The laser beam fusion cutting process is described with a three-dimensional free boundary problem for the motion of one phase boundary. Its variational formulation is presented. An ODE approximation of the PDE problem is derived, which describes the dynamics of the process by means of characteristic dynamical variables : the position of the boundary, the heat content within the material and the surface temperature. For the spatially one-dimensional case an analytical solution of the PDE problem is presented and the accuracy of the corresponding ODE approximation is investigated. Analysis of the ODE approximation revealed the separation of time scales for the dynamics of the boundary and the heat content. The results of the time scale separation and the nonlinear response of the cutting process onto external modulation of the processing parameters are compared with the experimental measurements of the thermal emission.