  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 [24].
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)[15]. 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.
