Geodesic
On a~technique to identify solvable discrete-time many-body problems
Teoretičeskaâ i matematičeskaâ fizika, Tome 172 (2012) no. 2, pp. 198-223

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The starting point is an $N{\times}N$ matrix, $U\equiv U(\ell)$, evolving in the discrete-time independent variable $\ell=0,1,2,\dots$ according to a solvable matrix evolution equation. One then focuses on the evolution of its $N$ eigenvalues $z_n(\ell)$. This evolution generally also involves $N(N{-}1)$ additional variables. In some cases via a compatible ansatz these additional variables can be expressed in terms of the $N$ variables $z_n(\ell)$. Thereby one obtains a system of discrete-time evolution equations involving only the $N$ dependent variables $z_n(\ell)$, which is often interpretable as a discrete-time many-body problem. Various peculiarities of this approach are investigated, including the possibility to manufacture nontrivial isochronous models (all solutions of which are periodic with the same period). These properties are illustrated via specific examples. In the process novel discrete-time many-body problems are exhibited.
Keywords: integrable discrete-time dynamical system, solvable discrete-time dynamical system, integrable discrete-time many-body problem, solvable discrete-time many-body problem
Mots-clés : isochronous discrete-time evolution. 
@article{TMF_2012_172_2_a2,
     author = {F. Calogero},
     title = {On a~technique to identify solvable discrete-time many-body problems},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {198--223},
     publisher = {mathdoc},
     volume = {172},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2012_172_2_a2/}
}
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F. Calogero. On a~technique to identify solvable discrete-time many-body problems. Teoretičeskaâ i matematičeskaâ fizika, Tome 172 (2012) no. 2, pp. 198-223. http://geodesic.mathdoc.fr/item/TMF_2012_172_2_a2/