One thing I absolutely love about Noether's theorem (you know, besides the whole "she literally figured out _why_ conservations occur") is the fact that symmetry in position implies momentum conservation, and symmetry in time implies energy conservation. Meaning that the relationship between space and momentum is very similar to the relationship between time and energy - something that Einstein *_also_* figured out through a _completely_ different route. And the fact that 2 different people could arrive at such a fundamental truth about the universe through such different means is mindblowing. Oh, one more thing: Angular momentum is conserved because of rotational symmetry - yet elliptical planetary orbits also have a conserved angular momentum. How exactly do the two fit in? Does it mean "same energy even though the orbit shifted slightly"?
_"Angular momentum is conserved because of rotational symmetry - yet elliptical planetary orbits also have a conserved angular momentum. How exactly do the two fit in? Does it mean "same energy even though the orbit shifted slightly"?"_ No, it means "same angular momentum". The position changes, and the velocity changes, but their (cross) product stays the same. This is true with either circular or elliptical orbits (or with no orbit at all). Remember that position and velocity are vectors, with both magnitude (distance and speed) and direction. In circular orbits, the magnitudes are constant, and only the directions change; in elliptical orbits both magnitudes and directions change. But in either case, the cross product is constant.
I think you can also see the elliptical orbit as a combination of rotation and translation (it pops right out if you use polar coordinates) so you could think of it as a combination of rotational symmetry (conservation of angular momentum) and translational symmetry (conservation of momentum). I'm not 100% sure about this it's just an idea that popped into my head and l would love for someone with more knowledge about this topic to correct or approve this idea
In the video she talks mainly about potential energy. In the case of an elliptical orbit the potential energy is not concerved, but kinetic + potential energy is conserved, so the TOTAL energy of the system is still conserved. If the planet is closer to the center the potential energy is lower but kinetic energy is higher, and vice versa. So now indeed your question arises what the symmetry is - it's not a circle! What you need to remember here is that not the shape of the path has to be symmetric, but 'the way the planet acts' (in Lagrangian physics terms: the action). If the planet would be rotated to another point, the path would also be rotated with it, so the planet would still act the same, but it is rotated a bit, including the path it takes. So the symmetry doesn't have to mean it takes the exact path as before, but that it follows the same path, with that path just being rotated over the same angle as the object was. Does that help?
God said "let there be a bang" and there was a bang. A big one. And the only one we have proof of. I argue that if even one atom were misplaced, nothing would have happened.
There is no proof whatsoever of the ridiculosly naîve big bang. The universe is moving alright, but not outward since there is nothing, no spatial dimensions "outside", it continuously moves whitin itself, known as the Donut Theory. From this movement of the spatial dimensions themselves emanates the red shift.
Even the big bangers are now saying there was no "bang". Nowadays, it's just a simple expansion of of space-time, optionally with inflation and even more optionally with effervescence.
This is the best introductory discussion of Noether's Theorem I have seen. Thank you for your excellent work! Here is a suggestion. It seems to me that your examples of systems whose energy has not changed after a particular transformation might be easier to understand for viewers (for me anyway) if you considered a small, continuous change rather than a large jump change. A system whose energy did not change after a jump (for example, an instantaneous rotation through 60 degrees) might turn out to have periodic energy dependence with period, in this example, 30 or 60 degrees. While a jump change is not ruled out, I do not think a discontinuous change would best exemplify the type of conservation law you are aiming to illustrate in this video. The real-world conservation laws that I am aware of (in classical physics with or without relativity) are all based on continuous quantities. You don't have to call it an infinitesimal change. You might just say the change is gradual, or something like that. P.S. I might be wrong about this. It was just a thought.
No I love the suggestion. If I could go back, that's a change I'd want to make. Hopefully I will give Noether's theorem another shot in a while- I wasn't too happy with that one. I really appriciate the suggestions for improving it!
+Jack McMillan So each symmetry to be characterized requires its own Lie algebra to be set up based on the relevant degrees of freedom in the neighborhood of a given point in the problem space -- something like that?
Hi there. I've got to ask... " So...a symmetry is when you change a system and some number computed from the initial condition doesn't change. And, A conservation law says that when a system changes, there's a number that characterized the system in the first instance that's the same in the second instance. This theorem seems...tautological... " I first posted this to the original video (just now), but I realize that's probably not going to get a good answer. Can you help me sort my confusion? Edit...perhaps I am under-appreciating the development of ideas that it takes to come to such conclusions...particularly when I am now speaking from the shoulders of all the others who had to toil so!
