Symmetries & Conservation Laws: A (Physics) Love Story
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- Опубліковано 1 гру 2021
- There is a deep connection in physics between symmetries of nature and conservation laws, called Noether's theorem. In this physics lesson I'll show you how it works. Get the notes for free here: courses.physicswithelliot.com...
The relationship between symmetries and conservation laws is one of the most profound and far-reaching connections in physics. The central result is called Noether's theorem, and it says that for every continuous symmetry of the Lagrangian or action for a system, you'll find a corresponding conserved quantity. Momentum conservation, for example, follows from a symmetry called spatial translation invariance, meaning that you can pick up your system and slide it over without changing anything about the physics. Likewise, angular momentum conservation follows from rotation invariance, and energy conservation from time translation invariance.
Get all the links here: www.physicswithelliot.com/noe...
Introduction to the principle of least action: • Explaining the Princip...
Intro to Lagrangian (and Hamiltonian) mechanics: • Lagrangian and Hamilto...
The Hamiltonian version of Noether's theorem: • The Most Beautiful Res...
Tutoring inquiries: www.physicswithelliot.com/tut...
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About physics mini lessons:
In these intermediate-level physics lessons, I'll try to give you a self-contained introduction to some fascinating physics topics. If you're just getting started on your physics journey, you might not understand every single detail in every video---that's totally fine! What I'm really hoping is that you'll be inspired to go off and keep learning more on your own.
About me:
I’m Dr. Elliot Schneider. I love physics, and I want to help others learn (and learn to love) physics, too. Whether you’re a beginner just starting out with your physics studies, a more advanced student, or a lifelong learner, I hope you’ll find resources here that enable you to deepen your understanding of the laws of nature. For more cool physics stuff, visit me at www.physicswithelliot.com. - Наука та технологія
Bro, whatever happens, don’t stop uploading videos please, your videos are the best ones I have ever seen on UA-cam. Hands down. Life changing honestly.
Thank you Aiden!
Totally agree!
Um, what if he's turned to meat jelly after his misadventure with an errand bear? Is he still supposed to upload then?
Definately the best video on noethers theorem I’ve seen. Clearly shows relation to lagrangian and simply derives conserved quantities from symmetry in a way that that is easy to see how it generalizes. I think I properly understand (and should be able to remember) the core of how it works now!
Thanks George! Very glad it cleared things up for you
Definitely
All of this from just some mass m that moves some distance x. What a magnificent piece of physics. Noether was a top shelf genius for sure.
Sir,you're GENIUS
THE BEST EXPLANATION
Glad it helped Prem!
Really excellent! I love that you're being quantitative about the physics, something sorely missing on physics UA-cam. Excellent balance of math and physics.
When I first understood Noether's Theorem I felt like I really started to understand physics. Thank you for an excellent introduction. 👍👍👍👍👍
Possibly the most beautiful theorem in all physics, deserving of a (Nobel Prize)^2
I always wondered, ever since I first started learning Physics at 15, why energy is conserved. At 17, in high school, I was introduced to momentum, and again, I wondered why it had to be conserved. I derived a partial answer for momentum specifically - I.e. if momentum is conserved then Newton’s Third Law must be obeyed, but nothing more generalizable. And now, age 25, I’ve finished a degree in Physics, I watch this video, and I finally understand why energy and momentum get to be conserved.
Anyone else notice that the related symmetries to conserved quantities that arise from Noether’s Theorem are the same pairs of quantities that are paired in Heisenberg’s uncertainty principle: energy/time and position/momentum?
I don't get it...but I love it 💯
Wow i dont expect youre here
Today, the algorithm has blessed me with a great new physics channel. All praise the algorithm.
This is such a great explanation. My professor rushed through this topic and I so glad that I have your videos to explain it to me properly.
Glad it was helpful Anna!
Phenomenal job! This level of detail is just was I was looking for in an explanation of Noether's theorem. Thanks man
Thanks Nick!
You give a different perspective to look into the subject which unique ofcourse . Plz upload more this types of stuffs.Do not make us wait more....we are eagerly waiting.
