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fluidistic
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Homework Statement
I must show that the equation ##\frac{\partial E}{\partial t}=t\nabla \times E## is invariant under time translations and I must also find its symmetries if it has any. Where E is a function of time and position.
Homework Equations
Already given.
The Attempt at a Solution
I assume that ##\vec E (\vec x ,t)## is a solution of the equation ##\frac{\partial \vec E (\vec x ,t)}{\partial t}=t\vec \nabla \times \vec E (\vec x , t)##.
I want to show that ##\vec E _T (\vec x , t)=\vec E (\vec x , t-T)=\vec E(\vec x , u)## is not a solution of the equation ##\frac{\partial \vec E _T (\vec x ,t)}{\partial t}=t\vec \nabla \times \vec E_T (\vec x , t)##
Let's go:
##\frac{\partial \vec E_T(t,\vec x )}{\partial t}=\frac{\partial \vec E_T(u, \vec x )}{\partial t}=\frac{\partial u}{\partial t} \cdot \frac{\partial \vec E ( u , \vec x)}{\partial t}=\frac{\partial \vec E ( u , \vec x)}{\partial t}=u \vec \nabla \times \vec E (u , \vec x )=u \vec \nabla \times \vec E_T (\vec x , t)=(t-T) \vec \nabla \times \vec E_T (\vec x , t) \neq t\vec \nabla \times \vec E_T (\vec x , t)##.
Is what I've done okay/valid?
For the symmetries, I guess I must check out if it has a translational invariance and/or rotation invariance.
Edit: I checked out and if I didn't make any error the equation is invariant under space translation. I think this means that if the equation represents a physical system then the linear momentum is conserved, but since it isn't time invariant, the energy isn't conserved. I'd have to check out if it's invariant under rotations...
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