[moring]

I find it incredibly frustrating that again and again, the Lagrangian gets introduced and then said it should be minimized without ever explaining the motivation behind doing so. What is the Lagrangian and why should it be minimized?

I totally get how L gets defined mathematically, how it is derived from Newton's laws (this part is typically well explained by textbooks), and why in the case of point particles, a curve that violates Newton's laws does not minimize L. But there is no understanding at all, just saying "okay, it checks out" on a math level.

It doesn't help at all that on a math level, L isn't actually minimized but its derivative set to 0, which isn't even equivalent to "minimized or maximized". Why doesn't a single textbook explain why maximizing L is also okay when they first stated "minimized"? Or why derivative=0 is sufficient? As a reader, I always get the impression that "well, of course they cannot explain that, because they don't even know why L should be minimized in the first place". It's just all formulas that are easy to verify but don't convey a single bit of understanding.

Just for comparison, I found quantum mechanics based on the Schrödinger equation and the Hamiltonian rather easy to grasp, because every piece of it has an easy-to-understand meaning, that also gets explained really well. Why is this seemingly impossible for the Lagrangian?

[pa7x1]

In science why questions are not answered within the framework that raises them, you need a larger encompassing theory that explains the phenomena within a larger scope to answer them. In this larger encompassing theory you can sometimes answer some previous why questions but possibly new why questions are uncovered.

In classical mechanics there is no explanation for the stationary action principle, it just is the procedure you follow to derive the equations of motion (it answers a how, not a why). You need Quantum Mechanics, in particular the path integral formulation of Quantum Mechanics, to answer why classical solutions correspond with action extrema.

Long-story short, in quantum mechanics all possible paths contribute equally to the probability of a particle going from A to B but they interfere with each other. It can be shown that most paths interfere destructively between themselves except when the action gets to a extreme (minima, maxima or saddle point) where constructive interference occurs.

Further reading:

https://en.wikipedia.org/wiki/Path_integral_formulation#Stat...

[Maro]

I don't know if this will help you, but I'll try to give you a possible way out. It's in line with Feynman's famous "shut up and calculate" advice on QM.

Field theory and QFT in themselves are mathematical frameworks, not physics; we can think of it as a piece of applied mathematics. It becomes physics when we plug in a specific Lagrangian, and then apply the framework, eg. we draw Feynman diagrams, regularize, renormalize, all that jazz, and we get eg. a cross-section out of it. So, when you ask "why should L be minimized?", the answer is, because this whole construction works. If you follow the complicated (and convoluted) playbook of QFT, you will be able to calculate physical quantities (like cross-sections, or the fine structure constant), and then when you do an experiment, you find that the numbers match.

This doesn't work for all Lagrangians. Most L(x, p, t) or L(phi, phi', ..) functions we come up with don't correspond to physics, and the numbers the framework emits do not line up with experiments.

You may be dissatisfied by this, this is a black-box picture. Over time physicists have developed a lot of intuition for parts of the theory, and come up with heuristic explanations what the parts mean. This is also how the whole thing was constructed, by analogy from Lagrangians in classical mechanics, where things can be reduced to Newton's equation of motion, which we know works. But in the end, the reason eg. L should be minimized, is because that's how nature is and that's what works, with specific Ls.

[yccs27]

Starting from quantum mechanics, the Lagrangian describes the phase change per unit of time. For most evolutions of a system, similar evolutions have very different phases and cancel out. But when the Lagrangian is stationary (derivative 0), they interfere constructively. The stationary point is often a minimum, but it could also be a maximum or saddle point.

[prof-dr-ir]

Meh :) in classical mechanics one cares little about maximization versus minimization; it's really just (stationarity of L) <---> (equations of motion) that matters. Practically speaking it's just a cute gimmick to quickly find the equations of motion, especially for constrained systems.

