Explaining the Principle of Least Action: Physics Mini Lesson

[@williamchurcher9645]

As a mathematician, your content is some of the best I have ever seen. I love getting my teeth into the equations and proofs, and I love seeing stuff like this as well as your introduction to string theory video. Keep it up!

[@PhysicswithElliot]

Thank you so much William!

[@jonetyson]

If you want to see actual theorems with all the hypotheses stated, read Arnold's book "Mathematical Methods of classical mechanics." I think there is some truth that it's best to learn physics from mathematicians, and vice versa. :P

[@ewwseww]

Totally agree with, as a mathematician!

[@RogerBarraud]

@@ewwseww Sure, but how about As A Millionaire? ;-)

[@douglasstrother6584]

"The Principle of Least Action" ~ The Feynman Lectures, Vol. II, Ch. 19 Richard Feynman inserts a "WOW! That's cool!" lecture in the middle of his electromagnetism lectures. It starts with a personal account of how he was introduced to this idea. I came across this before getting formally introduced to Lagrangian Mechanics.

[@hrperformance]

Thank goodness for all the genius minds that have contributed to this wonderful subject and all the amazing teachers like this guy. I am so excited to get to grips with this and more!!

[@alphgeek]

I got most of the "parts" of this in high school maths and physics late 1980s but not til now really see how they fit together. Thanks for the great explanation.

[@amritawasthi7030]

I've always wondered to know things related to physics but what pulled me out to learn was that i was more interested in mathematics without applications aka pure maths. But to be really honest you're a saviour for me in that case. I love the way you explain things. Thank you so much for being there and a very happy new year from my side. More health to you.

[@PhysicswithElliot]

Thanks Amrit! Glad it helped!

[@wowtbcmagepvp]

Wonderfully clean. I’ve seen this proof many times - and only this one connects all the dots effortlessly to why we are even doing this in the first place.

[@sourabhjogalekar3842]

path of least "action"- story of my life

[@probablyshadman]

Dr, please make a whole lecture series on Classical Mechanics (all of Lagrangian, variational calculus , Hamiltonians , phase space and all of it) This is the rarest thing in the whole online universe. A request from Bangladesh 🖤🇧🇩

[@PhysicswithElliot]

I'm working on a course covering Lagrangian mechanics!

[@jimwang3084]

Thanks for your video, they are very logical and simpler to understand, they have make me have another perspective to understand about the mechanics for my freshman year to learn deeper understanding in physics. Thank you!

[@PhysicswithElliot]

Glad to hear it Jim!

[@motif123456]

I like mathematical formalism of physics...this formalism is like a beautiful poetry indeed . Your channel is fascinating because you have explained advanced topics very lucidly.

[@PhysicswithElliot]

Thanks Rajarshi!

[@jinks908]

I know you've gotten a million comments on your videos all saying the same thing, but I don't care, here's one more. I just got my BS in mathematics but I want to go to grad school for physics and so I am now self-studying physics to try and accomplish this. You are seriously one of the best teachers I have ever come across. Given that there are literally thousands of physics/math lectures and videos online, and that I have seen a ton of them over the years, this truly puts you in a category with the best of them. It's such a privilege to be able to access this kind of content outside of a college classroom, and it's an absolute miracle that people like you make it available for free. You've earned yourself a Patreon supporter and a lifetime subscriber, sir! I truly thank you for work ✊

[@christopherjoneswa]

@jlinks908 I totally agree, there are sooo many teachers out there and (I appreciate all of their efforts!) so there exists an average effectiveness in concept delivery. Elliot's offerings really highlight the fact that most hover around the average. I didn't really notice too much until I saw a couple of his videos but he does such a good job of making these concepts accessible through the right balance of visual aids and strong delivery of clear information. @Elliot: really mean that, you've elevated the field for all! Good on you!

[@danko6yg411]

Elliot. 🎉. Wonderful! You explain things so clearly. You have a clear mind. You get straight to the point. You are concise. You enunciate with precision. And some of the material I have been exposed to, and did not understand very well, I understand better now. More please …. Thanks so much!

[@aleksybalazinski]

This channel is a gem

[@michaelshmilovich5551]

Dr. Elliot -- you're a mensch. Thank you for these very clear and visually intuitive videos. I'm not a physicist or a mathematician (my science background is mostly molecular biology). However, I'm trying to learn modern physics on the side and your vids coupled with Sean Carroll and Lenny Susskind's are VERY helpful.