I am not qualified to assist you with that. Sorry. Perhaps the following might be of help: physics.stackexchange.com/questions/4959/can-noethers-theorem-be-understood-intuitively
Hi there Legionary42. Maybe I can help you a little bit. Symmetry means something very specific in physics. Symmetry refers to a certain transformation of a system's dynamical information (which consists of positions, velocities and perhaps even time) which leaves the Lagrangian unchanged. These transformations are physically interpreted as a change of coordinates. Note something very VERY important though. The only kinds of symmetries Emmy Noether discusses in her groundbreaking paper are so called continuous symmetries, where each symmetry transformation can be built up by a large number of tiny infinitesimal symmetry transformations, such (to be more precise, the group of symmetries forms a Lie group). An example is rotational symmetries. A conservation law refers to a physical scalar quantity that doesn't change with time. Notice, a priori these 2 concepts seem to have no no connection whatsoever but Noether's theorem shows otherwise. There is no intuitive argument I can think of to explain this relationship and in fact I'm not even sure there exists one. A Lie group refers to something very very specific (namely a differentiable manifold, equipped with a group operation) & the fact that every "continuous symmetry" (i.e. what we intuitively think of as being a continuous symmetry operation) can be given enough structure for it to form a Lie Group is a highly unintuitive fact. Symmetry, the way we intuitively think of it, is just a map on the phase space which leaves the Lagrangian unchanged. This is not enough to derive a conservation law. So although we have intuitive notions of what we mean by symmetry and conservation laws, those intuitive ideas are not enough to establish Noether's theorem. We have to introduce a lot more structure to our model to establish the connection. Unfortunately this breaks down the intuition. Moreover I think I should point out that although in classical mechanics, you get conservation laws only from continuous symmetries, In Quantum Field Theory, even discrete symmetries have a conservation law. For instance think of the harmonic oscillator modelled by the potential (1/2) kx^2. You can check that the Lagrangian doesn't change if you swap x for (-x). There is no conservation law for classical mechanics for such a symmetry. There is one in QFT.
Thank you for this great explanation of Noether's Theorem. I've been reading a biography of Emmy Noether, and I have a physics background, but I didn't really understand her work at all. Now I understand it a bit more. Thank you. I subscribed to your channel and will definitely be checking out some more of your work. Thanks again! P.S. Emmy Noether is really a tragically unknown figure in physics. I hope more people learn about her. That's why I decided to read up on her, because she's one of the most influential women in math/science (one of the most influential of men and women, in fact) but she is still totally unknown to most. It's a shame. But your video surely has reached new audiences and for that you deserve praise.
I'm so excited you got around to this video. Were you just using conservation of energy as an intuitive reference? Because I think (traditionally in Noether's theorem) it's treated as a complete independent thing.
+The Science Asylum Traditionally its symmetry of the lagrangian, but there's an equivalent Hamiltonian formulation that I got excited about. Way more intuitive to me!
After 2 or 3 years later and also constantly learning new things, i finally feel like I understand what’s being said said in this video. Truly feels amazing finally able to grasp it. Thank you 😊
Wow, this is AWESOME! I finally UNDERSTAND why there are these random 'laws.' And on your roller coaster example, I'VE HAD THE EXACT SAME THOUGHT!!! Yeah, I solved the problem, but I'm just using some law, so to really explain anything I must explain where the law comes from! This is so cool! Great video. I cannot express how long I've been thinking this and how annoyed I've been with 'It's just the law' (I really feel like this should have been explained to me years ago, maybe in high school or something) But, it's never to late to learn, so thanks!
yes this was definately not in HSC AUstralian higher physics 2004 though clearly shouldve been. I think its because symmetry has buddhist connotations, and Australia has a Christian hangover.
To be honest even if you don't know about Noether's theorem, conservation laws are just mathematical derivations from basic Dynamics (Newton's laws, Schrodinger Equation, etc.) They are not "random" or artificial, just magic numbers given rise to by mathematics. Of course if you are still not satisfied and want to go deeper you can always fall back to the Principle of Least Action.
The coolest thing about this video is that you explained the intuition, motivation, and significance of Noether's theorem without invoking the Lagrangian at all. I watched this video when it came out (in high school for me) and thought it was an amazing concept. I'm commenting now after taking classical dynamics for my physics degree, and this intuition was really useful for understanding the lagrangian, even though usually, its the other way around. Thanks again! Love the new "self-teaching" physics video too btw!
As a theoretical physicist, I'm very happy that this video exists. It certainly is the most beautiful theorem in physics and i wish i could explain it that well in simple terms.
