# Velocity field of fluid and Maurer-Cartan form?

Chatting with an engineer, he suggested me to have a look to a certain book in order to understand what fluid mechanics is about (I know nothing about the subject). But this question is not about fluid mechanics in general, it's a bit more specific (and much, much more basic).

Starting to read the aforementioned book, I couldn't help stopping by the first sentence because I felt the need to 'translate' things into mathematiquese in order to understand. What is a fluid motion? Well, just like a rigid motion on $M=\mathbb{R}^3$ is a smooth map $\varphi:\mathbb{R}\to\mathrm{Euc}^{+}(3)$, we can take a fluid motion to be a smooth map $\varphi:\mathbb{R}\times M\to M$ such that the induced map is a diffeomorphism at every time, $\varphi:\mathbb{R}\to\mathrm{Diff}(M)$, $(t,x)\mapsto\varphi_t(x)$. (We also assume $\varphi_0=\mathrm{id}_M$). $\varphi$ is not assumed to be a flow in general, i.e. need not verify $\varphi_t\circ\varphi_s=\varphi_{t+s}$ and $\varphi_t^{-1}=\varphi_{-t}$. Then, the notion of velocity field $v(t,x)$ of the fluid was mentioned, which is assumed to be, at a time $t$ and point $x\in M=\mathbb{R}^3$, the velocity of a material particle passing through position $x$ at time $t$. The immediate thought was that $v(t,x)$ must just be the velocity $d\varphi_t(x)/dt$ of the curve $t\mapsto \varphi_t(x)$. But it does not live in $T_xM$. So, let's take it back to $x$, i.e. define $v(t,x)$ as $(\varphi_t^{-1})_{*}(d\varphi_t(x)/dt)$, so that now it lies in $T_xM$. This is still 'physically' wrong, as one can see by considering a waterfall with a horizontal part (with almost constant velocity) and a vertical part in which water accelerates. So, I came up with the following definition:

$v(t,x):=\frac{d}{ds}|_{s=t}(\varphi_s(\varphi_t^{-1}(x)))=\dot{\varphi}_t(\varphi_t^{-1}(x))$ .

First question:

Is my definition the one usually (implicitely or explicitely) taken in fluid dynamics?

If we assume the motion is affine, i.e. $\varphi_t(x)=A(t)\cdot x + \beta(t)$ with $A(t)\in\mathrm{GL}(3)$, we get:

$v(t,x)=\dot{A}\cdot A^{-1}\cdot x - \dot{A}\cdot A^{-1}\cdot\beta+\dot{\beta}$,

in which the linear term reminds me of the Maurer-Cartan form $\omega_{MC}=g^{-1}\cdot\mathrm{d}g$ on a matrix Lie group $G$.

Second question:

Does $v(t,x)$ actually have anything to do with a Maurer-Cartan form?

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This point of view has been discussed in the literature. You might, for example, look at Arnol'd's remarks about treating fluid flow as dynamics in the group of (volume preserving) diffeomorphisms in his "Mathematical Methods of Classical Mechanics" and the references that he cites there. –  Robert Bryant Jan 11 '12 at 20:09
Thanks for the suggestion. I found in Arnold, Khezin Topological Methods in Hydrondynamics that 'my' definition was indeed the correct one; so question 1 is answered. –  Qfwfq Jan 11 '12 at 21:01
Also, looking at page 15 of the same book, it seems that my $\dot{A}\cdot A^{-1}$ is, in the case of a flow $\varphi:\mathbb{R}\to G=\mathrm{SO}(3)$, what they call "spatial angular velocity", which lives in the Lie algebra of $G$. In the case of a flow of (volume preserving) diffeomorphisms it might be linked to the velocity field $v(t,x)$... –  Qfwfq Jan 11 '12 at 21:40