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...
Збережено в:
| Дата: | 2012 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2012
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/148460 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1484602019-02-19T01:23:34Z Another New Solvable Many-Body Model of Goldfish Type Calogero, F. 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. 2012 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/148460 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Calogero, F. |
| spellingShingle |
Calogero, F. Another New Solvable Many-Body Model of Goldfish Type Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Calogero, F. |
| author_sort |
Calogero, F. |
| title |
Another New Solvable Many-Body Model of Goldfish Type |
| title_short |
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_sort |
another new solvable many-body model of goldfish type |
| publisher |
Інститут математики НАН України |
| publishDate |
2012 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/148460 |
| 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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:30:44Z |
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2023-05-20T17:30:44Z |
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