Outer Billiards with the Dynamics of a Standard Shift on a Finite Number of Invariant Curves

Bialy, Misha; Mironov, Andrey E.; Shalom, Lior

doi:10.1080/10586458.2018.1563514

Cited by 4 publications

(3 citation statements)

References 17 publications

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“…Equation (19) appeared in the context of bicycle kinematics in [11,41] and in the papers by Wegner, summarized in [47]. It also appeared in [27] in the context of billiards and flotation problems, and in [8], [9], [10] for magnetic, outer and wire billiards. This ubiquitous equation has a countable number of solutions but, except for 𝜋/2, there are no 𝜋-rational solutions [18].…”

Section: 2mentioning

confidence: 99%

Untitled

2022

Comm. Amer. Math. Soc.

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Twenty five years ago U. Pinkall discovered that the Korteweg-de Vries equation can be realized as an evolution of curves in centroaffine geometry. Since then, a number of authors interpreted various properties of KdV and its generalizations in terms of centroaffine geometry. In particular, the Bäcklund transformation of the Korteweg-de Vries equation can be viewed as a relation between centroaffine curves.Our paper concerns self-Bäcklund centroaffine curves. We describe general properties of these curves and provide a detailed description of them in terms of elliptic functions. Our work is a centroaffine counterpart to the study done by F. Wegner of a similar problem in Euclidean geometry, related to Ulam's problem of describing the (2-dimensional) bodies that float in equilibrium in all positions and to bicycle kinematics.We also consider a discretization of the problem where curves are replaced by polygons. This is related to discretization of KdV and the cross-ratio dynamics on ideal polygons.

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Section: 2mentioning

confidence: 99%

Untitled

2022

Comm. Amer. Math. Soc.

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Section: 2mentioning

confidence: 99%

Self-Bäcklund curves in centroaffine geometry and Lamé’s equation

Bialy

Bor

Tabachnikov

2022

Comm. Amer. Math. Soc.

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Twenty five years ago U. Pinkall discovered that the Korteweg-de Vries equation can be realized as an evolution of curves in centroaffine geometry. Since then, a number of authors interpreted various properties of KdV and its generalizations in terms of centroaffine geometry. In particular, the Bäcklund transformation of the Korteweg-de Vries equation can be viewed as a relation between centroaffine curves. Our paper concerns self-Bäcklund centroaffine curves. We describe general properties of these curves and provide a detailed description of them in terms of elliptic functions. Our work is a centroaffine counterpart to the study done by F. Wegner of a similar problem in Euclidean geometry, related to Ulam’s problem of describing the (2-dimensional) bodies that float in equilibrium in all positions and to bicycle kinematics. We also consider a discretization of the problem where curves are replaced by polygons. This is related to discretization of KdV and the cross-ratio dynamics on ideal polygons.

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“…Equation ( 18) appeared in the context of bicycle kinematics in [42,12] and in the papers by Wegner, summarized in [49]. It also appeared in [28] in the context of billiards and flotation problems, and in [9], [10], [11] for magnetic, outer and wire billiards. This ubiquitous equation has a countable number of solutions but, except for π/2, there are no π-rational solutions [19].…”

Section: See Proposition 2 Ofmentioning

confidence: 99%

Self-Bäcklund curves in centroaffine geometry and Lamé's equation

Bialy¹,

Bor²,

Tabachnikov³

2020

Preprint

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Twenty five years ago U. Pinkall discovered that the Korteweg-de Vries equation can be realized as an evolution of curves in centoraffine geometry. Since then, a number of authors interpreted various properties of KdV and its generalizations in terms of centoraffine geometry. In particular, the Bäcklund transformation of the Korteweg-de Vries equation can be viewed as a relation between centroaffine curves.Our paper concerns self-Bäcklund centroaffine curves. We describe general properties of these curves and provide a detailed description of them in terms of elliptic functions. Our work is a centroaffine counterpart to the study done by F. Wegner of a similar problem in Euclidean geometry, related to Ulam's problem of describing the (2-dimensional) bodies that float in equilibrium in all positions and to bicycle kinematics.We also consider a discretization of the problem where curves are replaced by polygons. This is related to discretization of KdV and the cross-ratio dynamics on ideal polygons.

show abstract

Outer Billiards with the Dynamics of a Standard Shift on a Finite Number of Invariant Curves

Cited by 4 publications

References 17 publications

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Self-Bäcklund curves in centroaffine geometry and Lamé’s equation

Self-Bäcklund curves in centroaffine geometry and Lamé's equation

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