Really interesting. I had encountered symmetries and conservation laws before, of course, but it had never occurred to me that they'd be related like this. It makes sense, and I think the rotational example is probably the best one in order to understand it. If an object behaves the same no matter where in its orbit it is, then logically it'll always come back to where it started, and angular momentum is conserved. It had never occurred to me though. Thanks for explaining it! On the second question, I don't really understand supersymmetry beyond that my physicist brother thinks it's hilarious that we haven't found any supersymmetric particles yet. I have some grasp of what supersymmetry is for, but not much at all about what it is. Looking it up, it seems to be about symmetry between bosons and fermions. But would energy be equivalent in that? If these supersymmetric particles exist they'd have to be of higher mass than particle accelerators can produce, so the boson version of a down quark, for instance, would have to have a much higher mass than it. And if it has a higher mass it has more energy, so then how could it be symmetric? Researching this has left me more confused than when I started... Maybe I'll ask my brother when he wakes up. On examples of symmetries, my music training is kicking in and I keep thinking of things like diminished seventh chords, augmented triads, and whole-tone or chromatic scales. those are translationally symmetric, at least if you don't count octave. I have no idea what, physically, is conserved there though. On the diminished seventh chords at least, harmonic function is conserved, which is actually an incredibly cool feature of those. (Seriously, diminished seventh modulations are my all-time favorite piece of music theory.) But that's not a physical property, it's an observational effect. If you're talking physically, you do go up in pitch as you go, and I believe higher frequency is higher energy, so not really symmetric. If you rotate it around so you're actually playing the same register of notes though I suppose you're conserving something. I don't know if that's valid though, since it requires accepting the octave as a fundamental unit. I may be cheating, but if I am I still don't have any useful answers so at least I'm not a very good cheater.
I've started commenting on videos with my 12tone account instead of my personal account (this one) to keep them a little more separate. I don't know if you recognize me anyway, but I've been watching and commenting for a while and I didn't want you to think I'd just disappeared!
This is an interesting question (I know I'm 6 years late to respond)! I don't have a full answer, but I will say the behavior of pitch classes in music works a lot like a cyclic group in abstract algebra! If you accept the octave as a fundamental unit, you can "quotient" out the octave and just focus on pitch classes, which is similar to if you took all the integers and divided them by 12, concentrating only on the remainder after the quotient. Physically, I think you could interpret this symmetry as a dilational symmetry in frequency (since moving everything a major third up for example is equivalent to multiplying the frequency by the third root of two). However, I can't think of a way you could apply Noether's theorem to this situation because Noether's theorem requires a *continuous* symmetry. For example, if you move forwards 10 seconds, 2 seconds, 0.01 seconds, 0.00001 seconds, etc., it doesn't matter: energy is still conserved in Newtonian mechanics. However, with the chords that you mention, the symmetry is *discrete* in that it only works for specific values of movement. If you move an augmented triad up a major third or an augmented fifth, it stays invariant, but if you move it up a whole tone, a semitone, a quarter-tone, etc., the symmetry breaks, meaning the conditions of Noether's theorem aren't met. Also, I'm not sure if there's an analogous principle of least action (which Noether's theorem depends on) in this situation, but I'm sure there's some way to draw a physical interpretation of this, maybe in studying the frequency spectrum of sound decay over time? This is just wild hypothesizing since it's 2 AM, but it would be interesting to see if there's a spectral equivalent of Noether's theorem since all the examples given (translation, rotation, time translation) are symmetries in space-time while your example is a movement in frequency space. I suspect that if you could formulate an appropriate Lagrangian, there might be some way to apply Noether's theorem, although it would probably have nothing to do with music. It's still an interesting thought experiment though! For an elaboration on the abstract algebra though, check out this cool stack exchange post about Messiaen's modes of limited transposition and group theory :) math.stackexchange.com/questions/4045800/music-and-maths-modes-of-limited-transposition Also, I love your video on diminished seventh modulations
I agree, I also find it extremely beautiful that conversation rules are connected with symmetries of time and space. When I've read about Emma Noether's theorem, it blew my mind!
I am currently a physics major in college. I absolutely love your work! It is so refreshing and clarifying yet introduces fascinating questions to ponder :)
This reminds me of the uncertainty principle, in that everything comes in pairs that multiply to give angular momentum. Momentum * Distance = Angular momentum * Angle (dimensionless in radians) = Energy * Time
More precisely, the pairs multiply to give "action" (which only happens to have the same unit as angular momentum). Planck's constant is a tiny quantum of action.
Thank you, ma'am. I watched this at least 5 times now (I'll probably watch it again, when my head stops spinning), and I feel like I understand much more than I did before, but also like I've only scratched the surface of the ideas presented here.
