EM Field Lagrangian: What Defines It?

[snoopies622]

According to this site

http://quantummechanics.ucsd.edu/ph130a/130_notes/node453.html

a good choice of Lagrangian for the electromagnetic field is

$$L = - \frac {1}{4} F_{\mu\nu}F_{\mu\nu} + \frac {1}{c} j_\mu A_\mu$$

where

$$F_{\mu \nu} = \frac {\partial A_\nu}{\partial x_{\mu}} - \frac {\partial A_\mu}{\partial x_{\nu}}$$

(I don't know why all the indices are at the bottom, but anyway...)

I take it that the components of $A_\mu$ are the generalized coordinates and their first partial derivatives with respect to space and time can be thought of as their corresponding generalized velocities.

This looks different from the kind of Lagrangian I've seen in classical mechanics in that

- the derivatives are not merely with respect to time

and

- it's not evidently some expression of kinetic energy minus potential energy.

But that's OK, right?

The site says that it's a good choice because the Euler-Lagrange equations that use it turn out to be Maxwell's equations.

Shall I conclude, then, that making good (correct) Euler-Langrange equations is what really defines a Lagrangian, and not the more limited definition I indicated above?

 

[Matterwave]

Yes, in general, the correct Lagrangian is the one which creates the right "equations of motion"...

 

[snoopies622]

Of course, that does make it strange to say that the principle of least action is a fundamental law of nature if we then define action (via our Lagrangian) to be in accord with laws of nature that we're already aware of.

 

[Born2bwire]

According to this site

http://quantummechanics.ucsd.edu/ph130a/130_notes/node453.html

a good choice of Lagrangian for the electromagnetic field is

$$L = - \frac {1}{4} F_{\mu\nu}F_{\mu\nu} + \frac {1}{c} j_\mu A_\mu$$

where

$$F_{\mu \nu} = \frac {\partial A_\nu}{\partial x_{\mu}} - \frac {\partial A_\mu}{\partial x_{\nu}}$$

(I don't know why all the indices are at the bottom, but anyway...)

I take it that the components of $A_\mu$ are the generalized coordinates and their first partial derivatives with respect to space and time can be thought of as their corresponding generalized velocities.

Hold on here a second. Are you familiar with four vectors? The indices are on the bottom for a reason and they do not denote derivatives. A is the potential four vector and j is the current four vector. You should take a look at the wikipedia pages for the four potential and four current (http://en.wikipedia.org/wiki/Electromagnetic_four-potential and http://en.wikipedia.org/wiki/Four-current) and better yet look at Jackson's textbook to understand what the notation means. If you work it out though it comes back out to be the familiar Lagrangian that you would find in classical EM.

 

[Pengwuino]

Hold on here a second. Are you familiar with four vectors? The indices are on the bottom for a reason and they do not denote derivatives. A is the potential four vector and j is the current four vector. You should take a look at the wikipedia pages for the four potential and four current (http://en.wikipedia.org/wiki/Electromagnetic_four-potential and http://en.wikipedia.org/wiki/Four-current) and better yet look at Jackson's textbook to understand what the notation means. If you work it out though it comes back out to be the familiar Lagrangian that you would find in classical EM.

To compare, 3-vectors have the index notation forms of, for example, $$V_i$$ where i = 1,2,3 to correspond to the x,y,z components. In this example, V is just any vector. Vectors are specific mathematical objects that transform in certain ways.

4-vectors on the other hand, are different. In the electromagnetic Lagrangian, A is the 4-potential that has 4 components just like a 3-vector has 3 components. The index notation is changed to $$A_\mu$$. Depending on what text you're using, you'll find that greek symbols denote the 4 indices of a 4 vector, $$\mu = 0,1,2,3$$ or $$\mu = 1,2,3,4$$ depending on what text you're in. Whereas a velocity (ie, 3 velocity) might have components $$V_i = (v_x, v_y, v_z)$$, a 4-vector (in this case, the 4 potential) has components $$A_\mu = (\phi, A_x, A_y, A_z)$$ which are the electromagnetic potentials. Again, depending on the author and metric used, there might be a minus sign somewhere and the $$\phi$$ might be the 4th component.

Also, as Born2bwire stated, $$A^\mu$$ and $$A_\mu$$ are different things! The former is a covariant 4-vector while the latter is a contravariant 4-vector.

 

[snoopies622]

Actually, all I meant was that in the first equation the author is summing over the $\mu$'s and $\nu$'s (contracting), and I thought the normal way to symbolize that is to place one index above and one below, instead of both above or both below.