Somehow most of your videos are everything I've been missing in my physics self education. Thank you, Elliot!
My pleasure!
Elliot! Your videos are amazing and keep getting better. Would love to see a video on Renormalization and topics related to Group Theory! Keep it up Elliot, you’re the best on UA-cam
Thanks Andrew!!
Wow, another lucent and concise video! Good job!
just what I was looking for! hands down one of the best videos on this topic
Thanks Vikrant!
I've just found this channel .. and is Amazing! . Liked & subscribed
Thanks a lot. Please keep it up you are helping a ton of people who have high standards when it comes to understanding
Glad it helped Reda!
This video really helped me to understand the topic. I was extremely confused about it!
Well done, been a long time for me, I still love Physics
Love the videos, especially with accompanying notes.
Another great video, you really get the precise proportion of broad explanations and mathematical examples.
I would love to hear you explain the related topics of center potentials and scattering theory, which are both commonly taught with other advanced mechanics topics such as those.
Thanks Ido! I will put scattering on my list
Love your videos! 😊🎉
You're one of the best channels out there explaining physics in a simple manner. I think people like Walter Lewin, Richard Feynman would be proud of you.
Beautiful explanation thanks
Amazing Elliot, well done on Noethers Theorum. 4 pages of derivation in 15 mins, well done
Thanks Neeraj!
You are the best physics teacher in the world! Crazy.
Was expecting to learn about Ring Algebra/Groups, instead, I got a refresher on classical mechanics. I loved that course, but it was difficult because it takes time to build up the intuition.
I am very glad to have found this Chanel
Awesome explanations, good job man !thanks from telaviv
Thanks for the great work and I wish that your channel will become big and for that I have no doubt because you're a very skilled physics teacher and scientist, it's only a matter of time.
Youcef Ammar-Khodja
Thanks Youcef!
Great explanation
Excellent lessons on QP and Mechanics
great video! Future topic suggestion: tensors.
Very interesting ,informative and worthwhile video.
Thanks Robert!
Amazing content! Thanks.
great work
Amazing explanation 👏 👌
awesome video!
Thank you Elliot.
Amazing videos!!
Really amazing 👏👏
I wish I'd had remarkable videos like this to learn from when I was doing my physics degree. Instead, it was mostly "shut up and calculate".
Holy shit, this is the most excellent crystal clear explanation of the subject. Well, at least for a layman with a background in math but no proper training in physics (like myself). Thank you so much.
Thanks George!
By far my favorite theorem! Would you be able to show the mathematics behind the more spicy symmetries in physics? Lorentz invariance, gauge invariance, and probability invariance I’ve all heard before but not seen the mathematics behind!
I will add spicy symmetries to my topics list!
Lorentz invariance is just the consequence of the universe being 4D afaik. Just as in 3D geometry, 3D length and rotation are invariant. It has more to do with geometry than symmetry if I understand correctly. Not sure about the other ones tho 😅 Would love to see a video on them 😊
@@feynstein1004 Not to nitpick but I think it could be good to explain a distinction here! Lorentz invariance refers specifically to scalar values, but the physical theories should be consistent for objects of higher dimensional objects like vectors. The more general feature is Lorentz Covariance, which says that the object transforms as a representation of the Lorentz group.
A representation of a group essentially means that there is way of representing a transformation of that group as a matrix. For example, the rotation matrix of a 3-dim vector is a representation of the group called the Special Orthogonal group of dimension 3, SO(3) sometimes called the Rotational group.
In the case of Lorentz Invariance, rotating a scalar value doesn't change anything, but rotating a vector may indeed change it. The speed of light, a scalar, is a Lorentz Invariant quantity, but the path light takes in the presence of mass is not invariant, it is covariant, it covaries (changes with) the changes in the field of spacetime. This is a distinction physicists sometimes conflate!
@@sebastianjovancic9814 Oh wow, I didn't know that. Thanks for the information 🙂I hadn't realized that the invariant spacetime interval in SR is a scalar quantity and as you said, vectors might not behave the same way.
