Hamilton, Lagrange and the Principle of Least Action

[thicknavyrain]

Trying to wrap my head around Lagrangian Dynamics and Hamiltons Principle of Least Action is doing a number on my head. Every answer leads to many more questions, like what IS the significance of the integral of the Lagrangian wrt time? Why should any deviation from this be equal to 0 and why does it's curve follow a minimum or maximum? Following the routes of inquiry is only leading to more and more questions about this seemingly huge principle underlying pretty much all motion, can anyone recommend a good learning resource/textbook or online series where the whole idea is explained from first principles? It's a tall order, but I'm not having much luck finding anything useful. Any suggestions?

[starslayer]

I think in this case you're looking at a first principle; the principle of least action is what underlies all of classical mechanics. There is no particular reason why the universe has to act this way, but it does - at a certain point, you just have to accept certain principles as true. You've hit one of those. If you want to learn more about Lagrangian/Hamiltonian dynamics in particular, you can nab basically any undergrad or graduate level classical mechanics textbook (I used Thornton & Marion when I took that class, but there's probably better out there).

Why should any deviation from this be equal to 0 and why does it's curve follow a minimum or maximum?

If you take the first as true, the second part of this sentence necessarily follows.

[eSOANEM]

I think in this case you're looking at a first principle; the principle of least action is what underlies all of classical mechanics. There is no particular reason why the universe has to act this way, but it does - at a certain point, you just have to accept certain principles as true.

I agree with what you're saying however that doesn't mean that I (or thicknavyrain) is comfortable with it. If I could easily see how it produced more intuitive "laws" such as objects falling into low energy states however this is certainly not obvious from the Lagrangian, in fact, the opposite appears to be true seeing as it is a -V rather than a +V.

[doogly]

The intro here (12 pages, nothing too nasty to chew over) is fantastic. Everything due to Baez is pretty fantastic though.
http://math.ucr.edu/home/baez/classical ... ssical.pdf

[SU3SU2U1]

Its just a different way of thinking about mechanics. Something must be special about the paths particles actually take- so we should be able to formulate some quantity which is extremal on those paths.

Also, its a useful way to organize thoughts about physics. Imagine if Einstein had started special relativity by considering a Minkowski space, and asking how to construct a Lagrangian/action that is a Lorentz invariant. The simplest such invariant is the length of paths in minkowski space.

\mathcal{S} \propto \int ds

Now, we notice two things. First, for this to be minimal, we'll need a negative sign (there is a maximum path length between two Minkowski). ds = \sqrt{dt^2 - dx^2}. so, we can refactor this as dt\sqrt{1-v^2}. This gives us an interesting insight into the \gamma factor. So (in order to recover the Newtonian limit) our Lagrangian must be

\mathcal{S} = -\int m\sqrt{1-v^2}dt

From here, you can use Noether's theorem to come up with the form of relativistic momentum, relativistic energy, etc. I find this more illuminating then some other treatments. It also leads naturally to GR, we simple replace the line element with the equivalent in curved space.

[soggybomb]

I recommend Classical Mechanics by John Taylor. He has a really good treatment of the principle of least action, and rigorously demonstrates why the principle of least action must be a valid statement. Basically what it boils down to is demonstrating that small displacements from the minimized path still lead to a an action that is equal to the minimized path.