pervect said:
I'm not too sure my thoughts will be helpful for the intended purpose (which I assume ultimately involves a simple presentation of special relativity), but perhaps they'll provide some insight. And they can be expressed pretty briefly. As long as special relativity has a non-trivial Hamiltonian, we can write Hamilton's equations:
\frac{\partial H}{\partial q} = \dot {p}
Now, if we can also identify H with energy, q with position, and ##\dot{p}## with force, then we have the work-energy theorem, the rate of change of the energy with position must be equal to the force.
This is an interesting approach, but if we examine it carefully I think it's not quite as much of a slam-dunk as it might seem. And it does have the disadvantage that it won't work as a presentation at the undergraduate level, since a physics major might not be exposed to Hamiltonian mechanics until after taking their upper-division SR course.
Spelling out the steps in more detail, I think we would have the following:
(1) The action has to be Lorentz-invariant and additive, and the only possibility that seems to present itself is ##S=(\ldots)m\int_{t_1}^{t_2} d\tau##, where ... represents a constant.
(2) Working backward from this, infer that the Lagrangian for a free relativistic particle in one dimension is ##L=(\ldots)m/\gamma##.
(3) Find the conjugate momentum, which is ##p=(\ldots)m\gamma v##.
(4) Interpret ##dp/dt## as a force.
(5) Calculate the Hamiltonian.
(6) Identify the Hamiltonian with the energy.
(7) Use Hamilton's equations to associate ##\partial H/\partial x## with minus the force.
The first thing to note is that this is much, much longer than the simple derivation I gave (the second of the three sections in my original post).
The next problem is that we have foundational issues in steps 1 and 4. Maybe there's an explicit uniqueness theorem one could give at step 1? Otherwise it's just a plausibility argument, with no a priori guarantee, for example, that the final result will come out consistent with Maxwell's equations in the case of a charge moving in a field. In step 4, we have to decide what is the best definition of force. One could argue that this definition is good, because by the time we get to step 7 we will have shown that it preserves the form of the work-energy theorem without correction. But this is a weak justification if we aren't at step 7 yet, and in fact we have a different definition of force, the four-force, which a priori would be preferable because it's tensorial.
At step 6, we have to check some technical criteria.
A general philosophical objection to the whole thing is that Hamiltonian mechanics lacks manifest Lorentz invariance at every step of the way. In particular, time is treated as a parameter rather than a coordinate.
