# Prove that this is a function of time

• Apr 14th 2013, 03:00 PM
Kiwi_Dave
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?

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

I have the following relationships from electromagnetism:

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

I have been battling this for over a week!
• Apr 14th 2013, 04:09 PM
topsquark
Re: Prove that this is a function of time
Quote:

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?

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

I have the following relationships from electromagnetism:

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

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

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

-Dan
• Apr 14th 2013, 05:30 PM
Kiwi_Dave
Re: Prove that this is a function of time
Here is the whole problem:

http://mathhelpforum.com/calculus/21...m-problem.html

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

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

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

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

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

Now term 1 is my problem

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

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

Using the equation for E I get:

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

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?

$q \frac{d \underline A}{dt} \cdot \underline r + q \underline E \cdot \underline r$
• Apr 14th 2013, 05:58 PM
topsquark
Re: Prove that this is a function of time
Quote:

Originally Posted by Kiwi_Dave
Here is the whole problem:

http://mathhelpforum.com/calculus/21...m-problem.html

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

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

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

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

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

Now term 1 is my problem

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

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

Using the equation for E I get:

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

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?

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

It looks like it ought to be a simple substitution problem. Just for clarity, what is $\zeta$? (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
• Apr 14th 2013, 06:15 PM
Kiwi_Dave
Re: Prove that this is a function of time
Zeta and Tau come from

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

giving

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

They come from

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

$\tau = (dT/d \epsilon)_{\epsilon = 0}$ 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