Noether's Theorem

[@PiercingSight]

"Having just discussed angular motion, it's time that we go full circle." lol

[@jezzabr]

Let's be honest...noone skipped to 6.45

[@gustavoalexandresouzamello715]

Really great content! I would love to see some video about the derivation of the Principle of Least Action.

[@colinmurphy9883]

I would love to make a video about the principle of least action! I'll definitely consider that.

[@deeptochatterjee532]

If the amount of energy in the universe is dependent on time, that's not using Noether's theorem, that's just using partial derivatives. If you wanted to use Noether's theorem you would have to show that the lagrangian has explicit time dependence

[@colinmurphy9883]

as I understand it, you are right. However, proving that the lagrangian is time dependent would require tensor calculus and general relativity, correct? I think if that is true, it’s fair to say that that is beyond the scope of this video. I should have notated it better. Thank you for your feedback.

[@Diamondketo]

Agreed, the proof at the end is unsatisfying Continuous symmetry (A) implies conservation of quantity thru time (B). Video did not use A at all to imply B but instead B to B?

[@javiercampoy1707]

Nice video, both from the informative and the divulgative perspective. Just a small tip. When shifting from one ecuation to another, it might be a bit confusing the way you use animations. Maybe just clearing the terms or making them appear from nowhere instead of moving the white "ink" from one point to another might be a bit less confusing. Anyway, great video, you earned a sub :)

[@rabiekheiri1833]

great video I have a question What is entropie?

[@colinmurphy9883]

You'll often hear entropy referred to as the amount of "disorder" in a system, and even more often, you'll hear that it is a quantity that must always increase. I think the best way to picture what it actually "is" is from the perspective of statistical mechanics. From that background, entropy is closely related to how many states tiny atoms that characterize the system could take on while still having the same macroscopic properties. For example, if you are an atom inside a lump of iron, your life is pretty much decided for you already. However, if you're an atom jostling around in a cloud, you could wind up in a thousand different configurations and the cloud may still have the same macroscopic properties. Therefore, the cloud has more entropy. Entropy must always increase because it requires much more work to put atoms back into set configurations then it does to let them out. It is much more probable that if you leave some gas in a room for a thousand years, and open the door and check on the gas, the atoms of the gas will be randomly spaced throughout the room (provided that you kept the room above 0K), and not in a uniformly-spaced cube. Therefore, in order to have entropy not increase in a given system would require that you do work on the system, which itself increases entropy. See what I mean?

[@colinmurphy9883]

Alexander Winemiller that’s a great way of thinking about it! I had never heard that before!

[@toomiscalbi4412]

Found u through reddit, subscribed!

Great content!! Have you considered making some series of videos in 3b1b essence of [...] style? I'd love an essence of quantum mechanics series explaining some QM basics for those of us that are studying an intro course in uni. Keep up the good job!!

[@fernandogarciacortez4911]

Came from reddit! Awesome!!

[@12jgy]

Good video! Just one tip, while you are going back and reviewing what we already saw, it might be good to show the equations and the manipulations you're talking about again. And by the way, I'd love to see a video on the Principle of Least Action (Maybe make a series on Lagrangian and Hamiltonian Mechanics?). Another series I think would be cool is one on Special and General Relativity (and maybe on it introduce some of the concepts of Tensor Calculus).

[@alegian7934]

Really clever concept. What wasn't clear to me, is how you pick the constant. For example, if the system is invariant to x, why is mx' conserved and not some random cx'? Why is it mass specifically?

[@colinmurphy9883]

The simple answer is: because of the Euler-LaGrange Equations. The better answer is probably something like this. The Euler-LaGrange equations are basically a restatement of Newton's Second Law. When we are taking the partial derivative of our Lagrangian with respect to time, you can think about what we are doing as checking to see if there are any external forces on our system. Therefore, if there are no external forces on our system (because our Lagrangian is invariant with respect to position) our momentum must be conserved. Lagrangian mechanics also applies to all "generalized" coordinates, which means we could follow the same procedure if we were, say, dealing with a rod rotating about its end. If we took the partial derivative of our Lagrangian with respect to angle, we would then see that the moment of inertia times our angular velocity is conserved. Hopefully, this clears things up a little bit.

[@alegian7934]

@@colinmurphy9883 Thanks for the reply! I guess the philosophical answer is just "because kinetic energy depends on mass". Since thats what we used in the E-L equation.

[@deeptochatterjee532]

Suppose you have a set of coordinates q and some lagrangian that is invariant with respect to a global (just means everywhere) continuous symmetry. Let that symmetry transformation be T. When you perform that transformation by an infinitesimal amount, your coordinates will correspondingly change infinitesimally. Generally, the degree to which these coordinates change will depend on the particle's position The change in the i_th coordinate then will be some function f_i(q)*δ. Here δ is some infinitesimal amount by which you perform the transformation (if it's translation, δ is how much you translate by. If it's rotation, δ is how much angle you rotate by) and q is denoting the position of the particle. There's a different f_i for each coordinate. If you work out Noether's theorem, it tells you that the conserved quantity implied by the global continuous symmetry is Q= [Sum over i] f_i(q)*p_i p_i is the i_th component of the momentum in this case, but more generally it is the conjugate momentum to the i_th coordinate. We define this as dL/dq_i = p_i. In the case where L=T-V, we find p_i = mv_i. So consider if the lagrangian is invariant to x translation. Then the x coordinate of the particle shifts by δ (no matter where you translate from), and none of the other coordinates will change. So f_i(q) = 1 when i corresponds to x and 0 whenever else. Thus, the conserved quantity is Q= p_x

[@frigorificoespecial6497]

Awww, great video! You did this video in an interesting dynamic way while also explaining things so well that even a dumdum like me can understand. Love it!