@@feynstein1004 Thank you for giving me an opportunity to try to teach this, it's a topic that fascinates me immensely! I highly recommend introducing yourself to group theory in the context of physics (which more accurately should be could representation theory since we mostly work with representations of groups in physics). It opens up ones understanding of quantum mechanics (in my opinion) and give you the tools to better understand where the fundamental forces come from, what particles are and more!
I’d love to see a video about Goldstone theorem and Goldstone’s boson!
I certainly hope to talk about symmetry breaking in future videos!
Amazing!
Greate job!
I don't understand the symmetry breaking at 9:35. If there's a potential U(x), then that -U'(x)*eta0 term is within the EOM, which we set to 0 in the previous example. The d/dt (mv*eta) still appears to be a constant, leading to the same result. What am I missing?
Amazing style, I can imagine myself as an undergrad having that felling after a great lecture… btw what software do you use for your board? I’d love to try it . Thanks 👍
Why do we use dot notation to regard n-th order differentials instead of the standard apostrophe "prime" such as f'(x)?
Wonderful
Hello Elliot, could you possibly do the last proof (conservation of energy / time invariance) in a sequel to this video?
please!! this is the most complicated and the reason I ended up in this video
Hi! Can someone explain (4:54) why we wrote U'(x)ε(epsilon)? why are we multiplying on ε?
Congratulations! When will be the video on Thermodynamics!?
Ok sir. But I have a few question. Do Lagrangian can change from an inertial observer wrt ground to an accelerating observer? Just like in deriving the Unruh Temperature for "thermal bath" of an accelerating observer. The action S is invariant for both observers but their time would be different. So to make S invariant, Lagrangian should also be different?
In classical mecanics we have the principel of d'Alambert. Very nice too. Who will understand your video, if he has not understood before?
I noticed the products of the symmetries and conserved quantities have units of action ([tE] = [xp]). Any relationship between Noether's theorem and the uncertainty principle?
Hey Elliot. Since watching this video, I've been trying to find if there are any symmetries associated with the laws of thermodynamics or stat. mech., but I've come up short. If I could conceptualise the Lagrangian and EOM or analogues I'd give it a go. Do you know anything you could point me to in that direction? Does it even make sense to expect them to have symmetries e.g. second law isn't a "conserved" quantity and so on? Cheers.
Symmetries are certainly important in stat mech. For example, you could consider a ferromagnet described by a bunch of little magnetic dipoles with rotational symmetry. As you cool it down all the little magnets tend to line up in a particular direction, which is called spontaneous symmetry breaking.
I love this explanation. How about gauge invariance next?
I'm sure I'll talk about gauge theories at some point!
please make a video for convolution, correlation and diffraction(fraunhoffer and fresnel) with different types of apertures and how fourier transform has to deal with all of these things .........I know i am asking for so many things but these are most confusing for most of us studying physics ........BTW loved your videos ....THANKS
Thanks Sagar!
Great video, thanks 👏
Are You able to make video about N masses (mass = m) lying on x - axis (like: fist on x_1= 0, second on x_2 = L and so on) being connected with springs (const factor = k)? The main goal is to find the speed of wave that can be carried on the spring net. Best Regards
Thank you Elliot for linking symmetry to conservation laws. When I studied physics more than 30 years ago, I don't remember doing this. Even though I didn't get everything you were saying here, it's still very helpful to go through it fast once to get the gist of it. I was wondering why Lagrangian would start by KE - PE. What's the significance of it? I know if you + them, you get the total energy and that's useful. But why minus them?
My video explaining the principle of least action might help! ua-cam.com/video/sUk9y23FPHk/v-deo.html
Can you please make a video on parity and how it is different from symmetry?
This may sound like a dumb question but: you tacitly assumed that if the time-derivative of a quantity was zero than that quantity was conserved. Sounds good. But time is just one dimension in space-time. Is there anything that tells us about quantities that are "conserved" in one or more space dimensions?
Thankyou so much. But Sir I have a doubt why we use lagrangian in noether theorem.
Please reply
Thanks
I think, in importance, Noether's theorem falls along with the principle of least action.
