# Lagrangian for electromagnetic field derivation

• I
• TimeRip496
In summary, the author presents a clever way to put the Lorentz force law into a specific form by using the partial derivative to pick out the jth component in the dot product. By adding a term containing the scalar potential inside the partial derivative, the final equation is obtained. The quantum derivation of this is even more illuminating, using local SU(1) symmetry.
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 jth 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)

dextercioby
TimeRip496 said:
" 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.

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

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

## 1. What is the Lagrangian for the electromagnetic field?

The Lagrangian for the electromagnetic field is a mathematical function that describes the dynamics of the electric and magnetic fields. It is derived from Maxwell's equations and is used in the study of electrodynamics.

## 2. Why is the Lagrangian used in the derivation of the electromagnetic field?

The use of the Lagrangian in the derivation of the electromagnetic field allows for a more elegant and concise formulation of the equations of motion. It also provides a deeper understanding of the underlying physical principles.

## 3. How is the Lagrangian for the electromagnetic field derived?

The Lagrangian for the electromagnetic field is derived by applying the principles of Lagrangian mechanics to the equations of motion for the electric and magnetic fields. This involves finding a function that satisfies the Euler-Lagrange equations and is consistent with Maxwell's equations.

## 4. What are the advantages of using the Lagrangian approach in deriving the electromagnetic field?

The Lagrangian approach offers several advantages, including a more intuitive understanding of the physical principles, the ability to easily incorporate constraints and boundary conditions, and the ability to generalize to more complex systems.

## 5. Are there any limitations to using the Lagrangian in the derivation of the electromagnetic field?

While the Lagrangian approach is a powerful tool, it does have some limitations. It may not be suitable for certain systems with non-conservative forces or when the system has a large number of degrees of freedom. In these cases, other methods, such as the Hamiltonian approach, may be more appropriate.

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