Velocity gauge formulations

Using the formalism of response theory [24,31], the scalar rotatory strength for a transition from the ground state 0) to an excited state n) can be evaluated as the residue of the linear response function. In the velocity gauge formulation, nR is given by the equation... 

It can be shown that the trace of the tensor G, and hence the computed optical rotation of a sample of randomly oriented chiral molecules, is independent of the origin provided that the linear response function satisfies O Eq. 5.46 and that the commutator of O Eq. 5.47 is fulfilled. Consequently, approximate linear response calculations of the length gauge optical rotation depend on the chosen coordinate origin. On the other hand, the trace of the velocity gauge formulation of the electric dipole - magnetic dipole polarizability... 

We can generalize this procedure If a function

Lagrangian description, and if

Eulerian description. The choice of the form is arbitrary but will be influenced by any advantage of a problem formulation in either description. For example, in solid mechanics, the Lagrangian description is commonly used, while in fluid mechanics the Eulerian description is popular. This is because in solid mechanics we can attach labels (e.g., visualize strain gauges at various points) on the surface of a solid body, and each material point can be easily traced from the reference state to the current state. On the other hand for a fluid we measure the velocity V or pressure p at the current position jc, therefore the Eulerian description better represents the fluid (note that for a fluid it is difficult to know the exact reference point X corresponding to all the current points jr). 

---