What is the inconsistency between Maxwell's electrodynamics and newtonian mechanics? As far as I understand, when a modification of a theory is made it is because some observation required this modifcation. Quantum Mechanics is a nice example of that: observations of microscopic phenomena showed that classical mechanics was giving the wrong predictions, so a new approach was required.
Now, another case is special relativity. It is often said that special relativity was required because newtonian mechanics was inconsistent with Maxwell's electrodynamics.
I must confess though that I've always failed to see what is that inconsistency. What I want here is to find a motivation for the requirement of special relativity. I want to understand what led Lorentz and Einstein to see the need of a new theory of spacetime.
So what is the inconsistency between newtonian mechanics and Maxwell's electrodynamics which led to the development of special relativity?
 A: The obvious difference is that Newton's equations retain their form for all inertial reference frames when Galileo's Principle of Relativity is used, but Maxwell's equations are not invariant under this transformation.
Instead one must use the Lorentz transform, which recognizes that there is a fixed speed for light, $c$. This limit was recognized by Maxwell when he first worked out the form of electro-magnetic waves; the theoretical value matched well with the then best experimental results for the speed of light.
The final result was the theory of Special Relativity, and the modification of Newton's Laws of Motion so as to make them Lorentz invariant.
A: 
I must confess though that I've always failed to see what is that inconsistency. What I want here is to find a motivation for the requirement of special relativity.

There would have been no inconsistency if the luminiferous aether existed. Newtonian mechanics needed a medium for all its wave manifestations.

Luminiferous aether or ether ("luminiferous", meaning "light-bearing"), was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space, something that waves should not be able to do. The assumption of a spatial plenum of luminiferous aether, rather than a spatial vacuum, provided the theoretical medium that was required by wave theories of light. 

If the Michelson Morley experiment had found the aether, no problemo.
So the inconsistency came because the experiments showed that there is no luminiferous aether through which, light included, everything waded, in the Newtonian mechanics.
Thus the special relativity Lorenz  transformations which at first appeared only in electromagnetic theory, were postulated by Einstein to also describe mechanics at high velocities, so as to have a consistent framework for physics; which was prophetic , predicting the nuclear age revolution .
A: One of the solutions to Maxwell's equations has the form of a wave equation, where the speed at which the waves propagate is $c$, where (in SI units) $c = \sqrt{1/(\varepsilon_0 \mu_0)}$, and $\varepsilon_0$ (permittivity of the vacuum) & $\mu_0$ (permeability of the vacuum) are constants in Maxwell's equations.
But, if Newtonian mechanics is correct (really: if Galilean relativity is correct), $c$ can not be a constant, as you can always choose a moving frame in which the speed of the waves will be less than, or more than, $c$.
This means one of two things:


*

*either Maxwell's equations are true only in some privileged frame of reference (I call this the 'rest frame' below);

*or Galilean relativity (and hence Newtonian mechanics) is not correct, and in particular is increasingly far from correct for frames moving at relative speeds $v$ near $c$ while being an increasingly good approximation when $v \ll c$ (this must be true because we know it makes very good predictions for frames like this).


Well, this is perfectly testable.  First of all, the wave motions predicted by Maxwell's equations do exist in reality: they're electromagnetic waves, including light waves, radio waves &c.
So then the experiment you need to do is to measure the speed of these waves in frames which are moving relative to each other.  In fact you can do this in a single frame by measuring the speed of the waves in the direction the frame is moving (relative to some other frame) and perpendicular to it.  If Galilean relativity is correct, then the speeds will differ, and it will be possible to find the special 'rest frame' in which Maxwell's equations are correct.  if Galilean relativity isn't correct then it won't be possible to find such a frame: Maxwell's equations will be correct in all (inertial) frames.
This was done, of course, by Michelson & Morley, where the 'moving' frame is the frame of the Earth, and we know the Earth's frame must be moving because the Earth moves around the Sun so even if the Sun's frame is not at rest in such a way that the Earth's frame is momentarily at rest, then the Earth's frame won't be at rest half-a-year later.
And the result of the experiment was, of course, that Maxwell's equations are correct in all inertial frames -- the speed of light is the same as measured from any inertial frame -- and so Galilean relativity must be incorrect, and if it is incorrect then Newtonian mechanics is also incorrect, since it is built on Galilean relativity.

(Note I have not mentioned many subtleties involved with doing the experiment and many proposed workarounds such as aether-dragging &c, all of which really got ruled out later.  It's worth reading the history if you're interested.)
A: Two of Maxwell's equations combine to yield a wave equation with a fixed wave velocity, the speed of light $c$, for both of two observers in relative motion to one another, contrary to the behavior of waves in Newtonian mechanics.
A: Acoording to Carlos Rovelli in his Quantum Gravity, the main inconsistency is that Newtons theory is non-local whilst Maxwells theory is local. What this means in more down to earth language is that Newtons theory of gravity has action at a distance: if the mass of the sun suddenly halved we would instantly feel the effects here on earth. Whereas, Maxwells theory has an electromagnetic field mediating the transmission of light, hence we would see the sun dim after eight or so minutes.
What Einstein set out to do was to find a local theory of gravity, that means in essence, a field theory of gravity. It turns out that this field is the metric on spacetime. It's worth pointing out that Newton realised that this action at a distance was a failure of his theory, but he could see no way around it. Given that he had already invented calculus, it's probably a bit much to also ask him to invent non-Euclidean geometry too.
A: I have a high school textbook that suggests a thought experiment aimed at highlighting the inconsistency of galilean relativity and electrodynamics without resorting to maths.
The book is intended for students who cannot understand the maths of Maxwell Equations, and only know that some of their solutions are waves.
I am not entirely convinced of this example, which goes as follows: suppose you can have two protons move at the same constant velocity along parallel straight paths. What are the forces between  them?
Surely they repel each other due to the electric fields they generate. Also, in a frame of reference where the protons are moving, they can be regarded as tiny currents. Hence each of them would  generate a magnetic field which ends up attracting the other proton, as we expect from the interaction betwee two parallel currents. So the attraction due to the magnetic field "weakens" the repulsion due to the electric field.
Now if we choose a frame of reference where the protons are at rest, there is no "current", no magnetic field and hence no attractive force. So the net force between the protons is different. How can that be?
I do understand the idea of "tiny current", but the whole idea seems weak to me, probably because we have only two protons.
I have just started thinking about it, and I cannot really put my finger on what bothers me.
I'd like to have a simple intuitive example not involving maths, but I would not like an handwaving example that has only the apparence of being rigorous.
Perhaps one might consider two parallel beams of protons which, for some reason, proceed along parallel paths despite the mutual repulsion (it's a thought experiment, anyway).
Any insight into this is appreciated.
PS earlier editions of the same textbook went through the standard line of reasoning, involving the Michelson Morley experiment. I guess they concluded this example is better but, again, I'm not really convinced.
