Some considerations on the elastic theory for nematic liquid crystals

The elastic theory for nematic liquid crystals is critically analysed. After a review on the variational calculus formalism, the range of applicability of the Lagrangian method for the solution of practical problems is discussed. It is underlined that only a limited number of problems can be solved by means of a variational approach. The role at the Jacobi equation is also discussed. The importance of the non linear character of the K13-problem is analyzed in the framework of a simple molecular model. Finally, the principle of virtual work is applied to the elastic theory of nematic liquid crystals. Our analysis shows that the K13 elastic problem is an ill-posed one, since this problem can only be solved by means of a variational, or virtual work, approach by modifying the bulk elastic free energy and taking into account new terms quadratic in the second order deviatives. However it is necessary to remember that, in the proximity of a surface, a spatial variation of the density and of the scalar order parameter of the liquid crystal are expected, and hence a true elastic description is no longer possible.