Introducing sub-Riemannian and sub-Finsler billiards

Alvaro del Pino Gomez; Lucas Dahinden

doi:10.3934/dcds.2022014

Introducing sub-Riemannian and sub-Finsler billiards

Alvaro del Pino Gomez^*, Lucas Dahinden

^*Corresponding author for this work

Heidelberg University

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

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.

Original language	English
Pages (from-to)	3187-3232
Number of pages	46
Journal	Discrete and Continuous Dynamical Systems
Volume	42
Issue number	7
Early online date	2022
DOIs	https://doi.org/10.3934/dcds.2022014
Publication status	Published - Jul 2022

Keywords

Sub-Riemannian
billiards
geodesics
gliding orbits
horizontal curves
sub-Finsler

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10.3934/dcds.2022014

https://arxiv.org/abs/2011.12136Licence: Unspecified

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title = "Introducing sub-Riemannian and sub-Finsler billiards",

abstract = "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.",

keywords = "Sub-Riemannian, billiards, geodesics, gliding orbits, horizontal curves, sub-Finsler",

author = "{del Pino Gomez}, Alvaro and Lucas Dahinden",

note = "Funding Information: 2020 Mathematics Subject Classification. Primary: 53C17. Secondary: 53D25, 37C83. Key words and phrases. Sub-Riemannian, billiards, sub-Finsler, horizontal curves, geodesics, gliding orbits. The first author is supported by Deutsche Forschungsgemeinschaft (DFG) under Germany{\textquoteright}s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster), as well as by SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics” funded by the DFG. The second author was supported by the NWO 016.Veni.192.013 grant for the duration of the project. ∗Corresponding author: {\'A}lvaro del Pino. Funding Information: Acknowledgments. The second author would like to thank Aaron Gootjes Drees-bach for providing comments on the first version of the paper. The first author is supported by Deutsche Forschungsgemeinschaft (DFG) under Germany{\textquoteright}s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster), as well as by SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics” funded by the DFG. The second author was supported by the NWO 016.Veni.192.013 grant for the duration of the project. Publisher Copyright: {\textcopyright} 2022 American Institute of Mathematical Sciences. All rights reserved.",

year = "2022",

month = jul,

doi = "10.3934/dcds.2022014",

language = "English",

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pages = "3187--3232",

journal = "Discrete and Continuous Dynamical Systems",

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AU - Dahinden, Lucas

N1 - Funding Information: 2020 Mathematics Subject Classification. Primary: 53C17. Secondary: 53D25, 37C83. Key words and phrases. Sub-Riemannian, billiards, sub-Finsler, horizontal curves, geodesics, gliding orbits. The first author is supported by Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster), as well as by SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics” funded by the DFG. The second author was supported by the NWO 016.Veni.192.013 grant for the duration of the project. ∗Corresponding author: Álvaro del Pino. Funding Information: Acknowledgments. The second author would like to thank Aaron Gootjes Drees-bach for providing comments on the first version of the paper. The first author is supported by Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster), as well as by SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics” funded by the DFG. The second author was supported by the NWO 016.Veni.192.013 grant for the duration of the project. Publisher Copyright: © 2022 American Institute of Mathematical Sciences. All rights reserved.

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N2 - 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.

AB - 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.

KW - Sub-Riemannian

KW - billiards

KW - geodesics

KW - gliding orbits

KW - horizontal curves

KW - sub-Finsler

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M3 - Article

SN - 1078-0947

VL - 42

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EP - 3232

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

IS - 7

ER -

Introducing sub-Riemannian and sub-Finsler billiards

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Keywords

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