Here's a little derivation which I used to work out several examples, for the one-dimensional case. Let G be a generator satisfying the Leibniz rule, i.e. G[fg] = fG[g] + gG[f] For the 1-d case, if G is a first-order differential operator is can be written as G = v d/dq where v is some suitably integrable function of q (note d/dq is supposed to be a partial derivative wrt. q, but for the 1-d case that does not matter much anyway). Define w as the antiderivative wrt q of the reciprocal of v: dw/dq = 1/v and let W be the inverse function of w. Given these definitions, and k being some parameter, it can be shown that the operation exp[kG]{f(q)} = exp[ k v df/dq ] can be evaluated: exp[kG]{f(q)} = f(W( w(q) + k )) Particular cases include 1) v(q) = 1; G = d/dq; exp[kG]{f(q)} = f(q+k) i.e. the derivative wrt is the generator of translations wrt q. 2) v(q) = q ; exp[kG]{f(q)} = f(q exp[k]) i.e. q d/dq is the generator of dilations other interesting cases: 3) v(q) = q^(1-a) ; exp[kG]{f(q)} = f( (q^a + ak)^(1/a)) 4) v(q) = sech(q) ; exp[kG]{f(q)} = f(arsinh( sinh(q) + k)) one of my favourites, which is an interesting transformation, asymptotically mapping f(q) to itself for sufficiently large values of q, but skilngully avoids the origin for values of sinh(q) comparable to k. v = sqrt(q^2 - 1) is also interesting. I guess similar versions with the cisrcular functions may also be interesting. Note, I was copying my notes while typing that so I hope the derivation is correct and I didn't copy the mistaken version (yes, there were a few of those...)
Thank you so much for your explanation, I heard about this from Jack Kruse, a renowned brain surgeon who is very interested in quantum biology. Best to you.
The best way I can think of explaining symmetry ts than if in happens, and you observe it from different locations, angles and time, you will always observe the same event after accounting for the differences in location angle and time. In other words the laws of physics do not change based on when and where and observer is witnessing and event. Due to Noether's theorem, this means that we can only disprove the three conservation laws, if we disprove the very idea that the laws of physics are consistent over space and time.
Thank you! I have read about Noether's theorem and I know Einstein really like her work. I think I have a good understandings of the basics. Well done.
If you can, could you (or someone else) summarize Dirac/bra-ket notation briefly, or perhaps just provide links to a webpage that you think summarizes it in layman's terms. As a British High-school student watching your (really fantastic) videos out of simple curiosity, i think it would help immensely with my understanding of the quantum mechanics ones in particular. Thanks so much :)
I began watching this video thinking that Noether was a pun joining the words No Ether, but it is actually a real name! and really apt for the theorem associated with it too! coincidence or what!
Actually Noether was German and we'd usually write it like "Nöther", but due due migration over history, Umlaute in names got replaced with the respective vocal + e (ä=ae, ö=oe, ü=ue). Just want to mention it because I find it funny that English speakers would tear apart the name between exactly the two letters which once belonged together :D
Never heard it before,my friend suggested me to look for noether theorem ,I searched it and I'm glad I found it 💙💙💙. You explained it so well ,thank you 😇🙏
This clarified a lot about Noether's theorem - specifically what exactly counts as a symmetry. I still have two questions if anyone knows the answers: 1. I noticed that, in all three examples, the dimensions of the conserved quantity are ML^2/T divided by the dimensions of the symmetric property. I could easily see there being a mathematical reason for this, since the theorem is about quantities being conserved over time in systems where energy is symmetric, and the units of energy*time are ML^2/T, but I could also see it just being a coincidence. Which is it? 2. If we slide an object across the ground, it will lose momentum due to friction, but the object's energy wouldn't change if it was moved to a different part of the ground. Why doesn't this violate Noether's theorem?
This is a really good video demonstrating physical laws. For me, when trying to look this theorem up, I found wikipedia no help, being dropped into an infinite cycle of looking up unfamiliar terms. This video really showed the essence.
Mindblow at 7:57 when I realise that these line up perfectly with the quantum operators on the wave function! We work out the momentum of a particle by the rate of change of the wave function with respect to position (or translation!) Similarly, we work out the energy using the rate of change of the wave function with respect to time. I can only then imagine that the spin of a particle is calculated using the rate of change of the wave function with respect to its rotation in some sense? Also, please return to your videos! I am only discovering them now and seeing that you seem to be on some quite long hiatus. :-( A clearer explanation of what I was just saying: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-LZie2QC5Jbc.html
Was just about to apply this idea of conservation and symmetries to some logical systems that I have been examining; but then you asked it in the homework. But say we take (in first order logic) the logical expression X Y. Now we can see by the truth table that we could switch X and Y (X on the right and Y on the left) and we get the same truth table. However, if we tried that with X => Y, this is not equal to Y => X (If X not equal to Y of course). So an interesting question might be, what is conserved? Possibly Logic? Well I am going to look in to it, but before I do, I must thank you for bringing these ideas to my attention - your videos are amazing, and while I am sure you hear that all the time, I suppose it doesn't hurt to add to the pile of compliments ;).
