Understand Hamilton's Principle Intuitively

[tut_einstein]

While I understand how the Euler-Lagrange equations are derived by minimizing the integral of the Lagrangian, I don't intuitively understand why Hamilton's principle is true. Specifically, what physical quantity does the Lagrangian represent and what does minimizing it mean? I'd just like to get an intuitive feel for it, I understand it mathematically.

Thanks!

 

[vanhees71]

The deeper reason for why Hamilton's principle is true is mysterious within classical mechanics. It's just a mathematical observation that one can derive many equations of motion from a variational principle.

From the physical point of view you have to use quantum theory to understand why it holds true. If you formulate quantum theory in terms of Feynman's path integral in many cases it boils down to a path integral for the propagator of the particle over all paths with the action of the analogous classical system. In the case that the system is close to classical behavior, the action divided by $\hbar$ is very steeply rising around the classical path, along which it is stationary, and this means that the path integral is dominated by paths very close to the classical one, which explains why the equations of motion follow from the Hamilton's principle, which says nothing else than that the classical path is given by the stationary point of the action functional.

 

[Cleonis]

[...] I don't intuitively understand why Hamilton's principle is true. Specifically, what physical quantity does the Lagrangian represent and what does minimizing it mean?

The way I understand it the Lagrangian does not in itself represent a physical quantity. In classical mechanics the Lagrangian is a mathematical tool, with no direct physical counterpart.

As a starting point I recommend the following page (on Edwin Taylor's website), the page was created by Edwin Taylor and Slavomir Tuleja: Principle of least action interactive

The why of Hamilton's principle of least action is a recurrent question.
What does it mean to minimize the action?
I posted about this subject in november 2010, in a thread titled 'Lagrangian'

The mathematical discussion that I posted here on physicsforums is a shortened one, a more extensive discussion is on my own website.



Finding the one path for which the action is minimal is mathematically the same as finding the one path with the property that at all times the amount of change in kinetic energy is a match for the amount of change in potential energy.

The change in kinetic energy and the change in potential energy have to match; when an object traverses a potential field any gain/loss of kinetic energy must match the decrease/increase of potential energy.