Summary of the Ericksen-Leslie Dynamic Equations

It is convenient at this point to summarise the Ericksen-Leslie dynamic equations for nematics in the incompressible isothermal theory when the director inertial term is neglected. These are the most frequently used forms of the equations and we state them in the notation introduced in the previous Sections. They consist of the constraints 

It is also worth noting from (2.147) that when partial derivatives commute 

The balance laws (4.119) and (4.120) should be replaced by equations (4.116) and (4.103), respectively, if the director inertial term is to be included. 

Some simphfications are possible when the relations (4.112) hold for the forces and G. These relations are particularly useful when including the gravitational field and/or electromagnetic fields. In this case the balance of linear momentmn (4.119) simplifies to 

The one-constant approximation (2.62) for the elastic energy when strong anchoring is assumed is also useful for reducing the elastic energy to the form given by (2.67). In this case the simplified elastic energy approximation is 

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