[@dianedenonneville4259]

Principle of least action: they finally made a physics theory that reflects my life

[@LoreProkop]

Thank you so much! This and its companion videos were the clearest and most understandable explanation of the PoLA and world lines, plus special and general relativity that I have watched, and I watched a *lot*! The math was perfect and well chosen, easily followed, well done!

[@augustisalman8027]

Thank you is not enough. ❤ man to man. May he bless you in every good deed you do in every second of it along you life time.

[@neutralzone7775]

Thanks Elliot, for explaining the Principle of Least Action by using Quantum Mechanics, that last part of the video was so important for visualising all the math you did before. I’m actually mind blown. Now I have a better understanding how the epsilon proof before manifests in actual physics, because before all I could see is formulas and kinda had to accept it to be true. But you actually did it. HUGE thanks, subbed today!

[@nillchen]

Thank you, I learned this during my time at Duke but had forgotten about it :) Now it's back, very well explained!

[@jonetyson]

The statement that the action is minimized (or maximized) is refuted by the counter-example of a statue sitting in a temple for a thousand years: Temporarily moving the statue up to the roof or down to the basement for a sufficiently long time (say another thousand years) before returning it to the pedestal will change the action by an arbitrarily large amount in either direction (as the potential energy change gets integrated for an arbitrarily long time), dwarfing the finite change in action that occurs while the statue-moving company is on site. Furthermore, it is worth mention that in classical mechanics, what one really uses is the fact that the Euler-Lagrange equations are unchanged by changes of variables. This can be proved simply using the chain rule for derivatives, without recourse to the calculus of variations and any accompanying unnecessary assumptions. (Such a proof isn't a derivation, but it can remove any doubts in the more mathematically-inclined students, who may simply loose all interest at the first sign of unstated or missing hypotheses or lack of mathematical rigor.)

[@dukenukem9770]

Good stuff! I've never heard this account before. I can't wait for the relativistic generalizations!!!

[@PhysicswithElliot]

Come and get 'em! Special relativity kzbin.info/www/bejne/gYfOYoSEibx1rrM General relativity kzbin.info/www/bejne/nmO2dn5sf8-pabs

[@dukenukem9770]

@@PhysicswithElliot Sweet!!! I’ll watch immediately!!!

[@jackr1734]

Great way to tackle this kind of topics, keep it up man

[@PhysicswithElliot]

Thanks Jack!

[@utuberaj60]

Lovely made-easy intro to Least Action in the "least complicated" way Mr. Elliot. I have only heard this term before vaguely. This makes me want to know if this "Principle of Least Action" is related to Fermat's "Principle of Least Time"- which expplains Snell's Laws of refraction elegantly. Could you please make a video on this also?

[@PhysicswithElliot]

Thanks Rajagopal!

[@brainandforce]

Fermat's principle was the basis for the principle of least action.

[@tomkerruish2982]

Furthermore, Huygens' principle leads to Feynman's path integral formulation of quantum mechanics.

[@eastofthegreenline3324]

A nice treatment of this topic. Very much enjoyed it!

[@증걸대라쫌]

That's amazing... The reason why light goes straight is that all the path that light selected at the same time is cancelled out except the least action path. Only straight line survived....

[@youerny]

I have been looking for this video(s) for-ever. Finally they found me! Thank you so much. Just two micro questions: 1) dt at the beginning of the integrand expression is peculiar of this branch of maths? 2) what software are you using to draw? Tried several computer+tablet combinations, but this looks better than most of them Thank you again!

[@hOREP245]

with regards to "dt at the beginning of the integrand expression is peculiar of this branch of maths?". This is simply a style of integral notation, often used by physicists or just people who are taking integrals with respect to many variables. It can be nicer to work with, as you immediately see what variable is being used for the integration. I personally don't use it, but I know people who do.

[@rapp1584]

Your videos are always very well done. Thank you.

[@randymartin5500]

While Leonard Susskind's 10 part series on Classical Mechanics was good, it was too long winded at nearly 2hrs per lecture! . Dr. Elliot's series is excellent which covers a wonderful understanding of the exact same equations and derivations in less than an hour!

[@SynaTek240]

at 11:25 the change in L equation is missing a second dot above the first x

[@ssym2]

Thank you Elliot! Your videos are extremely helpful!

[@deepakjanardhanan7394]

Great. I hit a gold mine in you tube.

[@chrisjager5370]

It's amazing what you can do with "the integral of zero is zero"!

[@duttaalt]

Excellent video brother, you just gained a subscriber !

