variation of the electromagnetic lagrangian

[decraig]

Reference http://en.wikipedia.org/wiki/Lagrangian

 

Toward the end of the article is are sections titled,

"Electromagnetism in general relativity" and

Electromagnetism using differential forms".

 

I happen to use forms. Though the argument is just as valid in the standard notation up to a difference in parity. Both are generally covariant.

 

Through the usual method, we can derive maxwell's equation, and define electric charge assuming charge/current density is invariant with respect to the vector potential.

 

 

This is somewhat disappointing. Rather than obtaining the dynamical laws of electromagnetism, all we seem to get are maxwell's equations. However, the implicit assumption of the independence of charge/current with respect to the vector potential seems to be, itself, a dynamical constraint,

 

[math]\partial J / \partial A = 0[/math].

 

These are 16 constraints.

J is the charge/current density 4 vector, or charge/current density three-form

A is the vector potential, or covariant one-form.

 

I'm having a difficult time seeing if this leads to the lorentz force.

Edited December 21, 2013 by decraig

[ajb]

This is somewhat disappointing. Rather than obtaining the dynamical laws of electromagnetism, all we seem to get are maxwell's equations.

Aren't the dynamical laws of electromagnetism Maxwell's equations?

[decraig]

Maybe I shouldn't have used the term "dynamical." I tend to use arcane language. It's kind-of a fuzzy term.

 

In any case, it implies forces. Maxwell's equations don't give us any forces. The Lorentz force does.

 

 

But the background is not Newtonian physics but relativity, so we probably won't see dp/dt without some effort, but a Hamiltonian formulation. Just guessing.

Edited December 21, 2013 by decraig

[ajb]

In any case, it implies forces. Maxwell's equations don't give us any forces. The Lorentz force does.

Right, you have to work out how the EM field can couple to particles and then include these in the theory. In this sense you don't get an external force, you place it all in a Lagrangian and use the EL equations, but you do recover the Lorentz force.

 

In essence you Maxwell's equations are not quite enough, you need to add the interaction with charged particles.

[decraig]

I hope you see something I don't see.

 

F^*F the is the electromagnetic energy density term.

J^A is potential energy of field acting on charge, so we already appear to have an interaction term. ??

[ajb]

This is a source term, you still need to include what happens to a charged particle in the theory.

[decraig]

consider for a moment, Bruce. There are no source terms. This is bullshit.

 

No insult to your person implied. There is no one on this forum that respects you more than I.

 

I don't mean to imply the the mathematics is nonsciencical, but the physics in meaningless.The implication in such language is that charge causes fields. There is no causality in this: charge does not generate fields an more than fields generate charge. This is 20th century thinking.

 

 

 

Sorry. I got carried away.

Edited December 22, 2013 by decraig

[ajb]

The implication in such language is that charge causes fields.

 

 

There is a standard notion of a source in field theory. In EM this is an electric current.

 

You still need to add more to get the Lorentz force.

[decraig]

Thanks for your help. I only object to the designation "source", in presuming charge causes fields. This is not well-thought physics, but I know everyone thinks this way. They might invoke Biot-Savart. Given some effort, I think we could come up with something that says, with equal weight, that fields cause charge.

 

by the way, I found the electromagnetic lagrangian for a line integral for discrete charges in Jackson's classical electrodynamics text (chapter 12). It's a start. I want the lagrangian density for charge/current fields. I might have a program to follow. I take Jackson's relativistic Lorentz force equations for discrete charges and raise them to covariant densities.

Edited December 22, 2013 by decraig