For those of us who didn't major in physics... where did the whole "action" thing (let alone the thesis that it's minimized) itself even come from? The whole notion of "action" feels entirely foreign and unintuitive for someone who's just studied Newtonian mechanics. At least I've never managed to find a real world feel for what it is, unlike with force or energy.
I think of the Lagrangian, what we integate to get the "action", as some sort of energy related function. I don't really attribute much meaning to it other than the fact that minimizing it implies the equations of motion, which are something we can phyiscally grasp.
For a particle in one dimension,
L = L(x(t),v(t))
The solution to the minima is where the "gradient" of L with respect to x and v is zero. However, position x
and velocity v are not independent, so that "gradient = 0" equation implies:
dL/dx = d/dt(dL/dv)
- You can define dL/dv is the generalized momentum.
- You can think of dL/dx as a force.
This gives you newtons equation, but you can say you derived it.
F = d/dt(p)
Granted, we didn't really start from a more fundamental place. But then this starts to make more sense when you realize the world is governed by quantum mechanics. And this least action principal results from the fact that, in the classical physics regime, the only part of the "trajectory" (wave function) that gives a meaningful contribution is the part along with minima of the lagrangian.
Honestly it's a great question. Even the typical physics major isn't going to be able to give you a great answer because learning how this was derived isn't a part of any curriculum that I know of.
But if you can accept the principle of least time (that a light ray will travel along the path that takes the shortest time) then you kind of already accept that (ordinary, classical) light somehow knows the time it takes to go along all possible paths, then chooses the minimum.
The action is a kind of thing, discovered by Hamilton, by analogy, that plays the same role in mechanics as time does in optics. I image he just stared at the equations of optics for a while and had an "ah-ha" moment. He had been working on this stuff for decades, on top of being a pretty smart guy already.
It's extremely unintuitive that classical systems should minimize anything like a time or an action. Think about it: they travel along the minimum path, but how do they know that that path is the minimum? Do they sample the other paths to know?
Well interestingly, Feynman's thesis was about exactly this idea. What happens if you start from the assumption that particles just sample all possible paths (weighted by something having to do with the action/time)? It turns out you can get the Schrodinger equation (and optics equations) from that too. It partially explains how paths "find the minimum." It turns out they don't, but a nice cancellation happens that makes the minimizing path the most probable one.
> It's extremely unintuitive that classical systems should minimize anything like a time or an action.
If I get to define the measure arbitrarily, then I can always find a measure that something else is always a minimum of. So in that sense it's not surprising at all. The interesting question to me is why should that seemingly arbitrary measure be action? What does that even mean, physically? I have no intuition for it.
> Think about it: they travel along the minimum path,
I'm already stuck here. What would it even mean for a particle to have an "action" (whatever that is) that is not minimized? Like what would that look like, physically? I understand what it means for distance not to be minimized, but action isn't distance...
>What would it even mean for a particle to have an "action" (whatever that is) that is not minimized?
The particle doesn't have an action. The trajectory of a particle is what the action is defined in terms of. One way to think of it would be "it's a measure of how much the trajectory deviates from the one dictated by Newton's equations." Pretty much like what you said: "I can always find a measure that something else is always a minimum of."
About what a trajectory with non-minimal action would look like: it would be an arbitrary violation of the equations of motion for the system (ex: free particle moving in a zigzag instead of a straight line at constant velocity). Moving in a straight line at constant velocity is what Newtonian mechanics prescribes, and that trajectory will minimize action for the corresponding Hamiltonian.
> then you kind of already accept that (ordinary, classical) light somehow knows the time it takes to go along all possible paths, then chooses the minimum
But it doesn't, minimum (or more precisely extremum) is a local quality. Basically light goes by the path with zero derivative because otherwise neighboring pathes interfere. Feynman lectures touch on it relatively early [1] which I think is nice
> It's extremely unintuitive that classical systems should minimize anything like a time or an action.
Perhaps, they minimize the action as the primary driver (cause), and time (effect) is generated as part of the solution, as a definition of evolution ...
This margin is too small to contain my full explanation :)
Hamilton did not discover any "action". Like everybody since Maupertuis, a century earlier, Hamilton used "action" with the meaning "accumulated living force".
"Living force" is English for the Latin "vis viva", which is the old name of kinetic energy (the term "kinetic energy" was introduced only later, in 1854, by William Thomson, who in 1892 became Lord Kelvin; for a short time before 1854 "actual energy" was used instead of "kinetic energy" and opposed to "potential energy", as defined by Rankine in 1853). So "action", in the sense introduced by Maupertuis and used by everybody until the 20th century, meant integral of the kinetic energy (actually the living force was mv^2, so the double of the kinetic energy).
Hamilton has introduced a new physical quantity, never used by anyone before him, which he named just "function S". Unlike the principle of the minimum action, which is false except for certain restricted cases, Hamilton's variational principle about the "function S" is always true, including in relativistic mechanics and quantum mechanics.
Nowadays Hamilton's "function S" is usually called "Hamilton's action", because it has the same measurement unit like the traditional action, even if it is a different physical quantity. "Hamilton's" is frequently dropped, which does not cause much ambiguity, because now the traditional "action" is seldom mentioned.
Nevertheless, whenever history is discussed, a very clear distinction between "action" and Hamilton's "function S" must be maintained, otherwise it is impossible to understand the evolution of physics.
Hamilton has discovered his function S by starting from the system of equations of Lagrange and finding a way to deduce them from a simpler principle.