I've seen a lot of videos and read a lot of books explaining Noether's theorem. Considering the balance between completeness and simplicity, this video is by far the best.
Thanks for this video! I've heard of this before but I just noticed an interesting thing watching this. I realize that in a lot of formulas, you see position and moment together, and energy and time together. For instance, I've seen that the Heisenberg uncertainty principle can be expressed in terms of position and momentum, but can also be expressed in terms of time and energy. We also know that the momentum of a light wave is related to its wavenumber (cycles per distance), and the energy of a light wave is related to its frequency (cycles per time). Perhaps this is what you mean when you say that more appears in quantum mechanics. Very cool!
I knew that therr was something in Physics that had to do with symmetry implying a something, but I was not sure. I went to RU-vid to find a video explaining it. And yours was bu far the best. Thank you. I have subscribed to your channel, and I am working my way threw all of your videos now. I love them. Keep up the great work!
" The most profound and far-reaching idea" is quite ambitious even thoug Thank you for this explanation, such a brilliant carrier that Emmy had by the way.
Split your symmetry into more symmetries. Will make understanding the curves a lot easier and more intuitive. I think like this since I was a kid I have no academic training and I'm working on creating my own theory of quantum time relativity to disprove Newtonian mechanics and Einstein's theory of general relativity with just this concept xD I naturally split each rotational and translational 3D axis into symmetries to understand motions at the core of my perception. I research to understand what I don't know yet when it's not logical, but it's just like a game of pool with magnetic and other forces acting upon the symmetries, either linearly or exponentially, while observing the laws of conservation and other transformations/reactions based on "sacred" geometries and energy charges. Entropy takes care of slowly shifting the balance and we perceive that as time, but it is a local phenomenon and not a linear universal axis translation like so many people think of time as.
wow great video. i kinda had a thought abt something similar before. if we shrunk(or expand) everything in our universe by decreasing the size of atoms by a constant, would that be symmetric. i think it would be and if yes what if we could put normal atoms and minimized atoms in a room together how would they interact with each other? Another unrelated question: life(or intelligent life), exists and has a size of some huge number times the size of the atom. but isn't that sort of arbitrary.Could life evolve in the quantum world. but then it wouldn't exactly be " life" as we know it but something like that.All organisms we see are based on heaps of molecules(hail carbon). but could life form out of chunks of subatomic particles maybe a bit of quarks, muons, etc ???
Thank you for helping me understand what the theorems about. Although I barely understand what you were saying at all. I strangely get what you are trying to say. Then again, I am just a person who is trying to do a book report about Emmy Noether. Not helpful that I don't even take a physics class. But, then again, thank you very much!
Very nice, my understanding of conservation laws come from Chemistry, so this explanation really rounded out my understanding. In Chemistry, the conservation of matter means that no new matter can form, nor can it disappear. While this is workable and provable, we are guessing that it is not true based on the big bang theory. It is my guess that being that E = MC2 that saying matter and energy are the same thing, that you have proven that it only conserves if there is symmetry, that the size of space itself creates potential energy and that is the source of all matter.
this is very interesting, never thought that there was any reason for conservation laws, much less that they were connected to symmetries. from what i found about super symmetry it appears to be a type of transformation involving particle's spin value, which would imply that fermions and bosons are simply a transformation of this value. however because there are no superpartners that have been discovered, the symmetry is said to be spontaneously broken.
Since you asked us to brainstorm on applications of the theorem: what if the transformation was not applied to the object? Could the theorem be applied to an apple’s reflection in a mirror? The object is symmetric when transformed by an external agent, the conversation law is the conservation of edge conditions in the medium of translation. The physical object is no longer physical in the image but preserves the edge conditions in the virtual image. This symmetry also conserves angular incidence via reflection or refraction.
Could you infer that if there was symmetry and conservation of some law, then there must be a transformation being applied even if it is imperceptible? For instance, your hands are symmetric about the median plane of the body and they almost perfectly preserve the boundary conditions of the first hand, therefore could we infer that there is a transformation being applied, even though it is imperceptible (such as DNA)?
I've seen a few other of your videos, but this one earned the sub. Math and physics with a solid backing in both? Absolutely. As a CS/Math person (who spends most of his time where the two look like the ssme thing), I would almost venture to say that I'd prefer a mechanics built from conservation laws implied by symmetries, because symmetries are the building blocks of the algebra we use to solve the problems in the first place. I think I'll spend the afternoon poking at Noether's theorem with respect to multivectors and geometric algebra. That angular momentum conservation due to rotational symmetry is begging to be described in terms of bivectors, heck, kepler's laws probably drop out if the orbit's spinor varies in time...