Could someone explain why Time translation is related to energy conserwation but not with momentum? How the symmetry even related with given quantites?
The time translation/energy case is slightly more complicated so I didn't go into the details in this video. Under a time translation t -> t + a, you can Taylor expand x(t+a) = x(t) + a \dot x(t) + ... . Then the change in x is \eta = a \dot{x}, and m \dot{x} \eta = m \dot{x}^2 a.
That's not the conserved quantity though---the reason is that the Lagrangian is not invariant, it likewise transforms by a dL/dt. You will still have a symmetry if the Lagrangian changes by a d/dt term like this, though, because when you integrate it only affects the value of the action at the endpoints, and doesn't change the equations of motion. So this still qualifies as a symmetry.
So we get dL/dt = d/dt (m \dot{x}^2), and if you move the L to the other side you'll learn that E = 1/2 m \dot{x}^2 + U(x) is conserved!
Noether's theorem is incomplete if it lacks 1st, 2nd & 3rd derivatives of initial conditions and, depending on the perturbing encounter, of THAT too (maybe).
I know that I can't be the only person who thought *Noether's* *Theorem* was the name of the *Michelson-Morley* *Experiment* to detect the luminiferous aether? The results are in the name!
wonderfull
Eliot, you are an excellent teacher.
If I understand you correctly, Noether's Theorem is a result of the properties of the Lagrangian formulation. Or, you can deduce Noether's Theorem from the behavior of the Lagrangian, with a proper interpretation and identification of the underlying symmetry contained in the Lagrangian.
Thus, am I correct in saying that Noether's theorem is not a basic principle from which you can derive the Lagrangian (or the Hamiltonian or the Newtonian), but rather an interesting bit of insight on what these latter formulations imply? Unless of course Noether's Theorem can in other areas make predictions on nature's laws that are not derivable by any other means. Are there any such examples that illustrate this latter possibility? In other words, what's more basic here, Noether's Theorem or the Lagrangian? Noether's Theorem, or the principle of extremal action, from which the Lagrangian itself is derived, as I understand it?
Thanks for your excellent videos.
Addendum: I just read through all the comments here and Eliot's responses and got insight into my own questions, above. Although the questions are still worth asking, I realize that Noether's Theorem allows us to spring forward to much physical insight. For instance, with the example of Conservation of Energy, without Noether's Theorem, it's something we discover empirically, almost by accident. But Noether's Theorem tells us that no, it's a consequence of something (time symmetry) that is very basic. And it seems to me that this theorem has led some physicists to discover relationships that without it, could never be fully realized. Thank you Ms. Noether.
In fact when physicists are writing down a Lagrangian e.g. in particle physics, we often let the symmetries determine the form it should take by writing down all the terms that would be consistent with the symmetry!
Hello I am a physics undergrad, do you know if Noethers theorem is taught in later years (maybe grad school)? Watching your videos gives me something to look forward to in my 2nd or 3rd years, very inspiring thank you!
It's usually taught in a junior year upper level mechanics course!
What’s the writing program you’re using for your videos?
Procreate
Can you show us how do you get the energy conservation ? That was the only thing I wanted to know. I am trying understand what is energy and it seems like it has a deep relationship with time. Like as the momentum is the reason why x changes, energy is the reason why time evolves. So can I say the energy is the origin of time ?
From an earlier comment:
The time translation/energy case is slightly more complicated so I didn't go into the details in this video. Under a time translation t -> t + a, you can Taylor expand x(t+a) = x(t) + a \dot x(t) + ... . Then the change in x is \eta = a \dot{x}, and m \dot{x} \eta = m \dot{x}^2 a.
That's not the conserved quantity though---the reason is that the Lagrangian is not invariant, it likewise transforms by a dL/dt. You will still have a symmetry if the Lagrangian changes by a d/dt term like this, though, because when you integrate it only affects the value of the action at the endpoints, and doesn't change the equations of motion. So this still qualifies as a symmetry.