In quantum mechanics everything is different, but for that you should study path integrals. To re-establish the connection with classical mechanics you have to learn about the saddle point approximation; it might also help to read Feynman's book called QED.

[gpsx]

I'm not exactly sure from what perspective you ask this question. As far as I know people misspeak when they say to minimize the lagrangian rather than find the stationary points (min max or maybe saddle). Applied to classical physics in the absence of quantum mechanics I don't think there is a good answer. It is just a rule. But, when combined with qunatum mechanics, it explains how classical physics is an approximation to quantum mechanics.

In the path integral formulation of quantum machanics, the evolution of a wave function from state 1 to a later state 2 can be thought of as occuring as a superposition of all possible paths between the two states, with each path contributing a factor of exp(iS) where S is the lagrangian. This means all paths either obeying classical physics or not.

In situations where classical physics is valid these contributions from different paths changes very quickly. The contributions from neighboring paths cancel each other out except at stationary points in the lagrangian, where there is a zero change between neighboring paths. Hence, we see classical physics are the trajectories that are the stationary points of the lagrangian.

[nh23423fefe]

I found this playlist informative when I was having the same thoughts

https://www.youtube.com/playlist?list=PL2ym2L69yzkamORF9DGWR...

It isn't L that is differentiated and set to zero. Instead the variation of the action is zero. So L isn't minimized or maximized, the variation of the action is zero. This implies that the solution path y(x) when varied by dy is a stationary point of the action. So for this path all nearby paths have the same action.

ok but then

> What is the Lagrangian and why should it be minimized

the form of the lagrangian is derivable from the d'alembert principle, principle of minimum potential, and then hamilton's principle.

It seems to me the principle is that real system behave in such a way that is characterized by hamiton's principle (the variation of the action is zero for real paths), and then we operationalize that principle by the calculus of variations to get real paths which have the properties established by the principles (use the Euler-lagrange equations to find paths for a particular system)

[elashri]

It is not minimum or maximum when you take derivative L = 0. You get sudden points which could be minimum or maximum or not. In cases of classical Lagrangian with V is the gravitional Field. It would be minimum. If you want to derive optics laws (snell/reflection) you will find that the path it not a minimum path. Actually you will find that the light will take the path that minimize the time it takes from point a to b.

For more complicated theorie. You always tty to start from one point (usually symmetric or equilibrium) and try to build your theory's Lagrangian. Usually this involve some inputs from experiments (i.e. Standard Model Lagrangian). If we are Lucky enough then the true theory wouldn't be too far.

[zinclozenge]

Unfortunately, it's not an intuitive idea and it's deeply rooted in the history and development of classical mechanics.

One way of looking at the Lagrangian is via the Legendre transform from the Hamiltonian, which in my opinion is highly intuitive since it's total energy, H = T + V. I'm not going to go through the whole explanation of the Legendre transform. Instead, I want to point out that in the Hamiltonian formulation, you can think of the independent variable as momentum, mv. When you transform to the Lagrangian, you're making, among other things, a change of variable to the velocity, v. If you have a velocity independent potential, then it only really matters in the kinetic energy, and the Lagrangian picks up a sign change due to the transform.

So in the end, it's recasting an intuitive idea into a more mathematically tractable form. The action starts making more sense in the path integral formulation of quantum mechanics.

[lokedhs]

In Susskinds book The Theoretical Minimum, he explains this and suggests that the word stationarising would be more accurate, as the goal is to make the lagrangian stationary and not minimal.

I haven't read any other textbooks on the topic so I don't know how common this is. I do suspect you're right though, as he made a point of explaining this

[psychphysic]

I believe he also goes over the legendre transform which lets you move between Hamiltonian and Lagrangian mechanics.

Just occured to me that the above poster may believe that QM is Hamiltonian only and classical is Lagrangian only. That'd be a forgivable but fatal mistake to understanding the subject!

Hamilton Jacobi equation may also help them remember QM is distinctly less intrusive than classical mechanics.

