Discrete-time Calogero–Moser system and Lagrangian 1-form structure

Abstract

We study the Lagrange formalism of the (rational) Ruijsenaars-Schneider (RS) system, both in discrete time as well as in continuous time, as a further example of a Lagrange 1-form structure in the sense of the recent paper [28]. The discrete-time model of the RS system was established some time ago arising via an Ansatz of a Lax pair, and was shown to lead to an exactly integrable correspondence (multivalued map) [17]. In this paper we consider an extended system representing a family of commuting flows of this type, and establish a connection with the lattice KP system. In the Lagrangian 1-form structure of this extended model, the closure relation is verified making use of the equations of motion. Performing successive continuum limits on the RS system, we establish the Lagrange 1-form structure for the corresponding continuum case of the RS model.

The Lagrangian structure of Calogero’s goldfish model

“…Alternatively, Lagrangians (4.8) and (4.9) can be obtained by performing the full continuum limit on the action as in the case of Calogero-Moser system and RuijsenaarsSchneider system, see [6,7]. Furthermore, we find that the closure relation for these…”

“…In [6], the discrete-time Calogero-Moser system was naturally obtained by looking at the pole-solution of the semi-discrete KP equation. In contrast, the discrete-time Ruijsenaars-Schneider system was constructed from Ansatz Lax pair.…”

“…In contrast, the discrete-time Ruijsenaars-Schneider system was constructed from Ansatz Lax pair. However, in [7], the connect between the Ruijseenaars-Schneider system and the lattice KP systems was established. In the same fashion with the Ruijseenaars-Schneider system, we start to derive the discrete-time Calogero's goldfish from the Ansatz Lax pair.…”

“…The key idea of this theory is that the action of the systems is invariant under the variation on the independent variables resulting in the feature relation called the closure relation which can be considered to be representation of the multi-dimensional consistency in Lagrangians aspect. For the case D = 1, the concrete model called the rational Calogero-Moser system which is the many-body system in one dimension with a long range interaction [4,5], was studied in both discrete time and continuous time [6]. In this case, the Lagrangians satisfy the 1-form structure.…”

“…In this case, the Lagrangians satisfy the 1-form structure. Then soon after the rational Ruijsenaars-Schneider system (considered to be relativistic version of the Calogero-Moser system) was also studied in the full detail of its Lagrangian structure [7]. In the case of one dimensional many-body system with nearest neighbour interaction called the Toda-typed system was also studied in the discrete level [8,9].…”