, giving a new proof of their result. The projectively equivalent metric (in contrast to the usual one) turns out to be regular on the projective closure of hyperboloid, which allows usto extend Knoerrer's map to this closure.
The talk is based on a recent joint work with Lihua Wu.
Keywords : geodesics; integrability; scattering
MSC Codes : Moser revisited this problem in 1978 revealing the link with the modern theory of solitons.
Surprisingly a similar question for hyperboloids did not get much attention, although the dynamics in this case is very different.
I will explain how to use the remarkable results of Moser and Knoerrer on the relations between Jacobi problem and integrable Neumann system on sphere to describe explicitly the geodesic scattering on hyperboloids. It will be shown also that Knoerrer's reparametrisation is closely related to the projectively equivalent metric on a quadric discovered in 1998 by Tabachnikov and, independently, by Matveev and Topalov, giving a new proof of their result. The projectively equivalent metric (in contrast to the usual one) turns out to be regular on the projective closure of hyperboloid, which allows usto extend Knoerrer's map to this closure.
The talk is based on a recent joint work with Lihua Wu.
Keywords : geodesics; integrability; scattering
53C22 - Geodesics [See also 58E10]
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 15/11/2021 Conference Date : 04/10/2021 Subseries : Research talks arXiv category : Differential Geometry Mathematical Area(s) : Dynamical Systems & ODE ; Geometry Format : MP4 (.mp4) - HD Video Time : 00:59:39 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2021-10-04_Veselov.mp4 
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Information on the Event
Event Title : Differential Geometry, Billiards, and Geometric Optics / Géométrie différentielle, billards et optique géométriqueEvent Organizers : Glutsyuk, Alexey ; Izmestiev, Ivan ; Radnovic, Milena ; Tabachnikov, Sergei Dates : 04/10/2021 - 08/10/2021 Event Year : 2021 Event URL : https://conferences.cirm-math.fr/2383.html
Citation Data
DOI : 10.24350/CIRM.V.19820303Cite this video as: Veselov, Alexander (2021). Geodesic scattering on hyperboloids. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19820303 URI : http://dx.doi.org/10.24350/CIRM.V.19820303 |
See Also
- [Multi angle] Novikov's problem and tiling billiards / Author of the conference Paris-Romaskevich, Olga.
- [Multi angle] Billiards on regular polygons / Author of the conference Davis, Diana.
- [Multi angle] Degenerate billiards / Author of the conference Bolotin, Sergei.
- [Multi angle] Birkhoff-Poritsky conjecture for centrally-symmetric billiards / Author of the conference Bialy, Michael.
Bibliography
- VESELOV, A. P. et WU, L. H. Geodesic scattering on hyperboloids and Knörrer's map. Nonlinearity, 2021, vol. 34, no 9, p. 5926. - https://arxiv.org/abs/2008.12524
