I don't understand the problem the author sees with the conventional explanation. Yes, test particles of different charges behave differently, but this in no way implies that the law should involve the third derivative of position! The author says:

> I’ve never worked it out in detail, but wouldn’t be surprised if such an approach can be made to work. Essentially, it’d make acceleration into a free (possibly constrained) parameter of the particle, rather than something completely determined by the distribution of matter and fields. That free parameter would implicitly contain what (in the conventional approach) we think of as the charge information. Indeed, the new equations of motion would have a conserved quantity, corresponding to the charge. But the resulting force laws would be quite a bit uglier.

I don't see how this could be true. If it is, I'd indeed like to see it worked out.

Particles with different charges still follow F = ma. The force is different for different particles, sure, but the evolution of the position of the particle still follows a second order differential equation and not a third order one. Furthermore, going to a third order equation doesn't even solve the problem that the F is different for particles of different charge, that problem is still there the same as before. I don't think is a problem at all, by the way: the forces always depend on the particular situation we're considering, including on the mass, charge, and so on, and even more so for compound objects.

The question "Why does F = ma?" is a good question though. The conventional explanation only explains why classical mechanics is governed by a second order differential equation. It does not explain why F = ma is the right equation. If you interpret F as a general function of the state of the system then indeed F = ma doesn't tell you anything, because by picking the appropriate F you can get any second order differential equation. However, if we already have some prior idea of what force and mass is, which we do, then it's not clear why it should be this differential equation. We do have an intuitive idea of what force is: if you hang 1kg on a rope then the rope pulls with some amount of force, and if you put 2kg then it pulls with double the force. Similarly, if you stretch a spring by 3cm you have some amount of force, and if you stretch two of those springs simultaneously you have double the force. You could relate the force in the spring with force by a weight by finding the amount of weight you need to stretch the spring by that 3cm. We can get a definition of force by picking some reference force as being 1 unit of force, and we can similarly come up with an experimental definition of mass.

If we accept such a definition then F = ma has physical meaning. For instance, it says that if you double the force then you double the acceleration. That is, if you pull an object with two springs then it accelerates twice as fast as when you pull it with one spring, if you stretch the springs by the same amount in both cases. This is a nontrivial fact about the physical world. The law F = ma also tells you that if you apply a constant force then the velocity increases linearly. That is, if you pull an object with a spring and keep the spring stretched by 3cm, then if the object accelerates to speed v in one second then it will accelerate to 2v in two seconds. This is a nontrivial fact about the physical world too.