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 v1, v2 be two velocity fields on the circle and let R(ϵv1,ηv2) be the rotation and translation of the circle induced by the following sequence of motions:
S→S+ϵv1→S+ϵv1+ηv2→S+ηv2→S
We work to second order in ϵ,η. Then by
ˉPexp[∮Adt]=1+12∮∑ijFwiwjαi˙αjdt
we find that R(ϵv1,ηv2)=1+ϵηFv1v2 where Fv1v2 lies on the Lie algebra of rigid motions. F is most easily computed by matching the boundary condition ηv2(θ) on the surface of the circle deformed by ϵv1(θ). If we call the resulting velocity field v12, then Fv1v2 is related to the assymptotics of v12 at infinity. In fact it is not hard to see that following our prescriptions we find
Ftrv1v2=limr→∞∫dθ2π(v12−v21)Frotv1v2=limr→∞∫dθ2πr×(v12−v21)
where the translational and rotational parts of F are defined by
Fv1v2=(0Frotv1v2(Ftrv1v2)x−Frotv1v20(Ftrv1v2)y000)
The problem is that I can't see how those expressions for F were derived. I mean, on my understanding, Fv1v2 is the generator of rigid motion which gives the motion of the shape resulting from the deformation by v1 and v2. 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
Fi=−∫∞σijdSj
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