On the integrability of Birkhoff billiards

Kaloshin, Vadim; Sorrentino, Alfonso

doi:10.1098/rsta.2017.0419

Cited by 37 publications

(30 citation statements)

References 46 publications

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“…For a dynamical entropic version of Birkhoff Conjecture and related results see [39]. For a survey on Birkhoff Conjecture and related results see [35], [36], [13] and references therein.…”

Section: Historical Remarksmentioning

confidence: 99%

On rationally integrable planar dual and projective billiards

Glutsyuk¹

2021

Preprint

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A caustic of a strictly convex planar bounded billiard is a smooth curve whose tangent lines are reflected from the billiard boundary to its tangent lines. The famous Birkhoff Conjecture states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then the billiard is an ellipse. It was studied by many mathematicians, including H.Poritsky, M.Bialy, S.Bolotin, A.Mironov, V.Kaloshin, A.Sorrentino and others. In the present paper we study its following generalized dual-projective version stated by S.Tabachnikov. Consider a closed smooth strictly convex planar curve γ equipped with a dual billiard structure: a family of projective involutions acting on its projective tangent lines and fixing the tangency points. Suppose that its outer neighborhood admits a foliation by closed curves (including γ) such that the involution of each tangent line permutes its intersection points with every leaf. Tabachnikov's Conjecture states that then the curve γ is an ellipse and the leaves are ellipses forming a pencil. We prove Tabachnikov's Conjecture in the case, when the curve is C 4 -smooth and the foliation admits a rational first integral. To this end, we show that each C 4 -smooth germ γ of planar curve carrying a rationally integrable dual billiard structure is a conic and classify all the rationally integrable dual billiards on conics. They include the dual billiards induced by pencils of conics, two infinite series of exotic dual billiards and five more exotic ones.

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“…For a dynamical entropic version of Birkhoff Conjecture and related results see [39]. For a survey on Birkhoff Conjecture and related results see [35], [36], [13] and references therein.…”

Section: Historical Remarksmentioning

confidence: 99%

On rationally integrable planar dual and projective billiards

Glutsyuk¹

2021

Preprint

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“…An above-than-expected frequency of ellipses for the confocal pair was signalled in [11]. As mentioned above, irrational centers X k , k ∈ [13,16] sweep out circles for the homothetic pair. X 15 and X 16 are known to be stationary over the Brocard family [3], however the locus of X 13 and X 14 are circles!…”

Section: Discussionmentioning

confidence: 95%

“…For this reason, this pair is termed the elliptic billiard; [26] is the seminal work. It is conjectured as the only integrable planar billiard [16]. One consequence, mentioned above, is that it conserves perimeter L. An explicit parametrization for 3-periodic vertices appears in Appendix A.1.…”

Section: Review Of Classic Porisms and Proof Methodsmentioning

confidence: 99%

Family Ties

García

Reznik²

2021

Kog (Online)

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We compare loci types and invariants across Poncelet families interscribed in three distinct concentric Ellipse pairs: (i) ellipse-incircle, (ii) circumcircle-inellipse, and (iii) homothetic. Their metric properties are mostly identical to those of 3 well-studied families: elliptic billiard (confocal pair), Chapple's poristic triangles, and the Brocard porism. We therefore organized them in three related groups.

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“…Equivalently, a certain quantity, known as Joachimsthal's constant 𝐽, is conserved [4,10]. Uniquely amongst all planar billiards, the elliptic billiard is an integrable dynamical system, i.e., its phase-space if foliated by tori or equivalently, the billiard-map is volume preserving [15].…”

Section: A Review: Elliptic Billiardmentioning

confidence: 99%

Exploring self-intersected N-periodics in the elliptic billiard

Garcia¹,

Reznik²

2022

AMI

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This is a continuation of our simulation-based investigation of 𝑁 -periodic trajectories in the elliptic billiard. With a special focus on self-intersected trajectories we (i) describe new properties of 𝑁 = 4 family, (ii) derive expressions for quantities recently shown to be conserved, and to support further experimentation, we (iii) derive explicit expressions for vertices and caustic semi-axes for several families. Finally, (iv) we include links to several animations of the phenomena.

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On the integrability of Birkhoff billiards

Cited by 37 publications

References 46 publications

On rationally integrable planar dual and projective billiards

On rationally integrable planar dual and projective billiards

Family Ties

Exploring self-intersected N-periodics in the elliptic billiard

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