Another New Solvable Many-Body Model of Goldfish Type
A new solvable many-body problem is identified. It is characterized by nonlinear Newtonian equations of motion (''acceleration equal force'') featuring one-body and two-body velocity-dependent forces ''of goldfish type'' which determine the motion of an arbitr...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2012 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2012
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/148460 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Another New Solvable Many-Body Model of Goldfish Type / C. Calogero // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 16 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862546439880572928 |
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| author | Calogero, F. |
| author_facet | Calogero, F. |
| citation_txt | Another New Solvable Many-Body Model of Goldfish Type / C. Calogero // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 16 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A new solvable many-body problem is identified. It is characterized by nonlinear Newtonian equations of motion (''acceleration equal force'') featuring one-body and two-body velocity-dependent forces ''of goldfish type'' which determine the motion of an arbitrary number N of unit-mass point-particles in a plane. The N (generally complex) values zn(t) at time t of the N coordinates of these moving particles are given by the N eigenvalues of a time-dependent N×N matrix U(t) explicitly known in terms of the 2N initial data zn(0) and z˙n(0). This model comes in two different variants, one featuring 3 arbitrary coupling constants, the other only 2; for special values of these parameters all solutions are completely periodic with the same period independent of the initial data (''isochrony''); for other special values of these parameters this property holds up to corrections vanishing exponentially as t→∞ (''asymptotic isochrony''). Other isochronous variants of these models are also reported. Alternative formulations, obtained by changing the dependent variables from the N zeros of a monic polynomial of degree N to its N coefficients, are also exhibited. Some mathematical findings implied by some of these results – such as Diophantine properties of the zeros of certain polynomials – are outlined, but their analysis is postponed to a separate paper.
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| first_indexed | 2025-11-25T10:19:40Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148460 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-25T10:19:40Z |
| publishDate | 2012 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Calogero, F. 2019-02-18T13:04:40Z 2019-02-18T13:04:40Z 2012 Another New Solvable Many-Body Model of Goldfish Type / C. Calogero // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 16 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J35; 37C27; 70F10; 70H08 DOI: http://dx.doi.org/10.3842/SIGMA.2012.046 https://nasplib.isofts.kiev.ua/handle/123456789/148460 A new solvable many-body problem is identified. It is characterized by nonlinear Newtonian equations of motion (''acceleration equal force'') featuring one-body and two-body velocity-dependent forces ''of goldfish type'' which determine the motion of an arbitrary number N of unit-mass point-particles in a plane. The N (generally complex) values zn(t) at time t of the N coordinates of these moving particles are given by the N eigenvalues of a time-dependent N×N matrix U(t) explicitly known in terms of the 2N initial data zn(0) and z˙n(0). This model comes in two different variants, one featuring 3 arbitrary coupling constants, the other only 2; for special values of these parameters all solutions are completely periodic with the same period independent of the initial data (''isochrony''); for other special values of these parameters this property holds up to corrections vanishing exponentially as t→∞ (''asymptotic isochrony''). Other isochronous variants of these models are also reported. Alternative formulations, obtained by changing the dependent variables from the N zeros of a monic polynomial of degree N to its N coefficients, are also exhibited. Some mathematical findings implied by some of these results – such as Diophantine properties of the zeros of certain polynomials – are outlined, but their analysis is postponed to a separate paper. This paper is a contribution to the Special Issue “Geometrical Methods in Mathematical Physics”. The full collection is available at http://www.emis.de/journals/SIGMA/GMMP2012.html. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Another New Solvable Many-Body Model of Goldfish Type Article published earlier |
| spellingShingle | Another New Solvable Many-Body Model of Goldfish Type Calogero, F. |
| title | Another New Solvable Many-Body Model of Goldfish Type |
| title_full | Another New Solvable Many-Body Model of Goldfish Type |
| title_fullStr | Another New Solvable Many-Body Model of Goldfish Type |
| title_full_unstemmed | Another New Solvable Many-Body Model of Goldfish Type |
| title_short | Another New Solvable Many-Body Model of Goldfish Type |
| title_sort | another new solvable many-body model of goldfish type |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148460 |
| work_keys_str_mv | AT calogerof anothernewsolvablemanybodymodelofgoldfishtype |