Under the variational or Lagrangian approach the MFG system, we consider posed on the torus, is somehow decoupled. One can solve first the fixed point problem on the space of Borel probability on the continuous curves of the torus through a semi--Lagrangian approach, and subsequently the HJ equation, somehow disregarding the continuity equation. With some more assumptions on the Hamiltonian, it is possible to show that the variational solutions of above are actually solutions of the system in the usual sense, with continuity equation driven by a vector field depending on the solution $u$ of the HJ equation, but with the advantage of being defined without requiring any additional differentiability condition on $u$.