Vector helicity meaning

The helicity of a divergenceless vector held was already used by Woltjer in 1958 [45] in an astrophysical context. Moreau [46] showed soon after that it is a conserved quantity in certain hows of huid dynamics. Moffatt, in a seminal paper [44], coined the term helicity and clarihed its topological meaning. For a pedagogical review, see Ref. 47. 

In conclusion, the helicity of a divergenceless vector field is a measure of the linking number of the field lines. Even in the case that these lines are not closed, the notion of linkage has sense, because a mean value of an asymptotic linking number can be defined, and this value coincides with the helicity. 

This shows that there is still some linearity. In particular, there is a subset of knots that form a vector space and is therefore a linear sector of the model. It is the set of the knots with zero helicity or with unlinked lines. Note also that the theory is fully linear from the local point of view, as a consequence of the local equivalence with Maxwell s theory shown in Section V.A. By this we mean that the set of the electromagentic knots contains all the linear combinations of standard solutions around any point. 

The anisotropic rigid-body displacements of a molecule can be described in terms of translation (T, vibration along a straight-line path), libration (L, vibration along an arc), and screw (S, a combination of vibration and translation that may be regarded as vibration along a helical path). The mean-square amplitude of translational vibration is usually referred to as a system of Cartesian coordinates and unit vectors. S is the mean correlation between libration about an axis and translation parallel to this axis. Each of these three components can be expressed as... 

The new chiral terms are of first order and therefore describe spontaneous deformations. The Dj term means that there is a tendency for the c-vector to bend in the plane of the layer, which means that a coupled ferroelectric polarization (see next chapter) will have a tendency for a spontaneous splay. The D3 term is a spontaneous twist, describing the helical order of the SmC ground state. In the ground state, there is a pure twist in c. This means that c-Vxc = -q, and consequently,/c Vxc = 0. This reduces (4.60) near the equilibrium to ... 

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