A Freedericksz Transition in Bookshelf Smectic

Inserting this into (6.124) then reduces the c-equations to 

Integrating this expression with respect to x provides the multiplier 7 in terms of the solution 0(z), namely, 

z) and /x(z) given by (6.131) and (6.125), respectively. The vector 0 is unique apart from the addition of the gradient of an arbitrary scalar field. Its components 

The components of 11 can be obtained explicitly from the general vector formula (6.92) if desired, with, of course, a and c given by (6.120). It is a simple exercise to use (6.125), (6.130) and (6.131) to verify that V x = 0 where y is defined by (6.132), indicating that the a-equations (6.123) are fulfilled. It now follows that solving the equilibrium equation (6.129) for 0(z) will result in a full solution to the equilibrium equations for a and c given by (6.120). Although we have calculated all the Lagrange multipliers explicitly in this example, it is not often required in simple static problems to know IS explicitly since it is frequently sufficient to know that 7 and fS can be found if desired once a solution to the c-equations has been determined, as mentioned previously. 

It is possible to make the one-constant approximation Bi = B2 = B and suitably amend the work contained on pages 79 to 82 using (6.139) to obtain qualitative plots for the behaviour of 0(z) for H H analogous to those presented in Figs. 3.9 and 3.10 for nematics. 

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