I asked my professor about this. he said that symmetric of time mean you can do that same experiment at any given time the result will be the same, symmetric of space mean you can do at any given position in space, it will be the same (similar to Einstein's relativity). I have a very strong sense that if an object doesn't move, I cannot know its velocity, similar to if time doesn't past, I cannot know its energy. It sounds very difference from the symmetric above, but when I try to combine it (and possibly doing some math), my mind got very fuzzed and still stuck in the idea for month. some other conservations like the conservation of charge, came natural as the other way to speak that the divergence of a curl is zero (derive from Maxwell's EM equation) which is a kind of symmetric. Feynman's lecture has just been published public so i might spend time watching it.
This video is far better than all the ones I've seen. Explain what the usefulness and not restating that ball go straight if thrown in a vacuum. Kids new that without seeing a theorist! I'm now much less frustrated.
Thanks a lot for wonderful and simple way the concepts are explained. The next question would 1. What there are certain symmetries ? 2. How is co ordinate translation related to this .
These symmetries exist because the world is geometric and relative. Why is the world geometric? We don't know. It is relative because it is mostly empty at this time. During the early phases of the big bang the density would have been so high that relativity would have only be relevant at really high energies and short distances. All the low energy creatures would have been living in a thick soup of matter and radiation. The universe would have looked more like the bottom of the ocean than space, the final frontier. The important thing to know is that even a geometric universe is not enough. Geometry is a very subtle interplay between fully symmetric and fully anti-symmetric geometric objects. In particular, a fully anti-symmetric vector product (cross product) only exists in three and seven dimensions. It is this cross-product that enables stable planetary orbits (and only in three dimensions). Heat flow on dimensions other than three will, if I am not mistaken, also lead to near insurmountable stability problems with systems like star-interiors and it will severely limit the thermodynamics of living organisms in fewer or more than three dimensions. So basically everything that we see around us depends on this three dimensional geometric background. Don't get me wrong, there might very well be other interesting solutions for highly organized universes in different dimensional spaces, but they won't look anything like this one. And so when you go down the rabbit hole of these kinds of insights in physics, then you will discover that almost everything that we know about nature depends more of less on the existence of this three dimensional physical vacuum in which relativity is the dominant symmetry. The reasoning just gets ever more complicated. What I told you above can be discovered with little more than high school math. The consequences for quantum field theory, on the other hand, require much more demanding mathematical tools to unearth and we are far from done with that work.
Thanks for putting this up. I believe there is a more basic version of this that predates Noether's theorem by many centuries, because you can easily make an argument by contradiction, where you suppose that a certain quantity is _not_ conserved, under a certain transformation, and then it follows that by measuring that quantity, you could tell where you are on the particular dimension on which the quantity varies. If there is no way for you to tell where you are on that dimension, by measuring that quantity, that means that the quantity is conserved under transformations on that dimension. If you could tell, by measuring a certain quantity, that means there has to be at least some type of difference between one place and another place on that dimension, in other words, an asymmetry. For example, the temperature outside is not conserved, from one day to the next, and consequently, you can look at a thermometer for 30 days in a row, and (in case you had no other information) you could ascertain the position of the earth in its solar orbit. That tells you that the position of the earth in the summer can not be exactly the same as in winter, and in fact, the earth's solar orbit does not have rotational symmetry, at all, and in fact, the weather is the first way that people knew that (and the positions of the stars, of course). If you applied the same reasoning to linear momentum, which is conserved, on the other hand, you could ask, if you rode a train from one town to another town, whether you could tell how far you had gone by measuring how fast the train was moving. If you could tell where you were by how fast it went, that would mean that our path did not have a symmetry under a linear translation, precisely because you could tell the difference between one place and the next. (I am implying an idealized train, but you see what I mean.) If it were possible on such a train to observe, for example, "We're going faster, so we must be almost there," that would mean that there was a difference between your momentum in the first town and your momentum in the second town, in other words, an asymmetry under linear translation. I believe what Noether's theorem does is merely formalize and generalize this fact, which people had already been applying for many centuries.
This concept is really something. I havent been in school for 20 years and I am kind of angry they never discussed Emmy Noether at all. Wow. I dont see anyone ever talk about Emmy Noether thats sad. Granted I dont hang out in math science circles but I dont think I've ever heard of her on tv or in a book.
It's typically second or third semester material in university, but you are correct. We should make Noether's theorem part of high school science. It's the central piece of modern physics and it can be explained at the high school level.
