Circular, elliptic and oval billiards in a gravitational field

Costa, Diogo Ricardo da; Dettmann, Carl P.; Leonel, Edson D.

doi:10.1016/j.cnsns.2014.08.030

Cited by 15 publications

(11 citation statements)

References 36 publications

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“…Since consecutive trajectory segments are bisected by the ellipse normal, Graves' theorem implies these will be tangent to a confocal caustic [26]. 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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Section: A Review: Elliptic Billiardmentioning

confidence: 99%

Exploring self-intersected N-periodics in the elliptic billiard

Garcia¹,

Reznik²

2022

AMI

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“…A point mass bouncing elastically in the interior of an ellipse is know as the elliptic billiard; see Figure 1. Two quantities are conserved: (i) energy and (ii) product of the angular momenta about the foci [4,10]. The latter implies every trajectory segment is tangent to a virtual confocal ellipse, known as the "caustic"; still equivalently, a quantity known as Joachimsthal's constant J is conserved [21]; see Appendix A.…”

Section: Introductionmentioning

confidence: 99%

Invariants of Self-Intersected N-Periodics in the Elliptic Billiard

Garcia,

Reznik

2020

Preprint

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We study self-intersected N-periodics in the elliptic billiard, describing new facts about their geometry (e.g., self-intersected 4-periodics have vertices concyclic with the foci). We also check if some invariants listed in "Eighty New Invariants of N-Periodics in the Elliptic Billiard" (2020), arXiv:2004.12497 remain invariant in the self-intersected case. Toward that end, we derive explicit expressions for many low-N simple and self-intersected cases. We identify two special cases (one simple, one self-intersected) where a quantity prescribed to be invariant is actually variable.

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“…It would be natural to consider the particle moving in the quantum realm [7,37,39] or moving relativistically [11,12,13]. Other billiard systems consider modifications to the region of motion itself, for example, a hole or multiple holes within the region-these are the so-called "open billiards"; billiard systems where the boundary changes in time [18,19,21,24,25,26]; and billiard systems where the billiard moves under the influence of a constant force field, either magnetic [3,10,14,30,38] or gravitational [9,20,23].…”

Section: Introductionmentioning

confidence: 99%

Computational study of the dynamics of an asymmetric wedge billiard

Anderson,

Villet

2020

Preprint

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We introduce the asymmetric wedge billiard as a generalization of the wedge billiard first introduced and studied by Lehtihet and Miller in 1986. This is a billiard system in which the billiard ball moves under the influence of a constant gravitational field, colliding elastically with two wedge walls with the collisions obeying the reflection law. Collision maps are given from which derivatives and area-preservation (or lack thereof) were determined.Expressions for the fixed points of the collision maps were also calculated and discussed.Long-term dynamics were determined computationally from which we observed integrable, quasi-periodic and chaotic behaviour which were all dependent on the wedge angles.

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Circular, elliptic and oval billiards in a gravitational field

Cited by 15 publications

References 36 publications

Exploring self-intersected N-periodics in the elliptic billiard

Exploring self-intersected N-periodics in the elliptic billiard

Invariants of Self-Intersected N-Periodics in the Elliptic Billiard

Computational study of the dynamics of an asymmetric wedge billiard

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