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Superintegrability of the Calogero-Moser system: constants of motion, master symmetries, and time-dependent symmetries. (English) Zbl 0956.37041
The classical Calogero-Moser system is a system of \(n\) particles in which the interaction force between every two particles is given by the inverse of the square of their relative distance. It is known that the Calogero-Moser system is not only integrable (in the sense of Liouville-Arnold) but even maximally superintegrable, i.e. there exist \(2n-1\) independent integrals of motion. This system is also endowed with a great amount of different symmetries. In the paper some properties closely related with the superintegrability of the Calogero-Moser system and the abundance of symmetries are studied. The constants of motion can be grouped in families. One of these families directly related with the fact that the Calogero-Moser system can be presented as a Lax equation was studied by J. Moser [Adv. Math. 16, 197-220 (1975; Zbl 0303.34019)]. The present paper discusses the existence and properties of additional families of time-independent and time-dependent constants of motion. The master symmetries and the time-dependent symmetries of the Calogero-Moser system are studied.

MSC:
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
70F10 \(n\)-body problems
70H05 Hamilton’s equations
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