Classical/QM justification of principle of least action

[greypilgrim]

Hi.

Is the principle of least (better: stationary) action only an axiom in classical mechanics, or can it be derived from a more profound (classical) principle?

As far as I know, it can be derived from the path integral formulation of QM. Is this a more profound justification for the principle of least action? Or are we just moving in circles, since QM emerges from canonical quantization of a classical Hamiltonian where the principle of least action has already been used?

If QM is indeed a more profound justification of the principle of least action, does this imply that QM might be necessary for classical physics?

 

[vanhees71]

The QM argument is the only one I know that in some sense "derives" the action principle. Within classical mechanics it's just a very clever and elegant mathematical tool to express the empirically known equations of motion, which let's you analyze them in a simpler way than by just doing "naive mechanics". Particularly the symmetry principles a la Noether are of prime importance to all physics. Also from a practical point of view, it's much simpler to derive equations of motion in arbitrary generalized coordinates than the brute-force way to express the "naive" equations of motion in the generalized coordinates.

QM is of course the more comprehensive theory, and classical mechanics is an emergent phenomenon in the sense that it can be derived as a certain approximation being valid under certain circumstances. The path integral teaches us that it is an approximation valid where the typical relevant scale of action variables is large compared to $\hbar$.

 

[bhobba]

Here is how its done.

You start out with <x'|x> then you insert a ton of ∫|xi><xi|dxi = 1 in the middle to get ∫...∫<x|x1><x1|...|xn><xn|x> dx1...dxn. Now <xi|xi+1> = ci e^iSi so rearranging you get
∫...∫c1...cn e^ i∑Si.

Focus in on ∑Si. Define Li = Si/Δti, Δti is the time between the xi along the jagged path they trace out. ∑ Si = ∑Li Δti. As Δti goes to zero the reasonable physical assumption is made that Li is well behaved and goes over to a continuum so you get ∫L dt.

Now Si depends on xi and Δxi. But for a path Δxi depends on the velocity vi = Δxi/Δti so its very reasonable to assume when it goes to the continuum L is a function of x and the velocity v.

Its a bit of fun working through the math with Taylor approximations seeing its quite a reasonable process.

In this way you see the origin of the Lagrangian. And by considering close paths we see most cancel and you are only left with the paths of stationary action.

Go and get copy of Landau - Mechanics where all of Classical mechanics is derived from this alone - including the existence of mass and that its positive. Strange but true. Actually some other assumptions are also made, but its an interesting exercise first seeing what they are, and secondly their physical significance. Then from that going through Chapter 3 of Ballentine: QM - A Modern Development.

Thanks
Bill

 

[pr3dator]

The principle of least action originated from principle of least time called Fermat's principle.
Because the particles are waves.

 

[bhobba]

The principle of least action originated from principle of least time called Fermat's principle.
Because the particles are waves.

Particles are not waves - that's a very very old and way outdated view.

I gave the detail where it comes from. If you want greater rigor you need the principle of steepest decent and tomes on the path integral formulation give the detail - but is not necessary to understand the physics.

Thanks
Bill

 

[pr3dator]

Read carefull of words of Einstein.
All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta?' Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken. (Albert Einstein, 1954)
http://web.comhem.se/~u99400062/EinsteinQ.html
You are mistaken when you think of particles

 

[bhobba]

Read carefull of words of Einstein.

Your point being? You realize don't you since Einsteins time a lot of progress has been made in understanding QM? One can only assume you are not aware of modern developments that show all the poineers, Einstein, Bohr, Schrödinger etc etc, with the notable exception of Dirac, were wrong.

Read carefully the words of Wienberg:
http://www.fisica.ufmg.br/~dsoares/cosmos/10/weinberg-einsteinsmistakes.pdf
The other mistake that is widely attributed to Einstein is that he was on the wrong side in his famous debate with Niels Bohr over quantum mechanics, starting at the Solvay Congress of 1927 and continuing into the 1930s. In brief, Bohr had presided over the formulation of the Copenhagen interpretation of quantum mechanics, in which it is only possible to calculate the probabilities of the various possible outcomes of experiments. Einstein rejected the notion that the laws of physics could deal with probabilities, famously decreeing that God does not play dice with the cosmos. But history gave its verdict against Einstein quantum mechanics went on from success to success, leaving Einstein on the sidelines. All this familiar story is true, but it leaves out an irony. Bohr's version of quantum mechanics was deeply flawed, but not for the reason Einstein thought. The Copenhagen interpretation describes what happens when an observer makes a measurement, but the observer and the act of measurement are themselves treated classically. This is surely wrong: Physicists and their apparatus must be governed by the same quantum mechanical rules that govern everything else in the universe. But these rules are expressed in terms of a wave-function (or, more precisely, a state vector) that evolves in a perfectly deterministic way. So where do the probabilistic rules of the Copenhagen interpretation come from? Considerable progress has been made in recent years toward the resolution of the problem, which I cannot go into here. It is enough to say that neither Bohr nor Einstein had focused on the real problem with quantum mechanics. The Copenhagen rules clearly work, so they have to be accepted. But this leaves the task of explaining them by applying the deterministic equation for the evolution of the wave-function, the Schrödinger equation, to observers and their apparatus. The difficulty is not that quantum mechanics is probabilistic that is something we apparently just have to live with. The real difficulty is that it is also deterministic, or more precisely, that it combines a probabilistic interpretation with deterministic dynamics.'

I suggest you study some modern QM interpretations eg:
http://quantum.phys.cmu.edu/CHS/histories.html

That is not to say its correct - no interpretation is better than any other - it simply gives a modern take.

But all that is just bye the bye - its getting way off the threads topic and if you want to discuss it start another thread or, correctly, the mods will shut this one down.

You are mistaken when you think of particles

Of course - its now known its quantum fields.

But I suspect that is not your point and you have not been exposed to a modern treatment of QM such as Ballentine. I suggest you rectify that ASAP if you want to discuss QM.

Thanks
Bill