#euler-lagrange equations

[peeterjoot]

Progressing towards coordinate free form of the Euler-Lagrange equations for Maxwell's equation

Progressing towards coordinate free form of the Euler-Lagrange equations for Maxwell’s equation

This is the 6th part of a series on finding Maxwell’s equations (including the fictitious magnetic sources that are useful in engineering) from a multivector Lagrangian representation. [Click here for a PDF version of this series of posts, up to and including this one.]  The first, second, third, fourth, and fifth parts are also available here on this blog. We managed to find Maxwell’s equation…

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[peeterjoot]

Maxwell's equations with magnetic charge and current densities, from Lagrangian.

Maxwell’s equations with magnetic charge and current densities, from Lagrangian.

This is the 4th part in a series on finding Maxwell’s equations (including the fictitious magnetic sources that are useful in engineering) from a Lagrangian representation. [Click here for a PDF version of this series of posts, up to and including this one.]  The first and second, and third parts are also available here on this blog. Now, let’s suppose that we have a pseudoscalar Lagrangian…

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[peeterjoot]

Lagrangian for magnetic portion of Lorentz force

Lagrangian for magnetic portion of Lorentz force

[Click here for a PDF of this post with nicer formatting]

In [1] it is claimed in an Aharonov-Bohm discussion that a Lagrangian modification to include electromagnetism is

\begin{equation}\label{eqn:magneticLorentzForceLagrangian:20} \LL \rightarrow \LL + \frac{e}{c} \Bv \cdot \BA. \end{equation}

That can’t be the full Lagrangian since there is no \( \phi \) term, so what exactly do we get?

If…

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