Introducing sub-Riemannian and sub-Finsler Billiards
We define billiards in the context of sub-Finsler Geometry. We provide symplectic and variational (or rather, control theoretical) descriptions of the problem and show that they coincide. We then discuss several phenomena in this setting, including the failure of the reflection law to be well-define...
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| Main Authors: | , |
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| Format: | Article (Journal) Chapter/Article |
| Language: | English |
| Published: |
9 December 2020
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| Edition: | Version v2 |
| In: |
Arxiv
Year: 2020, Pages: 1-38 |
| DOI: | 10.48550/arXiv.2011.12136 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.48550/arXiv.2011.12136 Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/2011.12136 
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| Author Notes: | Lucas Dahinden and Álvaro del Pino |
| Summary: | We define billiards in the context of sub-Finsler Geometry. We provide symplectic and variational (or rather, control theoretical) descriptions of the problem and show that they coincide. We then discuss several phenomena in this setting, including the failure of the reflection law to be well-defined at singular points of the boundary distribution, the appearance of gliding and creeping orbits, and the behavior of reflections at wavefronts. We then study some concrete tables in 3-dimensional euclidean space endowed with the standard contact structure. These can be interpreted as planar magnetic billiards, of varying magnetic strength, for which the magnetic strength may change under reflection. For each table we provide various results regarding periodic trajectories, gliding orbits, and creeping orbits. |
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| Item Description: | Identifizierung der Ressource nach: 9 Dec 2020 Version v1 vom 24. November 2020, Version v2 vom 9. Dezember 2020 Gesehen am 05.10.2022 |
| Physical Description: | Online Resource |
| DOI: | 10.48550/arXiv.2011.12136 |