So we get dL/dt = d/dt (m \dot{x}^2), and if you move the L to the other side you'll learn that E = 1/2 m \dot{x}^2 + U(x) is conserved!
@@PhysicswithElliot thanks 😁
How is Noether's theorem relevant to the Differential Bianchi Identity that is the signature of energy-momentum conservation in space-time?
Noether's theorem applies to relativistic theories just as well, and energy and momentum are again the conserved quantities associated to translations in time and space. But different observers will get different values for them, related by Lorentz transformations.
Bro, your videos are soo damngood.
I am willing to produce the subtitles in PT-BR. my pleasure.
Edit: btw, I do a physics degree ;)
Are there any symmetries for which we haven't yet found the conserved quantities? Or vice versa, any conserved quantities for which haven't found their symmetries? I guess my larger question is whether or not we think we've found everything to which Noether's theorem could be applied.
It can be hard to find all the symmetries in a theory. For example, in studying the motion of a planet around a star using Newton's law of gravity, there's a very unobvious symmetry transformation that leads to an additional conserved quantity called the Runge-Lenz vector. Once you know that it's conserved it's extremely useful; I showed how you can use it to derive the orbit in just a line or two in a video a couple of months ago: ua-cam.com/video/KOek-B3Rvmg/v-deo.html
Dear Elliot, the equivalence between symmetry and conserved quantity stated at the beginning of the video is a bit misleading.
Noethers Theorem goes in one direction solely, i.e. symmetry -> conserved quantity, not vice versa. There is no proof that for any given conserved quantity there is a symmetry keeping the Lagrangian invariant.
Or am I wrong?
Though I really appreciate your series. Thanks a lot!!!
4:40 So epsilon is a dx? I'm confused, because there is no dt or dx on the right hand side of the equation, and you called epsilon "the change in x dot".
The variation takes the path x(t) and replaces it with x(t) + \epsilon(t), where \epsilon(t) is a small deformation that adds little wiggles to the curve you started with. Then \dot{x}(t) is likewise transformed to \dot{x}(t) + \dot{\epsilon}(t).
Particle physics video?
How about a video about Inertial Frames of Reference?
I like to say that if Isaac Newton wrote the first chapter on classical mechanics, Emmy Noether was the one who wrote the last.
Are conservation laws in particle physics (like conservation of lepton number) also reflected by a symmetry?
Yes particle conservation laws also have corresponding symmetries!
In highschool physics I learnt to analyze systems using conservation laws, eg. Momentum is conserved for no external force or angular momentum for no external torque. How does this "lens" of symmetry-laws (rather than conservation laws) give us different insights into physical systems?
As a mathematician, the study of symmetries directly translates to the study of groups. Which allows you to use much more sophisticated techniques to understand the physical systems. The short of this, is that the beauty of physics is that these laws aren't just something that works sometimes, but they are the ingrained rules of the universe itself.
For something more complex, I know that the concept of spin is derived from Lorentz symmetry (though I wouldn't be one to prove it). And in general almost all of physics falls from similar kinds of symmetries.
@@XZ1680 ok, so finding symmetry in physics laws lets you use mathematical approaches from studying group theory ( as in groups from abstract algebra with the fields and sets?) Which THEN let you gain deeper physical insights with regard to the original physics. Cool.
There's lots of different ways of answering that; the quickest is maybe just to say that Newton's way of thinking doesn't generalize very readily to quantum mechanics, but the Lagrangian and Hamiltonian methods do. Noether's theorem tells us how to think about symmetries and conservation laws in Lagrangian mechanics (and there's a similar version for Hamiltonian mechanics), and in quantum mechanics we use these symmetries to classify the states of a system.
But even in classical physics thinking about symmetries gives us deep insights into physics. Newton's laws for example emphasize conservation of momentum (and angular momentum), but conservation of energy is something you discover as a consequence and might look like an accident. But we learn from Noether's theorem that it's due to a symmetry under time translations, and is in fact closely analogous to momentum, just that one is related to time translations and the other to space translations.
