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Helicity integral in differential forms

  • #1
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Homework Statement



Let ##V^{3}(t)## be a compact region moving with the fluid.

Assume that at ##t=0## the vorticity ##2##-form ##\omega^{2}## vanishes when restricted to the boundary ##\partial V^{3}(0)##; that is, ##i^{*}\omega^{2}=0##, where ##i## is the inclusion of ##\partial V## in ##\mathbb{R}^{3}##.

(This does ##\textit{not}## say that ##\omega^{2}## itself vanishes, rather only that ##\omega({\vec{u}},{\vec{w}})=0## for ##\vec{u}##,##\vec{w}## tangent to ##\partial V^{3}(0)##.)

Then the ##\textbf{helicity}## integral ##\displaystyle{\int_{V(t)}{\vec{v}}\cdot{\vec{\omega}}\ dx\wedge dy\wedge dz}## can be constant in time.

Prove that the helicity integral is constant in time.

Homework Equations



The Attempt at a Solution



$$\frac{d}{dt}\int_{V(t)}{\vec{v}}\cdot{\vec{\omega}}\ dx\wedge dy\wedge dz$$
$$= \frac{d}{dt}\int_{V(t)}{\vec{v}}\cdot{\vec{\omega}}\ dx^{1}\wedge dx^{2}\wedge dx^{3}$$
$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{V(t)} {\vec{v}}\cdot{\vec{\omega}}\ \text{vol}^{3}$$
$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{V(t)} \nu^{1}\wedge \omega^{2}$$
$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{V(t)} \nu^{1}\wedge {\rm d}\nu^{1}$$
$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{V(t)} \nu\wedge {\rm d}\nu$$
$$= \frac{1}{o(x)\sqrt{g}(x)}\frac{1}{2} \frac{d}{dt}\int_{W(t)} (\nu\wedge {\rm d}\nu)+(\nu\wedge {\rm d}\nu)$$
$$= \frac{1}{o(x)\sqrt{g}(x)}\frac{1}{2} \frac{d}{dt}\int_{W(t)} (\nu\wedge {\rm d}\nu)-({\rm d}\nu\wedge \nu)$$
$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{W(t)} {\rm d}(\nu\wedge \nu)$$
$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\oint_{\partial W(t)} \nu\wedge \nu$$
$$= 0.$$

I think this is wrong because I haven't used the assumption that at ##t=0## the vorticity ##2##-form ##\omega^{2}## vanishes when restricted to the boundary ##\partial V^{3}(0)##; that is, ##i^{*}\omega^{2}=0##, where ##i## is the inclusion of ##\partial V## in ##\mathbb{R}^{3}##.
 
Last edited:

Answers and Replies

  • #2
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Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
 

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