I'm a newcomer to physics/chemistry/mathematics. My mind is blown @how amazing these things are. I even have a solar-powered Einstein figure that points to his head as if to say, "Use your brain!" But symmetry? That's a whole Noether thing...
What do you think of this idea? - I think this idea and your explanation are beautiful and powerful like Euclid's "Things that are equal to the same thing are equal to each other"(paraphrasing Lincoln's usage . . from the Daniel Day Lewis movie). I am a volunteer STEM educator and insights like these help keep me going. Thank you very much. Have you heard of it before? - Yes but the Wikipedia article didn't bring me the intuition that your video did.
Proof (in classical mechanics): A symmetry of a physical system is defined by: Say we have a Lagrangian. If, under some transformation q'= q+f(q)δ , the Lagrangian is invariant (remains the same), or equivalently, the change in the Lagrangian δL=0, then the system is said to be _symmetrical_ under that transformation. In canonical coordinates, from the chain rule, δL= ΣdL/dqi δqi over all i (unfortunately, due to circumstances beyond my control, I cannot write i as a subscript). With some manipulation, this can be written as: The time derivative of pif(qi)δ. (confusing, I know) Since δL=0, The time derivative of pif(qi)δ=0 Thus pif(qi)δ is conserved. Did I do this right? Tell me if there's anything wrong.
I think you can take it a step further and say that symmetries can also imply potential energy. For example, an object going up will have the same speed when it falls back down to the same height; therefore gravitational potential energy exists. Or an object moving at a spring will have the same speed when it gets repelled back to the same distance, thus elastic potential energy exists. I’m not sure what to call that symmetry though.
Some of the comments below indicate some confusion that has it's origin in an inaccuracy (false statement) in this video. A system is not considered "symmetric" if it has "the same energy" when it is (for example) "moved to over there," but rather when it "looks" in all relevant aspects (i.e. in a sense "IS") the same after it is "moved to over there." It's not the "energy" but the "being" (appearance in all relevant ways) that is " the same" after the transformation. (Energy by itself, is of course "just another conservation law" that is the result of "remaining the same" under a translation in time, and thus falls under the same framework of "symmetry corresponding to a conservation law," as also postulated (discovered?) by Emmy Noether...)
Well from Classical mechanics, related to what you said in descriptions, Lagrangian and Hamiltonian essentially is the same as Newtonian. But I think these 3 approaches have different perspective so useful in different situations. I think since the theorem can be directly related to the action of the system, it is natural to think of it in Lagrangian framework. But I believe Noether's theorem is more general in mathematical sense as a great contribution to abstract algebra, as you indicated in the video. Sorry I did not really say much actually but there is a LOT to say about it if one chooses to focus in such route.
Halliday once said that we use physics to make it more easier to understand the nature around us. And to help us, we use tools like symmetries. I aways agreed with him on that, and now that i watched your video i can totally see where and how those symmetries help us. Also, i think we humans use symmetries because of our power to recognize patterns. Thanks very much for adding this
A minor point but at around 0:34 you say that "Symmetries imply conservations" but what you have in the image is an equvalence and during the rest of the video it sounds like you're talking about an equivalence rather than an implication. Other than that this was a very good video for introducing the concept of symmetries and the relation to conservations, of which I didn't know; but you learn something new every day and I say thank you for that! :)
Position/translation corresponds to momentum Rotation corresponds to angular momentum Time corresponds to energy Is this related to Heisenberg uncertainty principle
Thanks for the great video, but can this explain other conservation quantities? Take charge conservation for example. Take an electron as the object (or say, the system) moving toward a proton (just like the translation example about 6:30 in the video), its energy is changed, but the charge still conserved.
Abrar Soudagar - It's hard to focus on what she's saying cause all I can picture is her talking with this big shit eating grin on her face! LOL! Seems like she's about to burst into laughter. I'm like: what's so funny??? Sorry, apparently I get distracted easy.
I see a lot of comments asking for certain explanations to be approached from different directions. Some even say the implication of one thing to another is in the "wrong" direction. However, all the implications mentioned in this video actually go both ways. It may be that people are taught the implication one way only, so upon seeing this video they think it incorrect to go the other. Rest assured the in the examples it may be done in either direction.