@@PhysicswithElliot And thus it's no surprise that in 4 dimensional spacetime the two are combined: K.Energy is just the 4th component of the 4-momentum vector, and you expect that you get a momentum component in 3 directions of space. It directly says that K.E. is the time dimension component that is momentum.
@@ryanbright2696 Yes, with group theory you can recognize the _pattern_ of relationships, and guess that you're seeing part of the complete group. A direct example is how Quarks were figured out.
Isn’t gravity asymmetric in time? The next moment is more dense.
So I recently watched the video where you look at the phase space for a pendulum, the key assumption is that energy is conserved. What happens when it's not? How does the phase space change, and does the fact that, when energy is conserved, each path is unique and the paths don't intersect change when energy isn't conserved?
The problem sheet I posted with that video is actually an example of that, where instead of just a pendulum oscillating on its own someone else is jiggling it back and forth. That drives energy into the system, and so the energy of the pendulum will not be a constant. Then the motion won't be constrained to a curve of constant energy.
@@PhysicswithElliot awesome I'll have a look. Those question sets and supplementary lecture notes are really a wonderful idea to complement your videos with, I love it.
@@ryanbright2696 So glad they're helpful!
Hmm but isn't time translation invariance only true locally, not globally? And thus so is energy conservation? The universe isn't symmetric in time iirc 🤔
Love your content bro.
I have a topic I wish to know. Why is Hamiltonian mechanics a thing when Lagrangian mechanics is superior?
in a lot of quantum mechanics they use the Hamiltonian opposed to the Lagrangian, although Feynman figured the way to do it.
so to answer your question, I don't know... lol
@@briannguyen6994 bro, u gave me hope then all of a sudden, pulled the rug under my feet 😂😂
Thanks Sphakamiso! The Lagrangian is usually more practical for solving classical mechanics problems, but both approaches teach us new things about the physics. And the same goes in quantum mechanics: each formulation gives us a different perspective and offers new insights into lots of problems
12:38 where did the l come from?
l is length of the spring when it's in equilibrium, and r is length at any given moment, so that r - l is how much the spring has been displaced from equilibrium. Then 1/2 k (r-l)^2 is the potential energy
Up to 3:00 mins phenomenal information for my type of thought process ..not good with formulas
Principle of least action is not valid; path of stabilization completely depends on 1st, 2nd & 3rd derivatives of entity upon initial influence. These may drive configurations to states that are more action than if these derivatives did not exist.
All these considerations lack one important aspect: MATTER, or more specifically, PARTICLES that move from one location that generates it to another towards which it is generated.
If we consider space as consisting of a Cartesian network of Centers of Generation (CoGs, for short) at equal rectilinear distances that generate particles towards the centers (CoGs) in successive cubes around it, then there exists a unique number of CoGs accessible from any given Center, which are not uniform, but involves a geometry that links possible sizes of the generated particles to the distances from that unique generating CoG to "accissible" (obviously only some are) other CoGs from it, which doesn't require any superimposed SYMMETRY assumptions, thus a "particle physical geometry and arithmetic" that derives from the physics of the generated particles alone, rendering mathematics a branch of physics, without any prior assumptions as to existence of any mathematical concept (numbers, lines, points, symmetry or any other) at all.
I find the mixing of Newton and Leibniz notation confusing
I'm frustrated by all this talk of derivatives being taken but without it ever being specified what variables they're being taken with respect to. I can't follow the math because of this. For some reason the derivatives just keep ending up being multiplied by a factor of epsilon/eta even though it was added to rather than multiplied by x, and I have no idea why
Nice video. Next time you talk about Noether's theorem you might try and pronounce Emmy Noether's name correctly. Have you heard about the German umlauts (Ä, Ö, Ü)? Well the "oe" in Emmy's name is just a different way of spelling "Ö".
If only you kept η(x, t) instead of the const η_0 at 8:42, that would open the gates to the observer dependency in QM. ;-)
I don't understand why the word "energy" is given so much importance. (even more, wikipedia uses it in the definition of physics) if it is nothing more than a consequence of symmetry like so many other magnitudes
Furthermore, in many countries there are energy offices or ministries, but never linear momentum ministry!