[@nabinchapagain7311]

If it is possible, could you make a video on tensors (from EnM)? Can’t seem to understand it at all.

[@bean8287]

So is it possible the the energy of matter and charges is dependent on time in such a way that makes the total energy of the universe independent of time?

[@sahilshah4141]

Great Video! Could you make one on the Hoyle state in 12C? Also, how can I support your videos?

[@Unidentifying]

Interesting, it could imply there is some external force working on the universe, to balance out and conserve the energy

[@prestieger]

is this for the breakthrough junior challenge?

[@koktszfung]

What field in physics do you like the most?

[@colinmurphy9883]

It's really quite hard to pick a favorite. I think the actual physics (e.g. what is going on) is most interesting when you're dealing with quantum mechanics because of our total lack of intuition of what "should" happen at that scale. I think the most elegant branch of physics, evaluated solely based on personal opinion, is general relativity. I think that the most fun branch of physics is fluid dynamics because problems that appear deceptively simple quickly turn into problems that someone will pay you a million dollars to solve (read: the Millenium Problems).

[@benlev3375]

Assuming laminar flow simplies the thermodynamics a lot. Too bad the differential equation is difficult to calculate analytically.

[@koktszfung]

@@benlev3375 I think simulation, or calculate numerically, is where the fun begins

[@toopteeps6653]

Do you use manim to animate this? Great work!

[@jonanderirureta8331]

A physics themed channel with the aesthetics of 3Blue1Brown? Suscribed

[@omkarphasale5285]

Great video.

[@richardzhou2659]

You're gonna blow up dude, I'm calling it

[@fabiangn8022]

Gracias.😊👍🏼

[@rondai4019]

Moreover, perpetual motion machines are possible and thermodynamics laws can be violated

[@EricTai845]

Love it! Keep up the great content!

[@Arkunter]

Woah dude amazing! Can you make your videos a bit more mathematically rigorous? Awaiting one on the principle of least action.

[@MartijnvanSchaardenburg]

This is great, but the way that each equation got totally destroyed when changing any part of it really made it a lot more confusing than it should be. When terms are equal to zero, just fade them out.

[@nnniv]

this is some high quality stuff right here

[@Neura1net]

Cool Video

[@nimarjeetbajwa4322]

You are soooooo GOOD thanks

[@discoveringthegardenofeden7882]

Mmm. No. That logic in the first minute and thus baseless mathematical assumptions following from it is so backward and flawed, I can't believe it is being taught. The reasoning is that Energy is not conserved because the universe expanded and the universe expanded because we assume, due to Einsteinian math, that energy must not be conserved. Edwin Hubble never concluded his observations signified expansion and he warned explicitly against that simplified interpretation of wavelength change which knows many causes. The reification of Einstein's mathematical equation leads to undue complication in the form of extra assumptions. The Emmy Noether idea was used to try to save the day by saying that ‘if not everything is conserved, some things can still be assumed to be, here is an equation for it’, although she phrases it in a more complicated mathematical manner. All this because of the faulty and disproven assumption of the Lorentz transform in Einsteinian relativity. The observation of the existence of Einstein rings, whose lack of a tapering off behavior does not follow the predictions of Einstein, but were already a prediction of Classical mechanics, disproves the assumption underlying the Lorenz transform. If places and matter in space can bend light, which they can due to difficult to reliably quantify distant density gradients, and annulling effects, we cannot reliably use light to say anything about distance and velocity because that would only introduce extra assumptions (or the introduction of calibrations, which are guesses). Hence it undermines any basis for Einstein's relativity logic that depends on the former to be true. You have to start from the recognition that the universe is lumpy, and so is all of space filled with density gradients. Measuring speed cannot be done reliably. Hence we cannot reliably compare distant reference frames. Einstein's relativity can simply not work. Because of the density gradients, what we can reliably know is limited to our direct surroundings. Because we cannot reliably say what is farther away and cannot exclude its influence on our local area, all local systems have to be assumed to be open. Either as a sink or source for energy. Hence, neither the idea that energy is conserved nor the idea that energy is non-conserved are correct. Because all locally observable systems have to be evaluated with regards to the intensity of their in and outflow. This would also makes sense of Hubble's ideas in another way: in our local open system we measure an expansion, our locality being an outflow, hence a source; but elsewhere in the universe, there might be a sink. The extrapolation however that the universe, beyond the observable boundary, must also be the product of the same singular event called a Big Bang, cannot be reliably made.. On the level of the Universe we cannot reliably conclude whether energy is conserved or not conserved. Conclusion: A New Physics has to start from the idea that all local systems are open regardless of any theories about conservation of energy which are just approximations and always wrong when measured at a finer scale, accounting for the in- and outflow into the system.

[@anupjotbadesha1791]

Reddit brought me here 😉