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June 30[edit]

What does the principle of least action actually imply?[edit]

The principle of least action must be a fundamental law of physics, as it appears in a diverse range of physical theories from quantum mechanics to general relativity to string theory. But what does the principle of least action actually imply about nature?

You would think you would find the answer in the articles principle of least action or action (physics) but you can't. Why can't you just fudge the action to create any equations of motion? Why then do you need a principle of least action at all? Does the principle of least action actually place any restrictions on the equations of motion? It must do, because (for example) Liouville's theorem is a statement about mechanical systems that obey the principle of least action (as opposed to arbitrary dynamical systems).

For that matter, what is an action, really? As far as I can tell, it is just a function of the space of candidate solutions which is stationary at the true solution. Now, it is true that, in classical mechanics, the action takes a certain form as an integral of a Lagrangian which depends only on certain properties (position and velocity) of points on the trajectory, and this may place restrictions on the possible actions. On the other hand, the Einstein-Hilbert action is not of this form (actually it probably is, but disguised). What makes the Einstein-Hilbert action an action?

I would like to know what the principle of least action actually says about nature. PeterPresent (talk) 14:24, 30 June 2018 (UTC)[reply]

In my opinion the PLA is an observation and a hypothesis and an axiom, in various fields, but it is not quite a law. It's one of those things where it is quite obvious the observable universe obeys it, yet I doubt anybody would be massively surprised to find an exception. Greglocock (talk) 20:11, 30 June 2018 (UTC)[reply]

At least in classical mechanics, the PLA (like in Lagrangian dynamics) is a theorem you can derive from Newton's equations of motion (F=ma etc). See calculus of variations for some discussion. In more generality, conservation laws arise from Noether's theorem. 173.228.123.166 (talk) 23:55, 30 June 2018 (UTC)[reply]

And you can derive the Einstein-Hilbert action from the Einstein field equations. But what does the Einstein-Hilbert action have in common with the classical action? What is an action, fundamentally? What is a Lagrangian in abstract mathematical terms? --PeterPresent (talk) 01:09, 1 July 2018 (UTC)[reply]

Was the article action (physics) of any help? This stuff is too advanced for me but I'm imagining an action as something like a path length on a surface, that is minimized on a geodesic resulting from some conservation law. This Mathoverflow thread might be interesting. It might be ok to ask on MO if there is a more formal notion. 173.228.123.166 (talk) 02:32, 1 July 2018 (UTC)[reply]

I'm no physicist and I don't understand any of the mathematics, so I don't know if this helps, but according to Andrew Thomas in Hidden in plain sight 2

The lagrangian is the difference between the kinetic energy of a system minus its potential energy.
We can calculate the lagrangian of a moving system at any point in time to get a series of values.
The sum of all these values is the action.
If we consider the value of a moving system over a period of time the value of the action will always be the lowest posible value as if nature moves objects so as to minimise the action - this gives us the principle of least action, so, for intance, if you throw a ball it will follow a smooth curve as that results in the smallest action.
We don't yet know why this value is always as small as possible but theorists are convinced that action must be incredibly important.

All this is in the introduction - which is a far as I've got up to now - but he goes on to say he will discuss later in the book more about the balance (and imbalance) of energies which is the key factor in determining the motion of objects. The Hidden in plain sight set of books are incredibly cheap as as ebooks from Amazon but they are very good. Richerman (talk) 00:01, 2 July 2018 (UTC)[reply]

I wonder if there is a reason they make physics students study the equipartition theorem of thermal and statistical physics right before they introduce the Lagrangian formulation of classical dynamic systems. Maybe, maybe, the formal mathematical study of different applications in physics provides insight into generalities of the universe that can only be appreciated after working hundreds of equations over the span of many months of intense study; and that these difficult insights cannot be more rapidly or succinctly conveyed - not even in the best writings by our greatest proponents of popular science. Nimur (talk) 12:38, 2 July 2018 (UTC)[reply]

I merely offered this in case it was of any help to the original poster. There's no need to be so dismissive. Richerman (talk) 17:23, 2 July 2018 (UTC)[reply]

Richerman, I think your post was unhelpful because it was pretty obvious that the OP was already familiar with the classical Lagrangian and its general relativity equivalent (and probably the quantum version using the Feynman path integral). So I think they were asking for a more abstract description of an action, the way Noether's theorem abstractly describes conservations laws. I took a stab at it and didn't reach an actual answer, but I think that was in the direction of what OP wanted. Nimur, at my school the more serious version of the intro physics course did mechanics, e&m, and thermodynamics in that order, and the Lagrangian was introduced in the mechanics part, fwiw. 173.228.123.166 (talk) 22:25, 3 July 2018 (UTC)[reply]