I'm looking at this reference (sorry it's a postscript file, but I can't find a pdf version on the web. This paper describes a similar procedure).
The topic is the Faddeev-Jackiw treatment of Lagrangians which are singular (Hessian vanishes) - similar to what Dirac does, but without the need to differentiate between first and second class constraints. Just looking at classical stuff here, no quantization.
Starting with the Maxwell Lagrangian
L=FμνFμν
where
Fμν=∂μAν−∂νAμ
we see that it's second order in time derivatives acting on A.
We choose to write it in first order form like this
L=(∂μAν−∂νAμ)Fμν−12FμνFμν
where we're treating Fμν now as an auxilliary, independent variable. Having defined this, Faddeev says
"we rewrite (the last equation) as:
L=(∂0Ak)F0k+A0(∂kF0k)−Fik(∂iAk−∂kAi)−12(F0k)2−12(Fik)2
"
My question is how does he arrive at this from the previous equation ? I don't see how just expanding the indices into time and space values ever gets me to A0(∂kF0k)
I can see how there's something special about A0, since when I write out the EOM for the first order Lagrangian, A0 drops out, which indeed it should do because we'll end up with it being a Lagrange multiplier. I just can't see how you end up with that term, with A0 multiplying ∂kF0k.
It's clearly correct since A0divE is just the Gauss law constraint.
This post imported from StackExchange Physics at 2014-03-22 17:26 (UCT), posted by SE-user twistor59