[@mikepenz]

Great articulation of a complex topic. What is the app you use to build your presentations?

[@smoorej]

Fantastic explanation. One question: towards the end when you say “assign a number to each possible path”, aren’t there technically an uncountable number of paths?

[@eamon_concannon]

7:37 You appear to be assuming that not only ε but also dε/dt is a very small number. Is this correct? We could use λε(t) instead of ε(t) where λ is a constant set small enough so that d(λ(ε(t))/dt (= λd(ε(t))/dt ) is very small for all t. Thanks a lot for the very well presented videos. I have subscribed.

[@PhysicswithElliot]

Yes you can say it like that

[@DebabrataSaha-k7o]

Fantastic lecture. subscribed. Thank you.

[@wayneyadams]

15:24 If we break apart the bracket on the left, we get called ket. Put together we have a braket (bracket). Who said Physicists don't have a sense of humor? 😀

[@meghadulshan7689]

10:00 can somebody plz explain me why epsilon t1 and epsilon t2 are zero?

[@sergio3713]

Hello. 3 questions: Why would the particle follow the minimazing action S trajectory? What is the physical content of the minimum action S? What is the physical content of an any value action S? Thanks!

[@tomkerruish2982]

This is touched on at the end, starting at 13:50.

[@sergio3713]

Hello! Not using Newton's F = ma, allow me ask again: 1 - Where does the { Action = Integral (K - U)dt} come from? 2 - Where does the {Lagrangian (K - U)} come from? 3 - Can I deduce that I must minimize the Action integral equation from minimizing the potential energy U? 4 - Can you elaborate? Thanks!👋

[@wadelamble5462]

In Relativity, the principle of least action becomes simpler -- it's just the shortest path in spacetime, or longest proper time. In D(t), is t proper time or observer time. Hmm....would love to know if others have any insights here.

[@Edufis-zw4hh]

Actually, as far as know the action should be an extreme (maximum or minumum), it does not need to be necessarily a minimum, but in most of cases it is a minimum. At least in classical mechanics they claim for that. May I correct? Thank you!

[@Woodsford123]

Brilliantly well explained. Thanks.

[@FB0102]

Can you explain why the standard Lagrangian is T-U? Intuitively, why that specific form (other than 'because it works')? Thanks

[@NothingMaster]

Now, explain the Principle of Most Necessary Action.

[@sachinrajpandey5242]

Will you please also make video on linear algebra and group theory?

[@snake4eva]

@PhysicswithElliot What experimental evidence is there to suggest that the path integral formulation is correct? By this I mean, what evidence is there to "show" that the particle traverses all paths? Also could you include the experimental evidence in your future videos along with the name of the experiment or researcher who first did the experiment?

[@nicholastzilinis3832]

what a great compliment to Landaus book

[@Fireflyy29]

Hello sir, at 10:15 I did not get why the integrand must be zero if the integration is zero, the integration can still be sum up to zero even for the nonzero integrand function. Somebody pleasse explain.

[@darkol93king34]

very good content. I love physics :)

[@michaeledwardharris]

Excellent work!

[@PhysicswithElliot]

Thanks Michael!

[@gandalfthefool2410]

This might be a dump question, but how does an elementary particle know what minimum action is/ shortest path is? We can deduce it by taking the integral on all possible paths, but how does a particle know? Doesn’t that require a particle to travel along all possible paths to find out?

[@ES-qe1nh]

It doesn't "know". Physics (the possible ways a body or particle can behave) dictates/limits it will behave that way

[@CarlosRodriguez-mx2xy]

Un modelo de elegancia matematica y virtuosismo didactico. Muchas gracias

[@PhysicswithElliot]

Gracias Carlos!

[@whilewecan]

Wonderful. I'm grateful.

[@Harley492]

Just got into principles of least action. From my 10 year old Son asking me about ballistics. If the basics of Quantum Mechanics and General Relativity were derived from the same least action principle, why are they at odds with each other?

[@simontaeter]

Hey! Nice video thank you. I just dont get why you can make the epsilon^2 disappear like that?

[@ricardovencio]

awesome class. Thank you.

[@ekisvioleolivaradamos6701]

I really enjoyed your video... Tnx I learned a lot...

[@johnsolo123456]

thanks for making this channel!

[@DeepLyricist]

Why do forces have to be derivative of u? I'm trying to focus on myself.

[@mrbank3453]

Bruh

[@wieslawpopielarski8974]

well U is potential energy so from school you can convert it into work which is W=∑F∆s=U (see integral instead of sum :) ). So to get F you simply compute derivative of ∂U/∂s = F

[@I-M-2.]

