RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



SIGMA:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


SIGMA, 2012, Volume 8, 046, 17 pages (Mi sigma723)  

This article is cited in 2 scientific papers (total in 2 papers)

Another new solvable many-body model of goldfish type

Francesco Calogeroab

a Physics Department, University of Rome ''La Sapienza'', Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy
b INFN — National Institute of Nuclear Physics

Abstract: 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 $z_{n}(t)$ at time $t$ of the $N$ coordinates of these moving particles are given by the $N$ eigenvalues of a time-dependent $N\times N$ 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 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\to\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.

Keywords: nonlinear discrete-time dynamical systems, integrable and solvable maps, isochronous discrete-time dynamical systems, discrete-time dynamical systems of goldfish type.

DOI: https://doi.org/10.3842/SIGMA.2012.046

Full text: PDF file (372 kB)
Full text: http://emis.mi.ras.ru/.../046
References: PDF file   HTML file

Bibliographic databases:

ArXiv: 1207.4850
MSC: 37J35; 37C27; 70F10; 70H08
Received: May 3, 2012; in final form July 17, 2012; Published online July 20, 2012
Language:

Citation: Francesco Calogero, “Another new solvable many-body model of goldfish type”, SIGMA, 8 (2012), 046, 17 pp.

Citation in format AMSBIB
\Bibitem{Cal12}
\by Francesco Calogero
\paper Another new solvable many-body model of goldfish type
\jour SIGMA
\yr 2012
\vol 8
\papernumber 046
\totalpages 17
\mathnet{http://mi.mathnet.ru/sigma723}
\crossref{https://doi.org/10.3842/SIGMA.2012.046}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2958984}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000306612500001}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84864837807}


Linking options:
  • http://mi.mathnet.ru/eng/sigma723
  • http://mi.mathnet.ru/eng/sigma/v8/p46

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Francesco Calogero, Ge Yi, “A new class of solvable many-body problems”, SIGMA, 8 (2012), 066, 29 pp.  mathnet  crossref  mathscinet
    2. Calogero F., Yi G., “Polynomials Satisfying Functional and Differential Equations and Diophantine Properties of their Zeros”, Lett. Math. Phys., 103:6 (2013), 629–651  crossref  mathscinet  zmath  adsnasa  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
    Number of views:
    This page:135
    Full text:19
    References:15

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019