Quaternions singularity

The applicability of homotopic theory becomes much less obvious for liquid crystal phases with more complicated order parameters such as biaxial nematics and cholesterics, which are both locally defined by three directors forming a tripod. This gives rise to a description of the line singularities in terms of the quaternion group, Q. This is particularly interesting because the quaternion group Q is non-Abelian, a property that... 

Example 5.1.10 As pointed out in the introduction (see Sec. 1.3.11) a three parametric description of rotations may lead to singularities in the equations of motion. Therefore f a description in terms of four parameters 1, 2,93,94 so-called quaternions or Euler parameters, is used. Quaternions have the property... 

The arbitrary rotation of rigid particles is described in the following equations. In this context, the mathematical concept of quaternions is used to represent the angular displacement of the particle within the three-dimensional space [22]. Apart from a higher computational accuracy and efficiency, the main advantage over ordinary rotation matrices is that the solution of quaternions allows an enhanced stability of the numerical solution as a result of non-existent singularities. 

Among the important recent advances in computing methods for molecular dynamics simulations of condensed matter are the singularity free algorithm for rigid polyatomics, developed by Evans and Murad (, , and the multiple time step (MTS) method, developed by Streett, et al. In the method of Evans and Murad, the equations of rotational motion are expressed in terms of four parameters, x h, C, C, called quaternions, defined by... 

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