# Thread: Prove that this is a function of time

1. ## Prove that this is a function of time

As part of a larger problem I need to prove that the following equation is a function of time. I guess that means I must get rid of the partial derivatives and replace them with total derivatives?

$\displaystyle ( q \frac{\partial A_j}{\partial x_i}\dot x_j-q \frac{\partial V}{\partial x_i} )x_i$

I have the following relationships from electromagnetism:

$\displaystyle \underline B = \nabla \times \underline A$ and $\displaystyle \underline E = - \frac{\partial \underline A}{\partial t}-\frac{\partial V}{\partial x_i}$

I have been battling this for over a week!

2. ## Re: Prove that this is a function of time

Originally Posted by Kiwi_Dave
As part of a larger problem I need to prove that the following equation is a function of time. I guess that means I must get rid of the partial derivatives and replace them with total derivatives?

$\displaystyle ( q \frac{\partial A_j}{\partial x_i}\dot x_j-q \frac{\partial V}{\partial x_i} )x_i$

I have the following relationships from electromagnetism:

$\displaystyle \underline B = \nabla \times \underline A$ and $\displaystyle \underline E = - \frac{\partial \underline A}{\partial t}-\frac{\partial V}{\partial x_i}$

I have been battling this for over a week!
Possible sources of time dependence:
1. It's not typical but V could be time dependent

2. A could be time dependent

3. $\displaystyle \dot{x}$ could be time dependent

Without knowing the rest of the problem I can't say anything more about it.

-Dan

3. ## Re: Prove that this is a function of time

Here is the whole problem:

Calculus of Variations Rund Trautman Electromagnetism Problem

A,V,x and x dot are all functions of time.

I think I need to show that a function F exists such that

$\displaystyle \dot F(t)= ( q \frac{\partial A_j}{\partial x_i}\dot x_j-q \frac{\partial V}{\partial x_i} )x_i + q A_i \dot x_i +2tq(\dot x_i \frac{\partial A_i}{\partial t}-\frac{\partial V}{\partial t})-2qV$

Now terms 2 and 4 are clearly only functions of t.

I can manipulate term 3 to get:
$\displaystyle - \underline E \cdot \underline {\dot x} - \frac{dV}{dt}$

Now term 1 is my problem

$\displaystyle \dot F_1(t)= ( q \frac{\partial A_j}{\partial x_i}\dot x_j-q \frac{\partial V}{\partial x_i} )x_i$

And if I'm not mistaken I do that by getting rid of the partial derivaties?

Using the equation for E I get:

$\displaystyle ( q \frac{\partial A_j}{\partial x_i}\dot x_j-q \frac{\partial V}{\partial x_i} )x_i=q( \frac{\partial A_j}{\partial x_i}\dot x_j + \underline E + \frac{\partial A_i}{\partial t} )x_i$

Now the RHS of this looks remarkibly (but not quite) like the total derivative of A. The next equation is not correct (I don't think) but it is the sort of thing I need to get to?

$\displaystyle q \frac{d \underline A}{dt} \cdot \underline r + q \underline E \cdot \underline r$

4. ## Re: Prove that this is a function of time

Originally Posted by Kiwi_Dave
Here is the whole problem:

Calculus of Variations Rund Trautman Electromagnetism Problem

A,V,x and x dot are all functions of time.

I think I need to show that a function F exists such that

$\displaystyle \dot F(t)= ( q \frac{\partial A_j}{\partial x_i}\dot x_j-q \frac{\partial V}{\partial x_i} )x_i + q A_i \dot x_i +2tq(\dot x_i \frac{\partial A_i}{\partial t}-\frac{\partial V}{\partial t})-2qV$

Now terms 2 and 4 are clearly only functions of t.

I can manipulate term 3 to get:
$\displaystyle - \underline E \cdot \underline {\dot x} - \frac{dV}{dt}$

Now term 1 is my problem

$\displaystyle \dot F_1(t)= ( q \frac{\partial A_j}{\partial x_i}\dot x_j-q \frac{\partial V}{\partial x_i} )x_i$

And if I'm not mistaken I do that by getting rid of the partial derivaties?

Using the equation for E I get:

$\displaystyle ( q \frac{\partial A_j}{\partial x_i}\dot x_j-q \frac{\partial V}{\partial x_i} )x_i=q( \frac{\partial A_j}{\partial x_i}\dot x_j + \underline E + \frac{\partial A_i}{\partial t} )x_i$

Now the RHS of this looks remarkibly (but not quite) like the total derivative of A. The next equation is not correct (I don't think) but it is the sort of thing I need to get to?

$\displaystyle q \frac{d \underline A}{dt} \cdot \underline r + q \underline E \cdot \underline r$
It looks like it ought to be a simple substitution problem. Just for clarity, what is $\displaystyle \zeta$? (It might not matter since all you are looking for is to prove that F is the same in the primed and un-primed coordinates, but I haven't worked the problem yet.)

-Dan

5. ## Re: Prove that this is a function of time

Zeta and Tau come from

$\displaystyle t'=t(1+\epsilon)$ and $\displaystyle x^{\mu'}=x^{\mu}(1+\frac12 \epsilon)$

giving

$\displaystyle \zeta ^{\mu}=\frac {x^{\mu}}{2}$ and $\displaystyle \tau = t$

They come from

$\displaystyle t'=T(t,q^{\mu},\epsilon)$

$\displaystyle \tau = (dT/d \epsilon)_{\epsilon = 0}$ and a similar equation for zeta.

This problem comes from Example 4, here: Emmy Noether&#39;s Wonderful Theorem: Dwight E. Neuenschwander: 9780801896941: Amazon.com: Books