Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere
We consider the two- and <i>n</i>-body problems on the two-dimensional conformal sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">M</...
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MDPI AG
2024-03-01
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| Online Access: | https://www.mdpi.com/2227-7390/12/7/1025 |
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| author | Rubén Darío Ortiz Ortiz Ana Magnolia Marín Ramírez Ismael Oviedo de Julián |
| author_facet | Rubén Darío Ortiz Ortiz Ana Magnolia Marín Ramírez Ismael Oviedo de Julián |
| author_sort | Rubén Darío Ortiz Ortiz |
| collection | DOAJ |
| description | We consider the two- and <i>n</i>-body problems on the two-dimensional conformal sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">M</mi><mi>R</mi><mn>2</mn></msubsup></semantics></math></inline-formula>, with a radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate radius–mass relationship, these particles experience an unrestricted and free movement in alignment with the geodesic of the canonical Killing vector field in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">M</mi><mi>R</mi><mn>2</mn></msubsup></semantics></math></inline-formula>. An even number of bodies with pairwise conjugated positions, arranged in a regular <i>n</i>-gon, all with the same mass <i>m</i>, move freely on a geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium. Also, a center of mass formula is included. A relation is found for the relative equilibrium in the two-body problem in the sphere similar to the Snell law. |
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| language | English |
| last_indexed | 2024-04-24T10:39:56Z |
| publishDate | 2024-03-01 |
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| series | Mathematics |
| spelling | doaj.art-39cd457cc58a47e9b77a33e5a66b498e2024-04-12T13:22:39ZengMDPI AGMathematics2227-73902024-03-01127102510.3390/math12071025Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal SphereRubén Darío Ortiz Ortiz0Ana Magnolia Marín Ramírez1Ismael Oviedo de Julián2Grupo Ondas, Departamento de Matemáticas, Universidad de Cartagena, Sede San Pablo, Cartagena de Indias 130001, ColombiaGrupo Ondas, Departamento de Matemáticas, Universidad de Cartagena, Sede San Pablo, Cartagena de Indias 130001, ColombiaUnidad Azcapotzalco, Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Cd. de México 02128, MexicoWe consider the two- and <i>n</i>-body problems on the two-dimensional conformal sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">M</mi><mi>R</mi><mn>2</mn></msubsup></semantics></math></inline-formula>, with a radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate radius–mass relationship, these particles experience an unrestricted and free movement in alignment with the geodesic of the canonical Killing vector field in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">M</mi><mi>R</mi><mn>2</mn></msubsup></semantics></math></inline-formula>. An even number of bodies with pairwise conjugated positions, arranged in a regular <i>n</i>-gon, all with the same mass <i>m</i>, move freely on a geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium. Also, a center of mass formula is included. A relation is found for the relative equilibrium in the two-body problem in the sphere similar to the Snell law.https://www.mdpi.com/2227-7390/12/7/1025conformal sphere <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm3444"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="double-struck">M</mml:mi> <mml:mi>R</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>the two-body problemrelative equilibriaantipodal points |
| spellingShingle | Rubén Darío Ortiz Ortiz Ana Magnolia Marín Ramírez Ismael Oviedo de Julián Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere Mathematics conformal sphere <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm3444"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="double-struck">M</mml:mi> <mml:mi>R</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content> the two-body problem relative equilibria antipodal points |
| title | Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere |
| title_full | Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere |
| title_fullStr | Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere |
| title_full_unstemmed | Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere |
| title_short | Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere |
| title_sort | asymptotic antipodal solutions as the limit of elliptic relative equilibria for the two and n body problems in the two dimensional conformal sphere |
| topic | conformal sphere <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm3444"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="double-struck">M</mml:mi> <mml:mi>R</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content> the two-body problem relative equilibria antipodal points |
| url | https://www.mdpi.com/2227-7390/12/7/1025 |
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