case of motion of dislocation loops in the slip plane. In this case we allow the loop function to depend explicitly on time $t$ such that equation $$\label{eq:2.4}
l(x_1,x_2,c_3,t)=c$$ with fixed constants $c_3$ and $c$ describes one and the same dislocation line during its motion in the slip plane. Letting $\zeta $ be the variable along the dislocation line, we may represent the level curve defined by in the parametric form $$x_1=x_1(\zeta ,t),\quad x_2=x_2(\zeta ,t).$$ Fixing $\zeta $ and taking the differential of we obtain $$l_{,1}dx_1+l_{,2}dx_2+l_{,t}dt=0$$ that yields $$l_{,1}\frac{dx_1}{dt}+l_{,2}\frac{dx_2}{dt}=-l_{,t}.$$ Since $\mathbf{v}=\frac{dx_1}{dt}\mathbf{e}_1+\frac{dx_2}{dt}\mathbf{e}_2$ is the velocity of the fixed point on the dislocation line with coordinate $\zeta $, we define the normal velocity of the dislocation line as follows $$\label{eq:2.5}
v_\nu =\mathbf{v}\cdot {\boldsymbol{\nu}}=\frac{1}{\sqrt{l_{,1}^2+l_{,2}^2}}(l_{,1}\frac{dx_1}{dt}+l_{,2}\frac{dx_2}{dt})=-\frac{\dot{l}}{\sqrt{l_{,1}^2+l_{,2}^2}},$$ with $\dot{l}=l_{,t}$. This kinematic quantity will be used in the case of non-vanishing resistance to dislocation motion.

Governing equations for single crystals deforming in single slip
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According to @Kroener92, the elastic deformation $\mathbf{F}^e$ and the dislocation densities $\rho_\perp $ and $\rho_\parallel $ characterize the current state of the crystal, so these quantities are the state variables of the continuum dislocation theory. The reason why the plastic deformation $\mathbf{F}^p$ cannot be qualified for the state variable is that it 