While trying to understand the essentials on variational mechanics, I have come into this article: http://jazz.openfun.org/wiki/Brachistochrone.
The problem of the brachistochrone dates back to Newton (17th century) and appears mentioned frequently as one of the problems that can be solved be means of this refined technique.
In fact, this problem appears to be the actual trigger for the modern physics (or better stated, its solution), and reveals a completely counterintuitive phenomenon:
The quickest path is not the straight line
Amazingly, the solution for a bead on a wire going in the least time from point A to point B:
Is the cycloid represented in the picture.
As standard human, one always tends to think that the shortest path would be the quickest one. It is wrong.
Analogously, when it comes to minimize the action (see previous posts on the subject), it is very hard to imagine that the stationary points would not come at flattening the curve for the action.
This is what makes so hard to understand the variational principles that rule over Lagrangian, Eulerian and Hamiltonian mechanics.
If one only had been told...
I have made a spreadsheet with openoffice (www.openoffice.org) where this can be numerically perceived also.
The spreadsheet is here.
I have made it with the help of a paper by Jozef Hanc: The original Euler's calculus-of-variations method: Key to Lagrangian mechanics for beginners.
I found it extremely appropriate for my case, and also absolutely clarifying. Thanks mr Hanc