# Application of Noethers to Lorentz force Lagrangian with boost

1. Oct 22, 2008

### Peeter

I've been slowly self studying Lagrangian topics, and have gotten to Noether's theorem.

I've tried application of a Lorentz boost to all the terms in the Lorentz force Lagrangian (which is invariant with respect to boost since it has only four vector dot products). Then using Noether's theorem to find the invariant with respect to the boost rapidity I eventually end up with the following pair of vector equations:

$$- ct \frac{d (\gamma \mathbf{p})}{dt} + \mathbf{x} \frac{d (m c \gamma)}{dt} = \frac{q}{c} \frac{d \left( -ct \mathbf{A} + \phi \mathbf{x} \right) }{dt}$$
$$\mathbf{x} \times \frac{d( \gamma \mathbf{p} )}{dt} = \frac{q}{c} \frac{d}{dt} \left( \mathbf{x} \times \mathbf{A} \right)$$

I've made up the exersize for myself so I have no back of the book solutions to check against.

Has anybody seen something like this before, and if so did I get the right result?

2. Oct 22, 2008

### atyy

3. Oct 23, 2008

### Peeter

okay, thanks (that ascii thread is hard to read but at least my result is similar). I'll have to massage things to match them up more closely.

What I didn't realize until I read that is that my rotation wasn't fixed as either hyperbolic or euclidean since I didn't actually specify the specific nature of the bivector for the rotational plane. So I ended up with results for both the spatial invariance and the boost invariance at the same time. Oops;)

Of the six equations that generated the above

$$x^\mu v^\nu - x^\nu v^\mu = \frac{q}{mc} \frac{d}{d\tau} \left( A^\mu x^\nu - A^\nu x^\mu \right)$$

the first vector equation above (taking space time indexes) is the conserved quantity for a boost, and the second for purely spatial indexes is the conserved quantity for spatial rotation. That makes my result seem more reasonable since I didn't expect to get so much only considering boost.