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<p>We consider a Kepler billiard with zero-energy in the plane defined inside a smooth closed connected simple curve which intersects all focused parabola at at most two points. We show that if it has an invariant curve consisting of 2-periodic orbits and there exists a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^{1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-first integral with non-vanishing gradient in the region between the invariant curve and the boundary curve, then the system is defined actually inside an ellipse with the Kepler center occupying one of the foci. This statement is obtained as a simple “translation” of the theorem of Bialy–Mironov [<CitationRef CitationID="CR3">3</CitationRef>] with Levi–Civita transformation.</p>

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Integrability of Kepler billiards at zero-energy

  • 期刊论文

  • 发表日期：

    2025年11月25日

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  • Lei Zhao

摘要

We consider a Kepler billiard with zero-energy in the plane defined inside a smooth closed connected simple curve which intersects all focused parabola at at most two points. We show that if it has an invariant curve consisting of 2-periodic orbits and there exists a \(C^{1}\) C 1 -first integral with non-vanishing gradient in the region between the invariant curve and the boundary curve, then the system is defined actually inside an ellipse with the Kepler center occupying one of the foci. This statement is obtained as a simple “translation” of the theorem of Bialy–Mironov [3] with Levi–Civita transformation.