Great content! So forces are just a glitch in human perspective?

[@nicholasthesilly]

I wouldn't say that. It's not like an optical illusion. "Forces" is a legitimate way of thinking about human-scale phenomena. It just doesn't work well for the very large or very small.

[@jakubtvrdy4934]

Hello from Czech republic, I guess the time will run out until you notice with this amount of subs, but I have to try... I have exam in theoretical physics tomorrow, could you please explain to me why are higher powers of ε in taylor and generally also in other parts of the integral not relevant for the final result? Thank you very much for your time if you notice

[@PhysicswithElliot]

Hi Jakub-- It's very similar to finding the minimum of an ordinary function; you're looking for the point where the first derivative vanishes. In the Taylor series, f(x+dx) = f(x) + f'(x) dx + ..., the first derivative shows up in the linear term, so that's the one we want to pick out.

[@jakubtvrdy4934]

@@PhysicswithElliot Thank you very much, God bless you.

[@samtux762]

Does the least action principle only relate to a trajectory, but not the speed along the path? It should only govern the path (if we compare a ball bouncing vertically and a yo-yo toy, they have the same path, but different speed along the path). Im I right?

[@filipelqj]

Perhaps YT comments is an unlikely place to look for an answer like this but, this explanation (which is great and similar to the same I had when I saw QM for the first time) implies that your functional (in this case your Lagrangian) is a function of only analytical functions. Which then excludes all other non-analytical functions solutions... Anyone can try to explain this to me?

[@PhysicswithElliot]

When the trajectory isn't smooth the action typically blows up. Singular trajectories are of interest in quantum mechanics though

[@filipelqj]

@@PhysicswithElliot thanks for the answer but not being smooth is not the only case. The classic function e^1/x2 is for example infinitely differentiable (expect at 0 of course) and not analytical. I see this on DFT as well. A bunch of assumptions of Functionals being well behaved and an some "arbitrary" considerations (for the sake of simplicity) that exclude several classes of possible solutions. My Mathematics side is in pain while my engineering/Physicist side says if I don't make those considerations then I don't have any hope of answering those questions in the first place...

[@davidcurco3844]

Thanks for this great video and your clear explanations. A doubt in the development of the equations: (7:47)... I understand that e squared vanishes, but why does edot squared vanish too? Even in your picture, the difference between the "red" and "blue" trajectories (e) changes sharply with time. So, edot squared vanishes as a consequence that e is small or as a consequence that a trajectory close to the "optimal" one is characterized both by e small AND edot small?

[@PhysicswithElliot]

Thanks David! You could instead write the variation as x(t) -> x(t) + c f(t) where c is a small parameter and f(t) is any function that vanishes at the endpoints. Then the requirement is that the change in the action under this transformation is zero to order c. When I wrote \epsilon(t) I've essentially absorbed this small parameter into the variation, and then counting powers of \epsilon or its derivatives is equivalent to counting powers of c.

[@davidcurco3844]

@@PhysicswithElliot Thanks for your clear and kind answer.

[@ProjectileGrommet]

Vsauce didn’t describe this well (yet) but this video did

[@whatitmeans]

What will happen for the classical Action in the following experiment: a particle that travels in a line and have a perfect ellastic collision with a wall (1D position vs time function).... it will travel back, so it first derivative will have a bounded "jump discontinuity", that will become a singularity in the second derivative...How will be the action principle work in this scenario?

[@PhysicswithElliot]

The kinetic energy doesn't change when the particle reflects off the wall, so it won't have any effect on the action

[@ZehuaSong]

LOVE THIS!!!!! Thank you

[@urnext6874]

I love this channel

[@mohamedmouh3949]

thank you so much very simple 🤩

[@meow75714]

at 10:01, don't get why E(t1) and E(t2) should be zero? Can you please clarify more simply?

[@anthonyjulianelle6695]

Really good video but a quick question: At 5:30 in the def of s (in green) is the, "dt" in the wrong place?

[@PhysicswithElliot]

Not sure what you mean!

[@lineardielectric]

The dt can go before the integrand. It's pretty standard in physics and makes multiple integral easier to interpret (at least it did for me)

[@jdarcy5714]

Could you show the detail math of an actual problem?. The equations for y = -x^2 + 5 and the equation y = x^2-8 intersect. the graph shows one path longer than the other. Could you use these 2 equations and show the details. thank you for the video. I've been trying to understand this for a while. I'm almost 80 and would really appreciate the help.

