  We analyze the relation of the notion of a pluri-Lagrangian system, which
recently emerged in the theory of integrable systems, to the classical notion
of variational symmetry, due to E. Noether. We treat classical mechanical
systems and show that, for any Lagrangian system with $m$ commuting variational
symmetries, one can construct a pluri-Lagrangian 1-form in the
$(m+1)$-dimensional time, whose multi-time Euler-Lagrange equations coincide
with the original system supplied with $m$ commuting evolutionary flows
corresponding to the variational symmetries. We also give a Hamiltonian
counterpart of this construction, leading, for any system of commuting
Hamiltonian flows, to a pluri-Lagrangian 1-form with coefficients depending on
functions in the phase space.
