Multiplicative dynamical systems in terms of the induced dynamics
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MULTIPLICATIVE DYNAMICAL SYSTEMS IN TERMS OF THE INDUCED DYNAMICS A. K. Pogrebkov∗
We realize an example of induced dynamics using new multiplicative determinant relations whose roots give the particle positions. We present both a general scheme for describing completely integrable dynamical systems parameterized by an arbitrary N ×N matrix of momenta and an explicit model that interpolates between the Calogero–Moser and Ruijsenaars–Schneider hyperbolic systems. We consider some special cases of this model in detail.
Keywords: induced dynamics, completely integrable system
DOI: 10.1134/S0040577920090081
1. Introduction The term induced dynamics was introduced in [1] as an example of a dynamical system that is integrable by construction. As was shown, such systems include models given by the dynamics of singularities of soliton solutions of the Korteweg–de Vries and sinh–Gordon equations and also the rational cases of the Calogero– Moser (CM) and Ruijsenaars–Schneider (RS) models. Some other examples demonstrating a nontrivial interaction of the particles of the induced dynamical system were also presented in [1]. Here, we present a further development of this approach with a formulation of general properties of an N -particle completely integrable dynamical system in R1 . As illustration, we derive a system that can be called the CM+RS model because it interpolates between the hyperbolic (trigonometric) versions of the CM and RS systems. Recalling the definition of an induced dynamical system (see [1]), we introduce the background phase space AN of an N -particle dynamical system with coordinates qi and momenta pi , i = 1, . . . , N , canonical with respect to the Poisson bracket {qi , pj } = δij ,
i, j = 1, . . . , N.
(1)
We assume that qi are either real or pairwise complex conjugate and that pi have the analogous properties. We consider a special case where the dynamics of the background system are trivial q˙i = pi ,
p˙ i = 0.
(2)
∗
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia; Skolkovo Institute of Science and Technology, Moscow, Russia, e-mail: [email protected]. This research is supported by the Russian Foundation for Basic Research (Grant No. 18-01-00273). Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 204, No. 3, pp. 436–444, September, 2020. Received August 31, 2019. Revised March 25, 2020. Accepted March 27, 2020. c 2020 Pleiades Publishing, Ltd. 0040-5779/20/2043-1201
1201
Let a function f (q1 , . . . , qN , p1 , . . . , pN ) on AN be given such that the equation f (q1 − x, . . . , qN − x, p1 , . . . , pN ) = 0
(3)
has M simple real zeros x1 , . . . , xM , where 0 ≤ M ≤ N . We also assume that there exists an open subset AN ⊆ AN such that M = N for any (q1 , . . . , qN , p1 , . . . , pN ) ∈ AN . We define the induced dynamical system as the system with the configuration space given by the real roots of (3). This system is dynamical: because of (3), all roots xi (t) are
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