Where does Hamilton's principle come from?

[Trave11er]

Does it have any deeper theoretical foundation or is it just true empirically?

 

[Dickfore]

Which Hamilton's principle?

 

[Trave11er]

"The system develops in such a way as to extremize the action functional"

 

[Dickfore]

Well, if you are familiar with the Feynman's path integral formulation of QM, the transition amplitude for a particular classical phase trajectory is proportional to:
$$\exp \left( \frac{i}{\hbar} \, S \right)$$
where S is the classical action of the system.

Now, in classical mechanics, there is no mention of Planck's constant. Formally, you go to classical mechanics by setting $\hbar \rightarrow 0$. In this limit, the integral with a huge complex exponential is evaluated by the stationary phase approximation, i.e. the dominant contribution to the integral comes from the phase trajectory that makes the action extremal:
$$\delta S =0$$
Intuitively, you can understand this by the rapidly oscillating phase factor. Any phase trajectory that is not extremal, has a counterpart with the opposite phase, canceling their contribution. The only one left is the extremal phase trajectory.

You may notice that the last condition is the Hamilton's least (extremal) action principle.

 

[Trave11er]

Thanks for the reply. Though I can't quite see, why is that only the extremal integral is not canceled out. It seems that there is infinity of possible values with their corresponding action integrals - why should the extremal one survive - can you go into, please?

 

[Cleonis]

Thanks for the reply. Though I can't quite see, why is that only the extremal integral is not canceled out. It seems that there is infinity of possible values with their corresponding action integrals - why should the extremal one survive - can you go into, please?

A more generalized formulation is to talk about a path (among the variation range) for which the action integral is stationary.

As we know, a quadratic function has an extremum, but a third power function doesn't necessaritly have any extremum. The case of a stationary action integral is like a third power function with one point where the derivative is zero.

For the sake of simplicity let's say the points of the graph of the third power function represent the variation range of possible paths.

For any point that is not the point where the derivative is zero the following property applies: if you evaluate two paths, infinitisimally close to each other, then the action integrals of those two paths come out differently, and the difference is proportional to the magnitude of the derivative at that point of the graph.

But at the point on the graph where the derivative is zero the outcomes of the action integral are "bunched up" so to speak. At the 'stationary point' of the 'principle of stationary action' there is a unique situation: for two paths, infinitisimally close to each other, the difference in their action integrals goes to zero the closer to the 'stationary point'.

Mathematically this is trivial of course. As I understand it Feynman emphasized this as expressing a crucial physics point.

 

[Trave11er]

Thanks a lot.