Electromagnetic lagrangian: correcting it with special relativity

Our August 28, 2024 post suggests that the electromagnetic lagrangian (or action) does not understand the kinetic energy of two overlapping electric fields E + E'. This means that Maxwell's equations do not understand it either.

Is there a generic way in which we can make a lagrangian to respect special relativity?

Moving elastic energy

Let us look at an elastic solid block. How does a physical model understand that moving elastic energy should possess kinetic energy and momentum?

    =====================  elastic block

    =====================  

    wedge  /\   ---> v               ● obstacle

                  ^    moving wedge

                  |    presses the block

                  F

Let us have an elastic block which a wedge presses upward with a force, causing a distortion which contains elastic energy. The block is very lightweight, so that the kinetic energy of its atoms is negligible.

If the wedge moves at a velocity v, there should be momentum associated with the moving elastic energy. Otherwise, the center of mass of an isolated system could move.

Since there is momentum, the wedge can bump into an obstacle, giving it momentum and kinetic energy. This proves that the elastic energy must contain kinetic energy, too.

How do we model this system in special relativity?

https://en.wikipedia.org/wiki/Strain_energy

Usually, the kinetic energy of elastic energy is ignored, and a lagrangian density is written simply for the elastic energy content:

       L  =  -1/2 E ε²,

where E is Young's modulus and ε is the strain. The elastic energy is so small that we do not need to care about its momentum and kinetic energy.

Writing truly special relativistic lagrangians

Is there some simple way to write the kinetic energy into the lagrangian density? Let us look at the stress-energy tensor of the elastic energy. Maybe we should treat potential energy just like a massive particle is treated?

https://en.wikipedia.org/wiki/Four-momentum

In quantum mechanics, any particle is an excitation of a field. It is natural to treat elastic energy in the same way as a massive particle?

https://en.wikipedia.org/wiki/Relativistic_Lagrangian_mechanics

The Wikipedia article contains a lagrangian for a charged particle q in an electromagnetic field (φ, A), but it ignores the kinetic energy of the potential energy of the particle q.

Conclusions

A correct special relativistic lagrangian for electromagnetism can be quite complicated to write. In many cases, the lagrangian found in literature is precise enough.

If an electron is under a large potential from a charged insulator, then its potential energy does have major effects. Our solution of the Klein paradox on October 31, 2018 relies on the extra inertia which an electron acquires at a steep potential wall.