# qualitative explanation of Principle of Least Action (vertical movement)

Consider the following situation I want to understand what the PLA means here from an intuitive and qualitative point of view.

I understand the mathematical approach. Combining $L(y,\dot{y})=\frac{1}{2}m\dot{y}^2 - mgy$ with the Euler-Lagrange equation $$\frac{\partial L}{\partial y}- \frac{d}{dt} \left(\frac{\partial L}{\partial \dot{y}}\right)=0$$ leads to $$-mg - \frac{d}{dt} m \dot{y} = -mg - m \ddot{y} = 0$$ or $-g=\ddot{y}$. Integrating then gives $$y(t)=y_0 + v_0 t - \frac{gt^2}{2}$$ and we have recovered the traditional equation of motion

On the other hand, considered non-mathematically (i.e. physically) the PLA seems to imply that Nature is thrifty in all its actions. Here, thrifty is taken in the sense that Nature avoids waste, avoids doing anything unnecessary or needless.

So what would be a superflouous operation of Nature in the above example? Let's look at the hypothetical wobbly curve.

• kinetic energy -> potential energy
• kinetic energy -> potential energy (slower)
• kinetic energy -> potential energy (faster)
• potential energy -> kinetic energy (object goes down)
• kinetic energy -> potential energy (object goes up)
• potential energy -> kinetic energy (object goes down)

Intuitively this doesn't look the most efficient way to reach its goal. What is the goal anyway? The goal (reaching height $H$ and passing through the lower heights $h\leq H$) could also been accomplished in more efficient way, i.e. by the parabola?

Nonetheless, in the wobbly curve energy is transformed and not lost. So this makes me question what does it mean to be efficient? Does the concept of "efficiency" here function on a meta-level, i.e. the way energy used/transformed must be efficient?

I can understand the mathematical context in variational terms (i.e. considering $P+\delta P$), but I don't understand the physical meaning. In particular, I want to connect the whole thing with my metaphorical interpretation of the PLA that Nature is thrifty, economical, doesn't do anything unnecessary.

• It's an interesting question and I'd love to hear any thoughts on it from a classical point of view. I fear it may just be that the PLA is a starting axiom that we can't really explain. Ultimately of course the PLA is not strictly true at the quantum level. There the classical path contributes the most because each path in the path integral contributes with the phase $\exp{- i S}$ and all the other paths tend to cancel out. Then your question becomes something like "Why is the path integral what it is?" Then you can talk about exponentiating operators, and Lie algebra and Lie groups, I suppose – gn0m0n Sep 8 '15 at 16:36
• If you like this question you may enjoy reading this Feynman lecture. – Qmechanic Sep 8 '15 at 19:50

On the other hand, considered non-mathematically (i.e. physically) the PLA seems to imply that Nature is thrifty in all its actions. Here, thrifty is taken in the sense that Nature avoids waste, avoids doing anything unnecessary or needless.

Hey, now, let's not get ahead of ourselves. The principle is actually one of stationary action, or action which is unchanged by small path-perturbations which leave the beginning and end points fixed. Valid trajectories can also be realized by maximal-action paths or saddle-point paths, in principle.

So, for example, if we have a light bulb over here and an omnidirectional detector over there, we will see that the biggest contribution is from light going straight from the bulb to the detector. However, there can be other contributions too, say, if there is a mirror nearby. Light doesn't just take the path of least-time, full-stop: there is now also a path which takes a little longer (to hit the mirror and reflect back to the detector) which also contributes. If the light is particularly coherent you might even be able to arrange for the light going those two different paths to destructively interfere at the detector with some other half-silvered mirrors etc.

Nature chooses the paths where little vibrations around the path do not affect the action-integral. You can view this as the action integral being sensitive to little quantum perturbations of the trajectories and classical-Nature preferring only those trajectories where the action is insensitive. In QED this happens because the particle has an associated wave-phase, called collectively the "amplitude" of detecting the particle, which requires constructive interference of a lot of "nearby" paths in order to build up to any substantial sum. The action turns out to just be the phase of the wave, so we need an "unperturbed action" to get "the same phase" to get "all the nearby paths constructively interfere to generate a big probability."

