Recently I've been studying locomotion at low Reynolds number. I already asked here about the computation of the gauge potential. Now I have a more objective question, which arose when reading the same article by Shapere and Wilczek. Basically, we can find in the article on page 567 the following excert:

To compute the field strength tensor $F$, which governs the motion resulting from infinitesimal deformations, we must consider closed paths in two-dimensional subspaces of shape space. Let $v_1$, $v_2$ be two velocity fields on the circle and let $\mathfrak{R}(\epsilon v_1, \eta v_2)$ be the rotation and translation of the circle induced by the following sequence of motions:

$$S\to S+\epsilon v_1 \to S + \epsilon v_1 + \eta v_2 \to S + \eta v_2 \to S$$

We work to second order in $\epsilon,\eta$. Then by

$$\bar{P}\exp\left[\oint Adt\right] = 1 + \dfrac{1}{2}\oint\sum_{ij}F_{w_iw_j}\alpha_i \dot{\alpha}_jdt$$

we find that $\mathfrak{R}(\epsilon v_1, \eta v_2) = 1 + \epsilon \eta F_{v_1 v_2}$ where $F_{v_1 v_2}$ lies on the Lie algebra of rigid motions. $F$ is most easily computed by matching the boundary condition $\eta v_2(\theta)$ on the surface of the circle deformed by $\epsilon v_1(\theta)$. If we call the resulting velocity field $v_{12}$, then $F_{v_1v_2}$ is related to the assymptotics of $v_{12}$ at infinity. In fact it is not hard to see that following our prescriptions we find

$$F_{v_1v_2}^{\operatorname{tr}} = \lim_{r\to \infty} \int \dfrac{d\theta}{2\pi}(v_{12}-v_{21}) \\ F_{v_1v_2}^{\operatorname{rot}} = \lim_{r\to \infty} \int \dfrac{d\theta}{2\pi}r\times (v_{12}-v_{21})$$

where the translational and rotational parts of $F$ are defined by

$$F_{v_1v_2} = \begin{pmatrix}0 && F_{v_1v_2}^{\operatorname{rot}} && (F_{v_1v_2}^{\operatorname{tr}})_x \\ -F_{v_1v_2}^{\operatorname{rot}} && 0 && (F_{v_1v_2}^{\operatorname{tr}})_y \\ 0 && 0 && 0 \end{pmatrix}$$

The problem is that I can't see how those expressions for $F$ were derived. I mean, on my understanding, $F_{v_1v_2}$ is the generator of rigid motion which gives the motion of the shape resulting from the deformation by $v_1$ and $v_2$. Now, how from this can one obtain the integrals presented on the article?

I believe it has something to do with the fact that the $i$-th component of the force acting on a body is given, because of conservation of stress tensor, as

$$F_i = -\int_{\infty} \sigma_{ij} dS_j$$

but I'm unsure on how to connect this with the problem at hand. How can I derive these formulas for the field strength tensor?

This post imported from StackExchange Physics at 2015-06-21 21:44 (UTC), posted by SE-user user1620696