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 ofan arbitra...
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2012-07-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Online Access: | http://dx.doi.org/10.3842/SIGMA.2012.046 |
| _version_ | 1819200152837029888 |
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| author | Francesco Calogero |
| author_facet | Francesco Calogero |
| author_sort | Francesco Calogero |
| collection | DOAJ |
| 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 ofan arbitrary number $N$ of unit-mass point-particles in a plane. The $N$ (generally complex) values $z_{n}(t)$ at time $t$ ofthe $N$ coordinates of these moving particles are given by the $N$eigenvalues of a time-dependent $Nimes N$ matrix $U(t)$explicitly known in terms of the $2N$ initial data $z_{n}(0)$and $dot{z}_{n}(0) $. This model comes in two dif/ferentvariants, 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 $tightarrow infty$ (''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. |
| first_indexed | 2024-12-23T03:27:42Z |
| format | Article |
| id | doaj.art-2f6ad3c5f37a4bb69c156d73d6ab4ce4 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-23T03:27:42Z |
| publishDate | 2012-07-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-2f6ad3c5f37a4bb69c156d73d6ab4ce42022-12-21T18:01:48ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592012-07-018046Another New Solvable Many-Body Model of Goldfish TypeFrancesco CalogeroA 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 ofan arbitrary number $N$ of unit-mass point-particles in a plane. The $N$ (generally complex) values $z_{n}(t)$ at time $t$ ofthe $N$ coordinates of these moving particles are given by the $N$eigenvalues of a time-dependent $Nimes N$ matrix $U(t)$explicitly known in terms of the $2N$ initial data $z_{n}(0)$and $dot{z}_{n}(0) $. This model comes in two dif/ferentvariants, 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 $tightarrow infty$ (''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.http://dx.doi.org/10.3842/SIGMA.2012.046nonlinear discrete-time dynamical systemsintegrable and solvable mapsisochronous discrete-time dynamical systemsdiscrete-time dynamical systems of goldfish type |
| spellingShingle | Francesco Calogero Another New Solvable Many-Body Model of Goldfish Type Symmetry, Integrability and Geometry: Methods and Applications nonlinear discrete-time dynamical systems integrable and solvable maps isochronous discrete-time dynamical systems discrete-time dynamical systems of goldfish type |
| 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 |
| topic | nonlinear discrete-time dynamical systems integrable and solvable maps isochronous discrete-time dynamical systems discrete-time dynamical systems of goldfish type |
| url | http://dx.doi.org/10.3842/SIGMA.2012.046 |
| work_keys_str_mv | AT francescocalogero anothernewsolvablemanybodymodelofgoldfishtype |