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Self propulsion at low Reynolds number: how to compute the connection $1$-form?

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I've been studying some approaches with gauge theory to some problems in Mechanics and I've found the problem of self propulsion at low Reynolds number a quite complicated one. The approach I'm asking about has been treated by Shapere and Wilckzek here.

The problem is basically as follows: one considers a region $D\subset \mathbb{R}^3$ filled with a fluid and a deformable body inside the region. One can say that the shape of the body at time $t$ is basically a parametrization $S : U\times \mathbb{R}\to D$ of the boundary of the body with $U\subset \mathbb{R}^2$

The question is: since at low Reynolds number there are no net forces and torques on the body, just deforming itself is it possible that the body induces one net rigid motion? In that case, one considers the group $E(3)$ of rigid motions on space and it's Lie algebra $\mathfrak{e}(3)$. If $M$ is the space of all shapes, that is, functions $S : U\to D$ then $E(3)$ acts on the right on $M$ through $S\cdot g (u,v) = S(u,v)\cdot g$.

Two shapes are considered equivalent when they are the same regardless of location. The space of unalocated shapes is $M/E(3)$. In that case one has a principal bundle structure. A sequence of deformations is a path in $M/E(3)$ and the question may be answered by a connection $1$-form which allows the lifting of such path to a path containing locomotion as well.

Indeed, if $\omega$ is the connection $1$-form and $\Omega = D\omega$ the connection $2$-form, then since if $h$ is the projection on the horizontal part $\Omega(X,Y) = -\omega([hX,hY])$ we have that $\Omega$ gives how much one infinitesimal loop of transformations induce a infinitesimal rigid motion.

Although that is all clear, for me is not clear how does one compute the connection $1$-form. I mean, one needs to solve Navier-Stokes equations and find the fluid velocity right? But after that, how does one compute $\omega$?

Before approaching this problem I've tackled another two: reorientation of a falling cat and motion of a car. In the falling cat problem, conservation of angular momentum lead to the connection $1$-form. In the car problem, the no-slipping contact constraint leads to the connection $1$-form.

Now, on this particular problem, how does one really compute $\omega$? I've read the article sometimes but I couldn't get the idea.

This post imported from StackExchange Physics at 2015-05-02 20:47 (UTC), posted by SE-user user1620696
asked May 2, 2015 in Theoretical Physics by user1620696 (160 points) [ no revision ]

Mathematically spioled physical problems :-)

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