derive from this variational principle a new set of equilibrium equations, boundary conditions, and constitutive equations for these unknown functions. In case the resistance to dislocation motion is significant, the variational principle must be replaced by the variational equation that takes the dissipation into account. The constructed theory is generalized for single crystals having a finite number of active slip systems. We provide also the simplifications of the theory for small strains. As compared to the continuum dislocation dynamics proposed in [@Hochrainer2007; @Zhu2014] our theory is advantageous not only in the computational cost due to its simplicity, but also in its full consistency with the LEDS-hypothesis. In the problem of single crystal beam having only one active slip system and deforming in simple shear, the energy minimization problem reduces to the two-dimensional variational problem. We solve this problem analytically for the circular cross section and asymptotically for the rectangular cross section. We will show that this solution reduces to that found in [@Berdichevsky-Le07] for the crystals with thin and long cross-section.

The paper is organized as follows. After this short introduction we present in Section 2 the three-dimensional kinematics for single crystals deforming in single slip. Section 3 formulates the 