[psychphysic]

Physics does not answer why questions it's about modeling and predicting.

QM and esp. Schrödinger equation has way too many shortcomings for someone to claim it's easy to understand.

You're question about maximise, minimise and deritive set to 0 is a bit bizarre. Is it genuine? If so it's that it's about stationary points.

[Koshkin]

> Physics does not answer why questions

Well it does, but often it takes a more fundamental theory than where the question was posed.

On the other hand, an answer may involve logic and/or a mathematical derivation which is not so different from calculation, so where intuition ends and “shut up an calculate” begins is not all that clear and may depend on the level of expertise.

[psychphysic]

Since we don't (yet) have a fundamental theory it stands.

Not only that but the shut up and calculate mentality is still the most dominant one within physics.

Should string theory turn out to be the fundamental theory then it's possible the 'Why' will be that's the geometry of our universe. Which is not more for filing than tweaked constants.

Unfortunately no physics won't tell us Why. That's a question for philosophy.

[Aardwolf]

I would love to get intuition about this, but everytime I try to read about it I get lost in the math behind Lagrangians etc... and never get the intuition.

Specifically, I'd love the intuition behind why it must be so that if the laws of physics are time/position invariant, it must be impossible to create or destroy energy/momentum.

This because I can perfectly imagine a universe where you can create/destroy energy or momentum at will, yet have this work with the same physics at any time or position (e.g. a battlemage in some RPG able to cast fireball spells with heat and momentum out of nothing, where it works the same no matter at what position or point in time the battlemage is). So what about Lagrangians is preventing this, in an intuitively imaginable way?

[pa7x1]

> Specifically, I'd love the intuition behind why it must be so that if the laws of physics are time/position invariant, it must be impossible to create or destroy energy/momentum.

The geometrical intuition is that the momentum operators are the generator of spatial translations (i.e. acting with the momentum operator P_x on a system is the same as applying an infinitesimal displacement in the direction x). https://en.wikipedia.org/wiki/Translation_operator_(quantum_...

And the Hamiltonian is the generator of time evolution (i.e. acting with the Hamiltonian on a system shifts it in time an infinitesimal amount). This is quite literally what the Schrödinger equation says, btw. H |Psi> ~ d/dt |Psi>

If the physics of a system (which are given by its Hamiltonian) are invariant under translations then it must be the case that a shift in time (Hamiltonian generates shifts in time) of the momentum (generates shifts in space) of a system is 0.

As the Hamiltonian gives us the energy of the system, if the system is invariant under time translations then its energy is conserved. Using the previous argument.

Rinse and repeat for any symmetry of the system. For instance, angular momentum operators are the generators of rotations. If your system is invariant under rotations, then it conserves angular momentum. Invariance under relative movement (relativistic invariance) gives conservation of center of mass. Etc... https://en.wikipedia.org/wiki/Symmetry_(physics)#Conservatio...

[Cleonis]

Specifically about the Lagrangian of Classical Mechanics (Hamilton's Action) I have discussed that on physics.stackexchange https://physics.stackexchange.com/a/670705/ The ideas are expressed in diagrams. What is expressed in the diagrams is repeated/corroborated in mathematical expressions (stackexchange supports Mathjax). The visualizations leave no room to get lost.

[317070]

I think there is the underlying notion that among any set of possible paths, it is always possible to find for each path a quantity for which a particular path is optimal.

That seems a bit trivial to state, but reasoning on the space of those quantities is more flexible. It doesn't mind the non-linearity between parameters and resulting paths as much.

So when you are trying to find the laws of physics, reasoning on that higher level space of quantities-that-will-get-minimized seems to simplify things.

[Koshkin]

Read The Lazy Universe by Coopersmith.

(Also, the Fermat’s Principle of least time is perhaps more intuitive and has a firm grounding in the wave description of how light propagates.)

