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  Expansion of gauge potential on infinite dimensional manifold

+ 2 like - 0 dislike

I'm studying geometrical approaches to locomotion at low Reynolds number by reading the article Geometry of self-propulsion at low Reynolds number by Alfred Shapere and Frank Wilczek and found a situation I'm not understanding from a mathematical point of view what is being done.

The idea itself is not that hard and I shall present it first to make things clear: we consider that a body inside a fluid is a three-dimensional subset of the fluid region and we consider the body's shape as a parametrization of its boundary by means of a function $S : U\subset \mathbb{R}^2\to \mathbb{R}^3$.

In that case, the idea as I understand is to consider a sequence of such shapes $t\mapsto S_t$ parametrized by time (that is a curve of shapes) which just contain deformations with the shape "held fixed" and try to find a curve of rigid motions resulting from the deformations.

If one considers $M$ the space of all shapes, this is a space of functions. A sequence of shapes will be one curve $\gamma : I\subset\mathbb{R}\to M$. Considering then the euclidean group in three-dimensions (that is, the group of rigid motions in $\mathbb{R}^3$) which we denote $E(3)$ there is a natural right action of $E(3)$ in $M$ in which a shape is translated and rotated by some element $g\in E(3)$. Indeed, if $S\in M$ then $S\cdot g$ will be the function

$$(S\cdot g)(u,v) = S(u,v) \cdot g$$

Where the right action on the right side is the natural action of $E(3)$ in $\mathbb{R}^3$. Given the space $M$ then we can also consider the quotient $M/E(3)$ where shapes related by rigid motions are considered equivalent.

If $\pi : M\to M/E(3)$ is the natural projection, $(M,\pi, M/E(3))$ has the structure of a principle bundle with structure group $E(3)$ and the problem at hand becomes the problem of finding a connection which allows the horizontal lift of sequence of unlocated shapes (just deformations) into sequences of deformations and movements.

In page 7 of the document, the author considers infinitesimal deformations, from which he will describe a way to obtain the connection (which he calls the gauge potential). He defines then one infinitesimal deformation by parametrizing a sequence of unlocated shapes by

$$S_0(t) = S_0 + s(t),$$

where $S_0$ is one initial shape and $s(t)$ is infinitesimal. He then says we can expand $s(t)$ as

$$s(t) = \sum_{i} \alpha_i w_i$$

where $w_i$ is a "fixed basis of vector fields on $S_0$". From one intuitive point of view I imagine that a tangent vector to $M$ is equivalent to a vector field defined on the shape, so this makes sense.

Now, he considers the gauge potential as a vector field $A$ with the property that denoting $A_{\dot{S}_0(t)}[S_0(t)]$ the component of $A$ at the point $S(t)$ in the direction of $\dot{S}_0(t)$ this object belongs to the lie algebra of $E(3)$, that is $\mathfrak{e}(3)$ and we can find the curve of rigid motions $\mathfrak{R}(t)$ by solving the equation

$$\dfrac{d\mathfrak{R}}{dt} = \mathfrak{R}A_{\dot{S}_0(t)}[S_0(t)].$$

My doubt is then in what he does next: he considers this infinitesimal deformations $S_0(t)$ I gave and then he "expands the gauge potential" to second order in this way

$$A_{\dot{S}_0(t)}[S(t)] = A_{v(t)}[S_0]+\sum_i \dfrac{\partial A_v}{\partial w_i} \dot{\alpha}_i\approx \sum_j A_{w_j}\dot{\alpha}_j + \sum_i \dfrac{\partial A_v}{\partial w_i}\alpha_i\dot{\alpha}_j$$

where this $v(t) = S_0'(t)$. Now what is going on here? I simply cannot understand where this equation came from and what does it means. What is really happening here?

This post imported from StackExchange Physics at 2015-04-09 12:10 (UTC), posted by SE-user user1620696

asked Apr 6, 2015 in Theoretical Physics by user1620696 (160 points) [ revision history ]
edited Apr 9, 2015 by Dilaton

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