Variational symmetries and pluri-Lagrangian systems in classical mechanics

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