An especially beautiful constraint is "Schroedinger's equation has to be linear because otherwise you can build a (quantum) computer that can solve NP-complete problems in polynomial time". Schroedinger did not know about NP-completeness, but this notion does give us clues why the equation is what it is.
It is always instructive to see how a "fundamental" law can be derived or at least constrained from other laws. Thermodynamics and the notion of entropy would not have been invented if engineers did not "rederive" what was considered fundamental at the time.
The grandparent post is right in the sense that reductionism always reduces, ultimately, to a guess (in this case, Lagrangian mechanics is the guess).
You're right, though, in that a basic framework can be established, using minimal assumptions and axioms, in which most of classical physics and even much of quantum physics can be derived via (mostly) rigorous mathematical arguments. But that framework itself is, at the end of the day, a guess. It's simply the best guess we have at the given moment in time that fits the evidence (e.g., by allowing us to derive from it known rules and patterns that past generations have verified fit the evidence).
This is because in order to derive something, you need to first start from a set of assumptions. You can pick any assumptions you want, that is your privilege, as it is mine. If you want F=ma to be an assumption, you can go on and derive things from that.
But truly, F=ma is not the most general assumption in theoretical physics.
If you look in any decent book on classical mechanics, you will find that F=ma in fact is derived from a much more general principle, the Lagrangian.
That is the same Lagrangian that is the subject of the OP, which is used to derive the Schrodinger equation; and it's not just a coincidence.
I don't think I said it couldn't be derived. I said there wasn't a "fundamental reason why the Schrödinger equation is X". Using Lagrangians to derive it then begs the question "but why must electrons obey the Lagrangian?".
> This is because in order to derive something, you need to first start from a set of assumptions. You can pick any assumptions you want, that is your privilege, as it is mine. If you want F=ma to be an assumption, you can go on and derive things from that.
Well, F = ma was a bad example, I grant you. It's actually a definition of what a force is, nothing more. But it's relevant, because it has a similar purpose (except in a different field of physics). A better example of a fundamental law would be action-reaction or something.
But yes, you're free to pick any assumption you like. But if you assume X, which came about because of assumption Y, it shouldn't be a shock that X can be used to derive Y. At that moment, they are just different notations for the same assumption. You might argue (and hell, I might even agree) that Lagrangians are so much more mathematically pleasing, so make a better assumption. But that doesn't change the fact that you're dressing up assumption Y as assumption X.
> But truly, F=ma is not the most general assumption in theoretical physics.
> If you look in any decent book on classical mechanics, you will find that F=ma in fact is derived from a much more general principle, the Lagrangian.
The Lagrangian is great for solving many problems. But it is definitely not more general than Newtonian mechanics. It can't deal with friction, or quite a few other non-conservative forces. On the plus side, solving oscillating systems is much easier. And it's so much nicer when not using Cartesian coordinates.
> That is the same Lagrangian that is the subject of the OP, which is used to derive the Schrodinger equation; and it's not just a coincidence.
While this is correct, Lagrangian mechanics refers to concepts such as energy which are defined from forces. In particular, Lagrangian's deal with systems with only conservative forces (but total energy may change with time). So it's really a circular argument to say "this concept can be derived from this even more abstract concept, which was actually defined from the first concept and at the end of the day is a guess."
There's nothing wrong with guessing fundamental laws. That's how the scientific method works, after all. You ask a question, guess the answer, predict what your guess would imply and test your predictions.
Sure it is. It's true that for certain systems, you can derive one from the other so in that sense they might be "just different notations" for the same physics (I'm disregarding non-conservative forces, which aren't really considered in fundamental physics). But the Lagrangian formalism really consists of two parts: Hamilton's principle and a choice (postulate/guess) of Lagrangian (or Lagrangian density). When we say that Lagrangian formalism is more general than Newtonian mechanics, it means we can describe physics using the Lagrangian which we can't get to via Newton's laws. For example, you would be hard-pressed to derive general relativity from Newton's laws, but if you start with the Einstein-Hilbert action, you can derive Einstein's field equations.
> A better example of a fundamental law would be action-reaction or something.
This is another example where the Lagrangian formalism is more general. In this case, a translation-invariant Lagrangian implies conservation of (canonical) momentum. But in more complex systems, Newton's third law might may fail when canonical momentum is still conserved (the prototypical example is the Lorentz force law).
> Using Lagrangians to derive it then begs the question "but why must electrons obey the Lagrangian?".
This isn't circular as much it is one less level of indirection. It's like if you say the reason a dropped object accelerates to the Earth is gravity, you have just shifted the question to "why must gravity behave as an inverse squared law". You can go further and say that Newtonian gravity is not fundamental, but is an approximation of from general relativity, where the dropped object isn't really accelerating. Again, the question is shifted to "why is general relativity described by the Einstein field equations". At each level of reduction you describe one phenomenon by something more fundamental. That you don't have a further explanation doesn't logically preclude that it's more fundamental.
Although I would agree that it's fair to say "F=ma is a definition of what a force is", I think of it more as a law of physics than as a definition.
I would also add that Newton's second law is more than merely the statement F=ma.
The original formulation is:
Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd. [1]
So in addition to stating what the word force means, the second law is also saying there are these things called forces which move things in a certain direction.
The way I think of F=ma personally, is that it means the laws of physics are associated with differential equations. So for me, the statement F=ma implies a huge number of other things, since without also having made the assumption that the laws of physics can be described in terms of vector calculus, the definition of force is rather useless and uninteresting.
[1] https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Newt...