• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,721 comments
1,470 users with positive rep
818 active unimported users
More ...

  Field strength tensor for locomotion at low Reynolds number

+ 3 like - 0 dislike

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
asked Jun 19, 2015 in Theoretical Physics by user1620696 (160 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights