Invariance of direction of vorticity

[spaghetti3451]

Homework Statement

The vorticity vector ##\vec{\omega} = \text{curl}\ \vec{v}##, defined as usual by ##\omega^{2}=i_{{\vec{\omega}}}\text{vol}^{3}##, is ##\textit{not}## usually invariant since the flow need not conserve the volume form.

The mass form, ##\rho\ \text{vol}^{3}##, however, ##\textit{is}## conserved.

From ##{\bf{\omega}}^{2}=i_{\vec{\omega} / \rho}\rho \text{vol}^{3}##, it can be shown that the vector ##\vec{\omega}/\rho## should be invariant; that is, ##\mathcal{L}_{\vec{v}+\partial / \partial t} (\omega^{2}/ \rho)=0##.
How can you use ##\mathcal{L}_{\bf{X}}\circ i_{\bf{Y}}-i_{\bf{Y}}\circ\mathcal{L}_{\bf{X}}=i_{[{\bf{X}},{\bf{Y}}]}## to prove that the vector ##\vec{\omega}/\rho## is invariant?

Homework Equations

The Attempt at a Solution

##\mathcal{L}_{{\bf{v}+\partial / \partial t}}\left(\frac{\omega^{2}}{\rho}\right)##

##= \mathcal{L}_{{\bf{v}+\partial / \partial t}}\left(\frac{i_{(\vec{\omega}/\rho)}\ \rho\ \text{vol}^{3}}{\rho}\right)##

##= \frac{1}{\rho}\bigg(\mathcal{L}_{{\bf{v}+\partial / \partial t}}\left(i_{(\vec{\omega}/\rho)}\ \rho\ \text{vol}^{3}\right)\bigg)##

##= \frac{1}{\rho}\bigg(i_{(\vec{\omega}/\rho)}\left(\mathcal{L}_{{\bf{v}+\partial / \partial t}}\ \rho\ \text{vol}^{3}\right)+i_{[{\bf{v}+\partial / \partial t},(\vec{\omega} / \rho)]}\rho\ \text{vol}^{3}\bigg)##

##= \frac{1}{\rho}\bigg(0+i_{[{\bf{v}+\partial / \partial t},(\vec{\omega} / \rho)]}\rho\ \text{vol}^{3}\bigg).##

How do you proceed next?

 

[mighty2000]

Next, we can use the identity ##\mathcal{L}_{\bf{X}}\circ i_{\bf{Y}}-i_{\bf{Y}}\circ\mathcal{L}_{\bf{X}}=i_{[{\bf{X}},{\bf{Y}}]}## to simplify the expression further. We have:

##\frac{1}{\rho}\bigg(i_{[{\bf{v}+\partial / \partial t},(\vec{\omega} / \rho)]}\rho\ \text{vol}^{3}\bigg)##

##= \frac{1}{\rho}\bigg(\mathcal{L}_{\bf{v}+\partial / \partial t}\circ i_{(\vec{\omega} / \rho)}-i_{(\vec{\omega} / \rho)}\circ\mathcal{L}_{\bf{v}+\partial / \partial t}\bigg)(\rho\ \text{vol}^{3})##

##= \frac{1}{\rho}\bigg(i_{(\vec{\omega} / \rho)}\circ\mathcal{L}_{\bf{v}+\partial / \partial t}(\rho\ \text{vol}^{3})-i_{(\vec{\omega} / \rho)}\circ\mathcal{L}_{\bf{v}+\partial / \partial t}(\rho\ \text{vol}^{3})\bigg)##

##= 0.##

Therefore, we have shown that ##\mathcal{L}_{{\bf{v}+\partial / \partial t}}\left(\frac{\omega^{2}}{\rho}\right) = 0##, which proves that the vector ##\vec{\omega}/\rho## is invariant.