[dhoe]

Imagine you find _the_ fundamental law of the universe, say "the gralb must be flombed". It's the fundamental law, so there's no other law explaining why it exists. All other laws follow from it and therefore feel understandable.

(This is mostly theoretical, as the explanation with phases cancelling in the sibling comment is satisfactory to me at least. But nevertheless there will be some law without a "more fundamental reason").

[Cleonis]

It is in fact possible to explain Hamilton's action within the context of classical mechanics.

On physics.stackexchange I have discussed that, in an answer posted in oktober 2021. That discussion is illustrated with animated GIF's. The animated GIF's are composed of successive screenshots of interactive diagrams that are on my own website.

https://physics.stackexchange.com/a/670705/

Stackexchange has mathjax support, and support for uploading images, that is why I refer to my post on physics.stackexchange

The following is to give you an idea of what I discuss.

We have that if F=ma is granted as axiom then the Work-Energy theorem follows as theorem.

(As we know: the derivation of the Work-Energy theorem is subject to the following condition: it is only applicable if it is possible to define an unambiguous expression for potential energy. In order to have a well-defined expression for potential energy the force that is involved needs to be a conservative force.)

The Work-Energy theorem implies the following: In the process of interconversion of potential energy and kinetic energy: the rate of change of kinetic energy always matches the rate of change of potential energy. (If the potential energy is decreasing then the kinetic energy is increasing at the same rate)

In terms of exploring a variation space of trial trajectories:

The true trajectory has the following properties: A property of derivative with respect to time: - At every point along the trajectory the derivative of the kinetic energy with respect to time matches the derivative of the potential energy with respect to time.

A property of derivative with respect to _position_: - At every point along the trajectory the derivative of the kinetic energy with respect to _position_ matches the derivative of the potential energy with respect to _position_.

I want to highlight this: In mathematical models that describe changes taking place we are accustomed to taking the derivative with respect to _time_.

But: When we are representing the physics taking place in terms of _Energy_ it is powerful to take the derivative with respect to _position_.

In classical mechanics: When you insert the Lagrangian in the Euler-Lagrange equation then the operation that the Euler-Lagrange equation performs is that it takes the derivative of the Lagrangian with respect to position.

You are looking for the point where the _derivative_ of the Lagrangian with respect to _position_ is zero.

when that derivative is zero the derivative-of-the-kinetic-energy-with-respect-to-position _matches_ the derivitive-of-the-potential-energy-with-respect-to-position.

For the concept of stationary action minimum or maximum is immaterial. Stationary action is about identifying the point in variation space such that at every point along the trajectory the derivative-of-the-kinetic-energy-with-respect-to-position matches the derivitive-of-the-potential-energy-with-respect-to-position

Finally: There is a concept that I will refer to as Jacob's lemma. (This concept was introduced by Jacob Bernoulli in the course of presenting his solution to the Brachistochrone problem. The Brachistochrone problem had been presented by Johann Bernoulli, as a challenge.)

Jacob's Lemma was stated decades before Euler started development of calculus of variations. Jacob's Lemma is crucial to understanding calculus of variations.

Take the Brachistochrone curve. Divide it in subsections. Then each subsection is in and of itself an instance of the Brachistochrone problem. This process of subdivision can be repeated indefinitely. In the end you have a concatenation of infinitissimally short subsections, and we know that each of those subsections is an instance of the Brachistochrone problem.

This tells us that a differential equation must exist that solves the Brachistochrone problem. (It does not narrow down what that differential equation is, but at least you have logical proof that it _must_ exist.)

In fact, Jacob Bernoulli succeeded in solving the Brachistochrone problem using a differential calculus approach.

The Euler-Lagrange equation takes the variational formulation, and converts it to differential equation form.

[blueprint]

The Lagrangian is a simple statement of the energy of a system. Have you heard of the relationship between energy and time? Have you heard of the principle of least time for photons? Have you considered that a "photon" is not a wave or particle of energy before it is detected, but rather the probability that one will be detected?