To understand the nature of ferroelectric instabilities under finite electric fields, e.g., domain switching and polarization reversal, as an origin of critical failure of ferroelectric devices, it is essential to evaluate critical electrical and/or mechanical conditions where the ferroelectric state becomes unstable. However, the instability criterion for such a multi-physics system where the mechanical and electric properties interact each other has not yet been proposed. In this paper, we developed an analytical method to rigorously describe the multi-physics instability criterion for arbitrary atomic structures under finite electric field and/or mechanical load. According to the proposed method, the instability starts when the minimum eigenvalue of the Hessian matrix of the potential energy reaches zero. The corresponding eigenvector indicates the displacement of atoms at the instability. We applied the method to 180° domain walls in ferroelectric PbTiOs under external electric fields, and our criterion can successfully describe the onset of instability, namely, the domain switching, as well as their displacement pattern at the switching. This clearly indicates the validity of our multi-physics instability criterion. This success opens the door to investigate the intrinsic mechanism of more complicated ferroelectric instabilities, such as a vortex polarization reversal in nano-structured ferroelectrics and a pinning effect of vacancies of the domain switching.