Lagrangian for electromagnetic field derivation

[TimeRip496]

" We can put the Lorentz force law into this form by being clever. First, we write
$$\frac{dA_j}{dt}=\frac{d}{dt}(\frac{\partial{}}{\partial{v_j}}(v.A)),$$
since the partial derivative will pick out only the j^th component of the dot product. Now, since the scalar potential is independent of the velocity, we can add on a term containing it inside the partial derivative:
$$\frac{dA_j}{dt}=\frac{d}{dt}(\frac{\partial{}}{\partial{v_j}}(v.A-q∅)),$$ "

I don't understand the rationale behind these. It seems like the author is trying to get to the final equation so he guess the steps in order to get to the final equation.

Source: https://www.sccs.swarthmore.edu/users/02/no/pdfs/lorentz.pdf
Page 2 below eqn(13)

 

[Mentz114]

" We can put the Lorentz force law into this form by being clever. First, we write
$$\frac{dA_j}{dt}=\frac{d}{dt}(\frac{\partial{}}{\partial{v_j}}(v.A)),$$
..
Source: https://www.sccs.swarthmore.edu/users/02/no/pdfs/lorentz.pdf
Page 2 below eqn(13)

The link does not work for me - waits until timeout.

 

[bhobba]

The link worked for me.

However it is classical - not quantum so really should be in the classical physics section.

However the quantum derivation is very very beautiful and much more illuminating.

You will find it in page 129 of the following book:
https://www.amazon.com/dp/3319192000/?tag=pfamazon01-20

The basis, as I think was first discovered by Schwinger, is local SU(1) symmetry.

The same kind of reasoning leads to the existence of the Higgs for example - and much more. It is well worth your study.

Thanks
Bill

 

[dextercioby]

I hadn't seen this non-covariant derivation anywhere, so a big thank you to the OP for bringing it up.