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Abstract <p>Consider the problem of motion of a material point under the action of an elastic force of coefficient <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8215_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(k\)</EquationSource> <!--LobJMat2560534Belozerov-m1--> </InlineEquation> inside a triaxial ellipsoid whose center coincides with the center of the force field. Such a dynamical system is Liouville integrable in the piecewise-smooth sense. For the attractive and repulsive cases (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8215_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(k&gt;0\)</EquationSource> <!--LobJMat2560534Belozerov-m2--> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8215_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(k&lt;0\)</EquationSource> <!--LobJMat2560534Belozerov-m3--> </InlineEquation>, respectively), we describe the Liouville foliation of the system in small neighborhoods of regular layers as well as layers containing nondegenerate critical points.</p>

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Non-degenerate Singularities of a Three-dimensional Billiard Bounded by an Ellipsoid in a Hooke Potential Field

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • G. V. Belozerov

摘要

Abstract

Consider the problem of motion of a material point under the action of an elastic force of coefficient \(k\) inside a triaxial ellipsoid whose center coincides with the center of the force field. Such a dynamical system is Liouville integrable in the piecewise-smooth sense. For the attractive and repulsive cases ( \(k>0\) and \(k<0\) , respectively), we describe the Liouville foliation of the system in small neighborhoods of regular layers as well as layers containing nondegenerate critical points.