Lagrangians that cannot be approximated to zeroth and first order derivatives, magnets edition

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Meds: Not taken

Looping curiosity: Magnets? what are they.

Gripe: Gauge Theory and the Magnetic Vector Potential

Post engaged:

First off, In before high order terms of the Langrangian are axed by dimensional analysis, I don't care about the usefulness of the theor. Today, I care about the nature of the universe and the constitution of the gods. Therefore, I want not of your gauged electrodynamics; I'm specifically curious about the action of the langrangian of ungauged maxwells at absurd potentials and minuscule position differences. At least trying to capture not only x and x_dot, (in our case charge and current) but a lagrangian of the form with x double dot, (accelerating charge) and maybe even x triple dot (a jerk in a charged particle). In recap,

Electric Scalar Potential = potential energy / unit charge
Magnetic Vector Potential = potential energy / unit current
?? ?? Potential = potential energy / unit accelerating charge
?? ?? Potential = potential energy / united jerking charge

^^ I keep getting told here that accelerating charges emit EMF. Does one here have any intuitions on non uniformly accelerated charges.

On a semi related note is it possible to create materials with continuous gradients of permeability or susceptibility? And would there be a function to follow that would be like as if bumping the derivative on the charge passing through that medium, I think that would be badass and useful for experiment.