Simply put, you can imagine that at the microscopic level there is a lot of "jiggle" and Nature abhors such "jiggle" at the macroscopic level of action-principles, and then you have some "thriftiness" of Nature with respect to action-jiggle. But you cannot just say that Nature abhors spending any more action than it has to, because just as with the least-time principle not being 100% right when there are two locally-least paths, the least-action principle will also be violated when there are two locally-least paths. Nature is obviously willing to spend a little more action on the problem if it has to, as long as that action is not jiggly with the microscopic jiggles everywhere.

• Thanks, I understand how mathematical variations can be translated into the physical language of "jiggle." But what is the physical interpepation of the (mathematical) action-integral. When you say "Nature chooses the paths where little vibrations around the path do not affect the action-integral," I read this as meaning that "physical process" affect a non-physical (mathematical) quantity. But this does not make sense. Unless the Action-integral itself has some physical meaning. – kristof2014 Sep 8 '15 at 17:50
• @kristof2014 I think you're going to have to elevate it to having fundamental physical meaning, in other words it is the physics as far as Nature is concerned, no matter what form (say, $\int_{\mathcal P} dt [K - U]$) it takes. So rather than "Nature just has these forces, momentums, etc." you say "Nature just has this big action principle which has parts we call 'forces' and others that we call 'momentums' etc." Otherwise you are stuck with "Why does Nature care about that?" (intractable) rather than "Why do we call this thing that Nature cares about, 'momentum'?" (tractable). – CR Drost Sep 8 '15 at 18:49
• So for example for a double pendulum Nature just assigns a parameter to every trajectory $\theta_1(t),\theta_2(t)$ by $S[\theta_1,\theta_2]=\frac 12\int dt[(m_1+m_2)\ell_1^2\dot\theta_1^2+m_2\ell_2^2\dot\theta_2^2+4m_2\ell_1\ell_2{\dot\theta_1}\dot\theta_2\cos(\theta_1-\theta_2)].$ If we choose a different way to represent trajectories, Nature still assigns those numbers to those paths, but we humans like $\theta_{1,2},$ so that's what we use. Nature then tells us that the momentums $\partial L/\partial\dot\theta_i$ for these positions are actually quite complicated, and the system is chaotic. – CR Drost Sep 8 '15 at 19:00

None English speaking person here, sorry by advance. Principle of least action is also call the principle of stationary action. And should be called like that all the time...

The action is none intuitive to understand... And honestly i'm not sure to understand it myself, so take what i'm going to say carefully, i might be wrong. The action's unit is J.s, and between point A and B, we integrate the Lagrangian which have a unit of energy by respect to the time. So the answer of the integral is the action, in J.s. But what we try to do is not to calculate the action, but to make it stationary to derive the equation of movements. This is the meaning of the Euler-Lagrange equation. You can see it like you have a quantity of energy to spend for going from point A to point B, and there is no reason to think that a enormous quantity of energy gonna be spend without a valuable reason. The object that you study will follow the path that not will minimise the action, but just make it stationary, means that it gonna spend just the right amount depending of the exteriors forces.

It's how i understand it. You try to find the right quantity of action that will be spend for the movements. And this right quantity of action has no reason to be big or small, it just have to be what the object need to go from point A to point B. So mathematically, that's means that you have to take the derivative of the position of the object in the Lagrangian and make it equal to zero. It's like saying that doing that, you're sure that the object gonna try to go directly to his destination without any unwarranted detour. Depending on the forces, the straight line can be not the better choice.

You can also look for the history of the principle which start with Fermat and his principle of least time, Bernoulli and is test problem "brachistochrone", Maupertuis with the right definition and Lagrange with the proper mathematical demonstration. It can help!