Show that the total energy of a system of charged particles is conserved

[Inferior89]

Homework Statement

We have a system with N particles with masses m1, m2, ... mN and electrical charges q1, q2, .. qN that interact electrostatically. We assume these interactions are instantaneous. Construct the formula for the total energy and show that this is conserved.

The Attempt at a Solution

This is what I have done so far:
http://i.imgur.com/xfIbB.png

I am not sure how to continue or if this is on the right track. It sort of seems to be going the right way but obviously the expression inside the square bracket won't be zero since this would mean that the energy is conserved for every individual particle which is not true. However, I am not sure how to show that the whole sum will be zero.

Any help is greatly appreciated.

 

[betel]

What does $$r_{ij}, v_{ij}$$ mean?
Try rewritting this in terms of the $$\vec{x}_i$$.

 

[hikaru1221]

There is one mistake in your solution: $$F \neq m\dot{v}$$.
We only have: $$\vec{F}=m\dot{\vec{v}}$$ or $$F=m|\dot{\vec{v}}|$$
Why don't you try deducing back to this form: $$dE_{kinetic}=dW=\vec{F}d\vec{r}$$? This is where it all begins That is, using vectors, instead of scalar quantities:
$$v_i^2=(\vec{v}_i)^2$$
$$d(\frac{1}{r_{ij}})=-\frac{dr_{ij}}{r_{ij}^2}=-\frac{r_{ij}dr_{ij}}{r_{ij}^3}=-\frac{\vec{r}_{ij}d\vec{r}_{ij}}{r_{ij}^3}$$