[@sergiolucas38]

Nice video :)

[@Killer_Kovacs]

But a particle cant move along all paths at the speed of light, isn't the energy necessary at the speed of light the maximum that a particle can carry?

[@jtinalexandria]

But how do we know the particle was traveling "from point A to point B" until after it's done that? And once it's done, well obviously it took the easiest path.

[@AA-iq6ev]

But how did the particle know what path to take? Like the light that hit water,. Suddenly it realise it goes slower instantly and need change directions instantly

[@johnsnow7090]

OMG, I finally understand ❤❤❤

[@paulsutton5896]

As always Lagrange pulls the Newtonian rabbit out of the hat. What I have never understood is why Lagrange ever suspected that his hat might contain such a rabbit.

[@nicholasthesilly]

Even before Newton, scholars already knew that light rays obeyed a similar principle. That is, they took the path with least travel time (Fermat did a lot of work on this.) Lagrange wanted to generalize this idea to all matter, and probably just tinkered until he found a way to do it.

[@thrunsguinneabottle3066]

@@nicholasthesilly I wish my own tinkering was so fruitful.

[@logicaldomain3272]

Isn't this basically first law of thermodynamics?

[@tombouie]

Thk you for your clarity of explanation; I never could understand the classic laws of thermodynamics especially the ?entropy? ( en.wikipedia.org/wiki/Thermodynamics#Laws_of_thermodynamics ). ?Does least-action a better description of thermodynamics than classical thermodynamics especially the dreaded ?entropy? .

[@ARCT3CH]

Very cool

[@user-pb4jg2dh4w]

Is that a uv lamp there ??

[@SixtysymbolsSymbols]

You are my idol bro 💔

[@BraulioRamirez-q2k]

thx!

[@jacobfrando2969]

Can someone tell me where the U’(x)ε comes from at 7:49?

[@jayprajapati9496]

That's Taylor Series expansion of U(x+e) = U(x) +U'(x)e

[@e_squared604]

coooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooool

[@dennisbernstein6831]

"Minimize is too strong of a word"------In fact, the claim of least action is actually wrong. Why propagate misconceptions? Simplicity makes a nice story, but a miss is a miss, and "minimization" is known to be wrong. The best that can be claimed is stationarity.

[@drbonesshow1]

The Principle of Most Action is moshing your way around a nightclub or stadium.

[@gaHuJIa_Macmep]

You don't explain why Lagrangian is defined as T-U, and not something else. This is the true understanding of what's going on here...

[@FB0102]

How would you answer that question?

[@gaHuJIa_Macmep]

@@FB0102 "the margins of this book are too narrow to write it down here..."

[@FB0102]

@@gaHuJIa_Macmep Oh, it fits. Its quite straightforward :)

[@huynguyenquang7435]

10:22 0 or 1? make up your mind.

[@nitind9786]

Even i don't understand why should that be 0. ?? 1 makes sense .. as then the integral over epsilon would vanish regardless of the value of start and end 't'.

[@meghadulshan7689]

can somebody explain me 9:08

[@a.nelprober4971]

7:40 why??

[@clieding]

Your wonderfully clear and admirable explanations are very enlightening but the speed of delivery is dizzying; your brain is on fire! I just try to hold on and enjoy the ride.

[@onlyphysics143]

nice lecture but i cant get the real essence of least action. still i don't understand

[@ordell3917]

You looks like elliot

[@dumbphysicist5142]

I don't understand why

[@paulboro5278]

It's a law of nature.

[@paulboro5278]

There is no Why.

[@dumbphysicist5142]

@@paulboro5278 why nature has this law?

[@dumbphysicist5142]

@@paulboro5278 why there is no why?

[@kierkegaard54]

10:22 I thought 0! was 1??? JK

[@nitind9786]

Even i don;t understand why should that be 0. ?? 1 makes sense .. as then the integral over epsilon would vanish regardless of the value of start and end 't'.

[@paulbizard3493]

👍

[@ihbarddx]

Let me tell you about a fundamental principle of pedagogy: The Principle of Fewest Expository Anachronisms. One result that falls out of this is that you don’t base an entire presentation on a concept for which the audience has no objective or intuitive understanding. Where did the Lagrangian and the Principle of Least Action come from? I’m quite certain that neither Lagrange nor Euler consulted Feynman’s Ph.D. thesis, thank you very much. After that, if you want to present a useful example, why don’t you address one where mass varies. (e.g.; a rocket)