00:01
Okay we're given this lagrangian which we're told is for a particle, a relativistic particle, an electromagnetic field.
00:12
So we want to derive the equations of motion and i'm just going to point out that this is a three -dimensional lagrangian.
00:21
So it's a lagrangian in three space.
00:23
It's not the invariant lagrangian that we would use in if we were using minkowski space for instance.
00:37
So if we write it in terms of indices and i'm thinking einstein's summation convention throughout.
00:46
Whenever you see a repeated index we sum on it from one to three.
00:51
This will just make life a little bit easier.
01:01
So first of all we have to take the partial with respect to v sub i, the velocity.
01:11
The first term looks like this and the second term looks like that.
01:25
So the minus two cancels the minus a half.
01:28
We got a c squared in the numerator and denominator like that.
01:34
That's our first term.
01:36
Second term looks like that.
01:39
And so our first term is gamma mv sub i.
01:43
Remember gamma is one over the square root of one minus v squared over c squared.
01:47
So there's our gamma, kind of usual relativistic thing, plus q a sub i.
02:01
And then we take its time derivative and then i'm just going to leave that first term as it is.
02:17
We'll talk about it in just a second.
02:19
The second term of i have to expand that total time derivative as a various partial derivatives.
02:29
So we get a partial with respect to t plus the velocity v sub j times partial a i with respect to x j.
02:38
Again summed on j.
02:41
The first term is the total time derivative of the relativistic momentum gamma mv, which isn't too surprising.
02:52
We're expecting in our equations of motion should have a time derivative of the momentum and so we're good.
03:01
I'm just going to leave it just like that...