- #1
spaghetti3451
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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?