Time translation invariant equation

In summary: It doesn't seem to be solving anything.In summary, the equation ##\frac{\partial E}{\partial t}=t\vec \nabla \times E## is invariant under time translations, but it may not be invariant under rotations.
  • #1
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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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  • #2
I think you probably understand what you are meant to do. But the way you have written the answer is really confusing as to which step you are doing and why. And yeah, I'd guess you are meant to check for spatial translation and rotation. Maybe rotations around the origin is enough.
 
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  • #3
Thanks for the feedback. Could you tell me please which steps (all?) are confusing and/or how would you do it? Maybe it will make me understand something that I don't currently.
 
  • #4
well, you do the entire thing in one line. But the order of logic seems weird to me. Mathematically it is right. But instead you could use several lines (one for each equality) and you could write why you can say that the two things are equal. This would make it easier to follow.
 
  • #5
BruceW said:
well, you do the entire thing in one line. But the order of logic seems weird to me. Mathematically it is right. But instead you could use several lines (one for each equality) and you could write why you can say that the two things are equal. This would make it easier to follow.
I see, thanks.
Let ##u=t-T##. I define ##\vec E _T (t, \vec x )=\vec E (t-T, \vec x)##. So that ##\vec E_T (t, \vec x)=\vec E(u, \vec x)##.
I want to see whether ##\vec E _T(t, \vec x )## is a solution to the equation ##\frac{\partial \vec E _T(\vec x ,t)}{\partial t}=t\vec \nabla \times \vec E _T (\vec x , t)##. If it is, then the equation is invariant under time translations. If it is not a solution, then it's not invariant under time translations.

I start with the left hand side of the equation: ##\frac{\partial \vec E _T(\vec x ,t)}{\partial t}=\frac{\partial \vec E (u, \vec x) }{\partial t}##.
Using the chain rule (I believe?), I can rewrite the last expression as ##\frac{\partial u}{\partial t} \frac{\partial \vec E (u, \vec x )}{\partial u}##. But looking at the definition of u, it is easy to see that ##\frac{\partial u}{\partial t}=1##. Thus, one reaches that ##\frac{\partial \vec E _T(\vec x ,t)}{\partial t}=\frac{\partial \vec E (u,\vec x)}{\partial u}##.

Now I use the equation given in the problem statement (considering t as a dummy variable), so I can write the right hand side of the last equation as ##u \vec \nabla \times \vec E (u, \vec x )## (I can do this because ##\vec E (\vec x , t)## is a solution to the equation given in the problem statement).
Replacing back u by t-T, I reach that ##\frac{\partial \vec E _T (t, \vec x) }{\partial t}=(t-T) \vec \nabla \times \vec E (t-T, \vec x) ## This is true because the curl acts on the spatial coordinates, not on time.
And so ##\frac{\partial \vec E _T (t, \vec x) }{\partial t}=(t-T) \vec \nabla \times \vec E _T (t, \vec x)##.

Therefore, if ##\vec E (\vec x ,t)## is a solution to the equation ##\frac{\partial \vec E (\vec x ,t)}{\partial t}=t\vec \nabla \times \vec E (\vec x , t)##, then ##\vec E _T (t , \vec x )## is a solution to the equation ##\vec E _T (t , \vec x )=(t-T) \vec \nabla \times \vec E_T (\vec x , t)##.
Thus ##\vec E _T (t , \vec x )## cannot be a solution to the equation ##\frac{\partial \vec E _T(\vec x ,t)}{\partial t}=t\vec \nabla \times \vec E _T (\vec x , t)## unless that ##T=0##.
So that the equation is not invariant under time translations.
What do you think now?
 
  • #6
excellent :) nice and clear to read. also, I agree with the method and answer.
 

1. What is a time translation invariant equation?

A time translation invariant equation is a mathematical equation that remains unchanged even when the time variable is shifted. This means that the equation still holds true regardless of when it is being evaluated.

2. How does time translation invariance impact scientific research?

Time translation invariance is a fundamental concept in many scientific fields such as physics and engineering. It allows scientists to make predictions and draw conclusions based on the assumption that physical laws and equations remain unchanged over time.

3. What are some examples of time translation invariant equations?

The Schrödinger equation in quantum mechanics, the Navier-Stokes equation in fluid dynamics, and the Maxwell's equations in electromagnetism are all examples of time translation invariant equations.

4. Can time translation invariance be violated?

Yes, in certain cases, time translation invariance can be violated. This often occurs in systems that involve non-conservative forces, such as friction or external forces, which can cause the equations to change over time.

5. How is time translation invariance related to time symmetry?

Time translation invariance is closely related to time symmetry, which is the idea that physical laws and phenomena remain unchanged over time. Time translation invariance is a specific type of time symmetry, where the equations remain the same even when the time variable is shifted.

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