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Linear stability in billiards with potential

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Holger R. Dullin (University of Bremen, Germany)

chao-dyn · Submitted 16 Dec 1996

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim A general formula for the linearized Poincar\'e map of a billiard with a potential is derived.

how it is stated · asserted 1.00 read the record →

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arXiv IDchao-dyn/9612023 (version 1)
Pith IDZA2GDLYH, Pith's permanent record of this paper and its review
CitationGermany), Holger R. Dullin (University of Bremen. "Linear stability in billiards with potential." arXiv:chao-dyn/9612023 (1996). https://pith.science/paper/ZA2GDLYH
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,273 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.11 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.77 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

A general formula for the linearized Poincar\'e map of a billiard with a potential is derived. The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the contributions from the reflections alone. For the case without potential this gives well known formulas. Four billiards with potentials for which the free motion is integrable are treated as examples: The linear gravitational potential, the constant magnetic field, the harmonic potential, and a billiard in a rotating frame of reference, imitating the restricted three body problem. The linear stability of periodic orbits with period one and two is analyzed with the help of stability diagrams, showing the essential parameter dependence of the residue of the periodic orbits for these examples.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 A general formula for the linearized Poincar\'e map of a billiard with a potential is derived. contribution 0.87 0.95 0.39 asserted 1.00
s1 The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the contributions from the reflections alone. result 0.66 0.85 0.50 asserted 1.00
s2 For the case without potential this gives well known formulas. interpretation 0.38 0.25 0.05 asserted 1.00
s3 Four billiards with potentials for which the free motion is integrable are treated as examples: The linear gravitational potential, the constant magnetic field, the harmonic potential, and a billiard in a rotating frame of reference, imitating the restricted three body problem. contribution 0.88 0.85 0.05 asserted 1.00
s4 The linear stability of periodic orbits with period one and two is analyzed with the help of stability diagrams, showing the essential parameter dependence of the residue of the periodic orbits for these examples. contribution 0.64 0.93 0.24 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 1.00.

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  • T body sentences
  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

A general formula for the linearized Poincar\'e map of a billiard with a potential is derived. The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the contributions from the reflections alone. For the case without potential this gives well known formulas. Four billiards with potentials for which the free motion is integrable are treated as examples: The linear gravitational potential, the constant magnetic field, the harmonic potential, and a billiard in a rotating frame of reference, imitating the restricted three body problem. The linear stability of periodic orbits with period one and two is analyzed with the help of stability diagrams, showing the essential parameter dependence of the residue of the periodic orbits for these examples.

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