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In light of the -mapping method, the contribution of disclination lines to the free energy density of 2-dimensional liquid crystals is studied in the single-elastic constant approximation. It is pointed out that, compared with the previous theory, the free energy density can be divided into two parts. One is the usual distorted energy density of director field around the disclination lines. The other is the free energy density of the disclination lines themselves which is centralized at the disclination lines and topoligically quantized in a unit of 1/2kπ. The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of the director field at the disclination lines, i.e ., the disclination strength. From the method of Lagrangian multipliers, the equilibrium equation and the molecular field of 2-dimensional liquid crystals are also obtained. It is shown that the physical meaning of the Lagrangian multiplier is just the distorted energy density.
In light of the -mapping method, the contribution of disclination lines to the free energy density of 2-dimensional liquid crystals is studied in the single-elastic constant approximation. It is pointed out that, compared with the previous theory, the free energy one is the usual distorted energy density of director field around the disclination lines themselves which is centralized at the disclination lines and topoligically quantized in a unit of 1 / 2kπ. The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of the director field at the disclination lines, ie., The disclination strength. From the method of Lagrangian multipliers, the equilibrium equation and the molecular field of 2-dimensional liquid crystals are also obtained. It shows shown that the physical meaning of the Lagrangian multiplier is just the distorted energy density.