Posted this on Veritasium's video on Quantum Entanglement was hoping you could help read it and probably comment thanks :) Before, when I had studied different aspects of QM such as spin and wave functions I thought of them as separate concepts and therefore never fully understood them until I saw this video and decided to compare it to that of "A Brief History Of Time" It was only then that I understood that these concepts and ideas were all intertwined which led to me to discuss the possibilities of FTL travel mentioned in this video. Firstly, this video does not include a mention to or explanation of (collapsing) wave functions but what it did mention is that these "entangled" particles do not have any hidden information within them which would therefore mean that examining the spin of 1 particle in a pair would directly influence the outcome of the results found in observing the other particle instead of just revealing hidden information in the particle as previously thought. Now let's incorporate the concept of collapsing wave functions.So the examination of a particle or said particle's spin collapses the probability of it's position into one state, now if knowing the state of one of these entangled particles doesn't just tell you the state of the other but has a direct influence on it therefore collapsing the wave function of one particle also collapses the other particle's two possible states (spin up or down). Now when one examines a particle such as one in the double slit experiment it INSTANTANEOUSLY collapses to one state(passing through only one slit) So therefore these entangled particles do carry "information" over an arbitrary distance faster than light if not instantly. Interested to hear what people think :) P.S it helps to not think of spin as the traditional sense of the word but rather as a way of saying different states a particle can be in (for explanation's sake)
Not entirely on topic but i feel like more and more evidence suggests that the universe isn't a bunch of a independent pieces as is intuitively thought, but rather one connected system and the idea of duality/supersymmetry seems to be most fitting at this point.
I have heard of invariants in Mathematics in combinatorics and game theory. The idea is to look for some mathematical quantity that is invariant under an operation. This helps a lot. For example, in the game Nim - the invariant is the XOR of all the number of stones in each pile !
since spin is conserved, there exists su(1) su(2) symmetry. since charge is conserved, su(3) symmetry exists. find really good su(4) su(5), su(6) symmetries... like super symmetries, then you can automatically have what physical property was conserved in that symmetry, then people will bow down to your intelligence.
I still dont really know if I understand this correctly. Is the basis of it that there is a direct correlation between symmetry and conservation? That if something is symmetrical is has conserved its energy? Also...not sure I understand why this is so important.
A theorem of great scope yet a simple idea, sounds exactly like something a mathematician would come up with. It should definitely be mentioned in the classrooms in some form. I've heard of super symmetry and CPT symmetry, stuff to look up. So a system like a falling apple doesn't have translational symmetry but it can still have time translation symmetry since the total energy is conserved.
About the symmetry in time: If we let a system evolve in time, for instance say mixing of black coffee and milk in an isolated cup. Obviously we see that the system is not symmetric in time yet the energy is conserved (I think!). How can we explain this?
Translation is like position. Hey can you notice something : we know from uncertainty principle that position and momentum can not be determined simultaneously and precisely also energy and time cannot be determined simultaneously. Look translational symmetry implies conservation of momentum and time symmetry implies conservation of energy. How beautifully are they related. Does this point toward a great truth of understandings of the universe of is it just a coincidence? Are they really related??
Universe has translational and rotational symmetry which means momentum and angular momentum are conserved. But Universe does not have time symmetry (it is changing with time) which would mean energy is not conserved!!!??? But energy is conserved in an isolated system. Any explanation?
I should have put that better. Empty space has translational and rotationa symmetry. Does this mean there is a conserved property associated with empty space?
I thought about what would be conserved under mirror symmetry (like turning a right hand into a left hand), but that symmetry is not continuous... at least not in 3D. Is there a dimensional limit to Noether's theorem? Could we apply it in a 4D space, for instance? (assuming a 3D-mirror symmetry would be continuous in 4D, which I _think_ it would) And then, what sort of stuff would be conserved there? Could we then project it back into 3D-space, even with some losses? I'm going to have to look all that up, aren't I? Yeah, yeah... :D Anyway, the video was amazing once more. I'd known of Noether's Theorem before, but hadn't thought about it as much. Thanks!
If an apple falls, its kinetic energy changes and potential energy as well. Thus total energy is conserved. I did not get the idea why we looked at only potential energy in that example and said that energy is not conserved?
Great video. There is a debate about whether or not time exists. My theory is that when we break the symmetry of empty space, time appears. If this is true, what property is conserved? Can we prove it mathematically? I would be grateful for your help. Richard
I didn't know that relation, you've oppened doors for new reflexions, thank's a lot ... it's as strong as the Heisenberg uncertainty, the symetry of Quantum properties and mass-energy-speed-time relation, it must be linked somehow … One thing I'm sure : your video is awesome, they all are, you manage to explain complex concepts in a very pleasant, elegant and graspable way and with nice little real life examples and illustrations.
So what's the generator of reflective symmetry? And what symmetry does conservation of charge correspond to? I know you mentioned that there's a mathematical way to translate back and forth, but alas, I think the maths may be a bit above my head. My guess would be that conservation of charge might be related to the fact that there's no intrinsic "zero point" of electric potential (voltage) - we define zero as ground for convenience, but if you raised the potential of the entire universe by 10V, nothing would change.
