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Zbl 0766.58028
Calogero, F.; Françoise, J.-P.
Integrable dynamical systems obtained by duplication. (English)
[J] Ann. Inst. Henri Poincaré, Phys. Théor. 57, No.2, 167-181 (1992). ISSN 0246-0211

Integrable classical $n$-body systems on the real line are considered. It is shown how a simple ``duplication'' idea can be used to generate new integrable systems from simpler ones. A discrete symmetry of the original system is used to reduce the dynamics to a smaller set of variables, which are more natural for the symmetric case. In these variables the resulting system is again a multi-particle problem with a modified Hamiltonian. Starting from the well-known class of integrable Calogero- Moser systems one may thus start with an initial configuration of particles on the real axis, symmetric around the origin. The symmetry reduction leads to a new integrable system. This idea extends to the situation, where two configurations of particles are placed symmetrically on the real as well as the imaginary axis. In appropriate coordinates the reduced system is real and describes the integrable interaction of two different types of particles (corresponding to their original placement on the real or imaginary axis). Iterated duplications and suitable reductions of the Calogero-Moser class are considered. The integrability is established in terms of the corresponding Lax representations. Explicit solutions are obtained and it is shown that all orbits have the same period.
[W.Oevel (Loughborough)]

MSC 2000:: *37J35 Completely integrable systems, etc.
37K10 Completely integrable systems etc.
35Q58 Other completely integrable PDE

Keywords: integrable systems; Calogero-Moser systems

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