It is a little weird that even if Lagrange had discovered when young, together with Euler, the Euler-Lagrange equations for variational problems, many years later, when he has written his works about mechanics, he has never attempted to use any variational techniques in the formulation of his equations of dynamics (and he dismissed the principle of the minimum action as seldom applicable), so this relationship has been discovered only later, by Hamilton. While Lagrange has been the first who has used correct definitions for the kinetic energy and the potential energy, he has named them just "fonction T" and "fonction V", similarly to Hamilton's use of just "function S".
> For those of us who didn't major in physics... where did the whole "action" thing (let alone the thesis that it's minimized) itself even come from?
It comes from Maupertuis’ work in the 18th century (about a century before Hamilton). The initial insight is that a physical object follows the “shortest” possible “path”, in the same way as light follows the quickest path as in Descartes’ law. The difficulty is that the path is in a phase space with more than our usual 3 dimensions, so the whole thing is a bit abstract and calculations are a bit counter-intuitive at first. The approach is still useful because it helps solving problems that are very difficult to solve using Newton’s equations, like systems with constraints or couplings between objects.
> The whole notion of "action" feels entirely foreign and unintuitive for someone who's just studied Newtonian mechanics.
Action is a tool to calculate these shortest paths, and because the actual trajectory corresponds to an extremum of the action, and most of the time to a minimum, the principle is sometimes called the “least action principle”. Fundamentally, that’s almost all there is to it. The rest is defining the action, and processing it to get equations of motion. Action kind of looks like a weird energy in classical mechanics,
It is foreign from Newtonian mechanics. If you want to understand how it works you need to consider Lagrangian mechanics, which was a generalisation of Maupertuis’ principle and paved the way for Hamiltonian mechanics (which are another step in abstraction). Newtonian mechanics are built on calculus and the concept of derivative; Lagrangian mechanics are built on variational calculus and the concept of functionals.
> At least I've never managed to find a real world feel for what it is, unlike with force or energy.
Action is actually quite similar to energy, conceptually. Energy is whatever gets minimised in a Newtonian system at equilibrium. Energy changes are governed by differential equations that we can solve to calculate simple trajectories. Action is a function that is minimised along the trajectory of a physical system.
This approach is extremely powerful. The same principle can be used to derive the equation of motion for systems following classical or quantum mechanics, or general relativity by “simply” considering different definitions for the action (or, equivalently, different definitions of what we call the Lagrangian function, which is more common). It’s a bit difficult to explain more in this format; if you want to dig deeper you should start by looking into Lagrangian mechanics.
Fermat's principle (least time) predates Maupertuis' but it's not obvious it's basically the same thing. Interestingly Maupertuis was motivated by placing time and distance/space on the same footing, predating Einstein by several centuries.
It's a great question that, as far as I can reason, has no answer. Newtonian vibes that are familiar to us will only take you so far, and intuitive interpretations of physical quantities often break down when you try to relate them to the scale, experiences, and stimuli of humans.
Let's take momentum, energy, and charge, things that you probably have a strong "real world feel" for. It's worth noting that our intuition for these quantities is actually pretty far-removed from their mathematical origin. Maybe you consider these as different loosely related quantities that pop up in different loosely related calculations, which is a useful and powerful mental model. Momentum is a thing that..."gives velocity to inertial bodies". Energy is a thing...that "does work". Charge is a thing that..."causes forces in the presence of an electric field". If you try to define the terms within each definition, you'll find yourself in some circular definitions, and it'll become unclear which definition, if any, is "most fundamental".
But these quantities are actually quite similar in the sense that they can all be defined in terms of action! Specifically, these are quantities that are conserved because there exists some nice symmetries in the Lagrangian (roughly speaking, a derivative of action). So our intuitive definitions of these things are really just less generalized/more specific understanding of structure that is emergent from action.
Can we look at a physical system and say "oh this one's got a lot of action" or "nature's doing a great job of minimizing the action over here"? No, but we can look at a physical system and say "wow, everything that's happening in here lines up with what I'd observe if this little quantity I defined just so happened to be minimized"
I think no matter how many Lagrangians we integrate or variational calculations we perform, we'll probably never gain a better intuition for action beyond "The Thing That Explains A Lot Of Seemingly Unrelated Physics When It's Minimized." To me, it's both deeply unsatisfying for its abstract and unintuitive nature, but also deeply profound for its universal explanatory power.
tldr; when it comes to action, reject real world feels and embrace mathematical structure.
I have created a demonstration of Hamilton's stationary action with interactive diagrams, (supported with discussion of the mathematics that is involved).
Interestingly: it is possible to go in all forward steps from Newtonian mechanics to Hamilton's stationary action. That is the approach of this demonstration. (How Hamilton's stationary action came into the physics community is quite a convoluted story. With benefit of hindsight: a transparent exposition is possible.)
The path from F=ma to Hamilton's stationary action goes in two stages:
1) Derivation of the work-energy theorem from F=ma
2) Demonstration that in cases where the work-energy theorem holds good Hamilton's stationary action will hold good also
Also interesting:
Within the scope of Hamilton's stationary action there are also classes of cases such that the true trajectory corresponds to a maximum of Hamilton's action.
In the demonstration it is shown for which classes of cases the stationary point corresponds to a minimum of Hamilton's action, and for which classes to a maximum.
The point is: it is not about minimization.
The actual criterion is that which both have in common:
As you sweep out variation: in the variation space the true trajectory is the one with the property that the derivative of Hamilton's action is zero. The interactive diagrams illustrate why that property holds good (it follows from the work-energy theorem).
Hamilton's stationary action is a mathematical property. When the derivative of the kinetic energy matches the derivative of the potential energy: then the derivative of Hamilton's action is zero.
(Ycombinator does not give control over the layout of the text I submit. I insert end-of-line, to structure the text, but they are eaten.)
