LockenJohny101 lol, my exam is tomorrow and this is not even in my course

same

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.

_"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?

Newton also sort of figured it out. F*t= change in momentum F*d= energy

​@@michaelsommers2356 how do I bookmark this comment?

power of math, so beautiful

Yea, you can kinda tell she can barely contain herself. so kewl.

@@RSolimov 'kewl'?

@@fukpoeslaw3613 cool but with a cooler spelling.

@@SteveGouldinSpain tewk me half a year, but finally I know!

@@fukpoeslaw3613 kewl. 😹

+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!

Looking Glass Universe I think I should sit down with the math then. Lagrangians are weird. Anyway, I'm glad you got excited about it. It showed :-)

:D

Hi crazies!

@@LookingGlassUniverse Nah, AdS/CFT is way more beautiful

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.

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!

+razorborne bro that some really nice text. :)

saleem khatib Thanks!

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

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.

I feel 100% the same! :)

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.

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.

+Maria Ciara Lalata Thank you!! Yeah, I should make a video about that :P

Wow that's so cool! I didn't realize that yet.

Conservation of electric and color charge comes from gauge symmetries of the corresponding fields. Acceleration is not conserved, and does not stay the same in free fall. It may appear to stay the same in some circumstances, but that is just because it changes too slowly to be notices. However, the acceleration of the Moon toward the Earth is not the same as the acceleration of the apple that hit Newton on the head near the surface of the Earth.

In statistics that's a different stuff. It's a function used to compute mean values of things snd measures of deviation from a certain value.

assuming that gauge symmetry is Noether's Theorm but from what I understand they cover the same things

+Xavier Sebastien I know! The first time I saw this theorem that's exactly what I thought!!

+Xavier Sebastien I thought that too! But, more it reminded me of the noyes (pronounced noise) game. So No ether=> Noether and No Yes to Noyes.

+Scott Mc Logic en.wikipedia.org/wiki/Emmy_Noether

NeedsEvidence Nice

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

Ding ding ding! Well spotted. I'm writing those videos now.

come back! your videos are awesome!

Really soon! Promise :)

+Looking Glass Universe woo! I'm excited!

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 :)

Great question! But here I assumed the speed was the same (ie I didn't drop the apple- it just magically appeared closer to the ground).

Lisa thanks for the question I was wondering about that too! And Looking Glass Universe thanks for the clarification :-) Presumably your scenario does not go into gravitational effects of the nearby planet (acceleration) etc. correct?

Symmetry is usually described using groups, I reckon the prof was talking about Lie Groups, there are probably the most important symmetries in physics. en.m.wikipedia.org/wiki/Lie_group Ps: they are pronounced as li group