Due to the non-linearity of Eq. ( ), the time-evolution of Φ(r, t; R) does not have a Schrödinger form, unlike the time-evolution of χ(R, t). As discussed in the Introduction, interesting applications of the exact factorization of the electron-nuclear wavefunction focused on introducing a trajectory-based perspective within this framework, such that the nuclear dynamics is ultimately approximated using classical-like trajectories while the electronic dynamics is treated quantum mechanically. The trajectory-based solution of the nuclear equation ( ) is introduced naturally, since the classical-like limit simply requires to interpret the TDVP and TDPES as standard (classical) electromagnetic potentials producing a classical force on the nuclei. Therefore, the main efforts in this context have been devoted to the calculation and approximation of the TDVP and of the TDPES from the solution of the quantum-mechanical electronic equation (5). We describe in Section II A how the concept of classical trajectories is introduced in the exact factorization and recall the approximations leading to the derivation of CTMQC, CTMQC-E and CTTSH.