I finally find least action principle satisfying

The action principle has always been deeply unsatisfying to me. Why cook up this quantity? Why should we minimise it? I finally found a satisfying answer, which is made more satisfying by Noether's theorem.

This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject.

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Sources:

This video is basically an amalgamation of the paper:
However, I do have to warn that the paper uses pretty unconventional notations, which were confusing to me at first, so I have changed the notations considerably.

Rabbit holes to dive into:

(Are there similar works to this? Please let me know.)
(Seems like this is quite a general argument for a 2D system - I don’t have access to the book referenced, so maybe there are more details for higher-dimensional systems)

Video chapters:

00:00 Intro
00:41 Alternative definition of action
04:36 Why action is integral of T - V
07:07 Gauge dependence of action
12:00 Why minimise S?
16:23 Noether's theorem

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[mathemaniac]

I really like the interpretation and it is the first time I found a novel explanation also pedagogical (though novelty is still THE reason I make this video). It naturally leads to the concept of gauge, one of the most important concepts in physics; and it also naturally leads to Noether’s theorem with very little effort. Even if you don't find it satisfying, I think the fact that action is gauge-dependent explains why it has to be unsatisfying. By the way, I found the explanation of differences in action ridiculously similar to the Aharonov-Bohm effect, but I’m not too well-versed in this experiment.

[gcarcassi]

Very nice presentation! Hopefully this will bring more attention to this result (our video on this same explanation has less than 2500 views in two years)

[memealert]

The first author of that paper, gabriele carcassi has a really really good youtube channel where he discusses the interpretation of quantum mechanics!

[georgenabraham]

Finally, I found someone on YouTube who shares my thoughts about the least action principle and answered this question. Huge thanks.

[nicholasbohlsen]

It's very interesting to me to see this description. The observation that you can get an action principle for the flow of a vector field which is divergence free is really common in my research field (plasma) where we use it to write an action principle for magnetic field lines. I'd never thought about doing the same thing for the actual particle trajectories.

[aieousavren]

This is one of the most insightful videos I have ever watched on the action principle in my entire life. I cannot put it into words to thank you enough.

There are literally dozens of details I could comment on in this amazing video—seeing the Hamilton-Jacobi equations jump out of a curl like that was utterly wild, seeing how the principle of stationary action relates to a flux integral through a surface in phase space, it's almost unbelievable.

And at the end seeing the Noether charge actually labeled on a real, noticeable thing, and seeing how "Oh wow this very specific quantity really is conserved as the path evolves over time, " and seeing the formula for it like that, I am almost speechless. So much to think about.

Thank

[sensorer]

I thought I saw this somewhere!
Gabrielle Carcassi(the first author of the paper this video is based on) actually has a great video covering this!

[kapoioBCS]

15:54 the action of a system is the integral of the tautological 1-form.

Tautological 1-form is defined on the contingent bundle (phase space).

[ariaden]

The nice thing is that Noether charge can be defined also for infinitesimal transformations that are not symmetries. For example, in expanding universe the time translation is not a symmetry, but you can still define energy as its Noether charge (or charge density for fields) and see how does it change, as it is no longer conserved. For nonphysical symetries (e.g. coordinate changes or gauge transformations) the corresponding Noether charge is zero for any trajectory (even those that voilate equations of motion).

[muonneutrino_]

Genius video!! Currently following a lecture series on symplectic geometry and hamiltonian/lagrangian mechanics, and this video came just in time. Thank you mathemaniac :)

[Achrononmaster]

I didn't notice any comment noting the following: why the action principle is more powerful and one could say even _more appropriate_ than the differential form for the EoM. The reason being our world is quantum mechanical. A Hamiltonian time evolution story cannot accommodate non-deterministic effects (Schrödinger–Dirac Theory is deterministic) which is why there is the _ad hoc_ "collapse" postulate in orthodox QM. There is no need for such a postulate in the Action formulation, one just computes transition amplitudes, not fictitious "wavefunctions, " and in the Feynman path integral all nondeterministic effects are accounted for, and the Action has to be gauged (one does so using the gauge-covariant derivative), the gauge is no longer entirely "unphysical".

[jakobr_]

Ooh that’s so interesting! The action casts a net in phase space to how much it’s crossed by the natural paths of the system at various starting points. Minimizing action means aligning that net parallel to the flow of the natural path!

[GustavoOliveira-gpnr]

That was brilliant! Finally the action principle is making sense to me!

[DistortedV]

Wow I never thought I'd learn so much physics from youtube having only studied psychology.

[avadakedavra]

Thanks for the most beautiful video i have seen this month!

[victorhakim]

This is such a refreshing insight. So beautiful and yes, satisfying.

[yangyang]

This is the most impressive video I saw about least action principle so far. so amazing and satisfying!

[TheJara]

I was just checking your channel for new video, Today I see it!! super thanks and then listening to your approach to math is part of our life now!!

[danielkerr]

I've never seen the flux visualization of the enclosed path before, that's great.

Personally, what eventually satisfied me was studying Lagrangian mechanics in relativity. For example, in special relativity, the action is the just the integral of the spacetime interval multiplied by the rest mass. Extremizing the action means extremizing the proper time, which I have no problem accepting as defining physical trajectories. A variance in proper time from the extremum would imply an external energy source/sink pushing or pulling that trajectory. Additionally, the Lagrangian isn't T - V either; that only emerges in the non-relativistic limit when the spacetime interval is parameterized in terms of the local time coordinate. In the relativistic setting, the symmetries and gauge freedom arises from transformations that leave the spacetime interval invariant, which I find to be more intuitive than the non-relativistic representation.

[leonid]

But I think the issue of why the action is T-V and why minimize it should be clear from GR already: If you write down the the action of a particle in curved space-time it is just the line element g_uv dx^u/dt dx^v/dt (i.e. just the square of space-time distance) where t is proper time. in the weak field and weak velocity limit of a Schwarszchild spacetime you will see T-V, the potential is encoded in g_00 and the relative minus sign comes from the Lorentz signature. And the physical statement is that we are maximizing proper time which is equivalent to finding a geodesic in space-time. That's a very intuitive physical statement. This video is great and has some out of the box thinking for sure, but at the end of the day, i'm still not sure what the physical statement is? Like in the GR perspective you could just summarize it as "particles follow geodesics in curved space-time". That alone explains T-V and why S (which is just the space-time distance) needs to be optimized.