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Discrete-time Calogero-Moser system and Lagrangian 1-form structure
We study the Lagrange formalism of the (rational) Calogero-Moser (CM) system,
both in discrete time as well as in continuous time, as a first example of a
Lagrange 1-form structure in the sense of the recent paper [19]. The
discrete-time model of the CM system was established some time ago arising as a
pole-reduction of a semi-discrete version of the KP equation, and was shown to
lead to an exactly integrable correspondence (multivalued map). In this paper
we present the full KP solution based on the commutativity of the discrete-time
flows in the two discrete KP variables. The compatibility of the corresponding
Lax matrices is shown to lead directly to the relevant closure relation on the
level of the Lagrangians. Performing successive continuum limits on both the
level of the KP equation as well as of the CM system, we establish the proper
Lagrange 1-form structure for the continuum case of the CM model. We use the
example of the three-particle case to elucidate the implementation of the novel
least-action principle, which was presented in [19], for the simpler case of
Lagrange 1-forms.Comment: 37 pages, 8 figure