# Your Daily Equation #25: Noether's Amazing Theorem: Symmetry and Conservation

https://www.youtube.com/watch?v=w7Q5mQA_74o

[00:00] hey everyone welcome to this next episode of your daily equation and today I'm going to focus attention on a very famous theorem proved by the mathematician Emmy Noether which has really had profound impact in physics ever since she established his theorem back in 1918 so here is the hero of today's episode I mean oh they're German mathematician born in me in the 1880s came to prominence in the early part of the 20th century widely recognized as this brilliant mathematician when the Nazis came to power in Germany she left Germany immigrated to the United States joined the faculty at Bryn Mawr College I think I pronounced it right Bryn Mawr Bryn Mawr College and she unfortunately died at a very early age in her 50s from from cancer tragically left us far too
[01:01] soon but she proved and contributed great many theorems to our understanding.
[01:09] him in mathematics in the areas of abstract algebra.
[01:12] I've studied her work outside of physics in Galois theory and mathematics but perhaps the most famous theorem at least to a physicist is what is known as no--there's theorem as if there's only one there are many but it's a theorem that gives us insight into the connection between conservation laws and symmetry as well as I'll now discuss here's a here's a little excerpt from the paper itself as I mentioned 1918 as you can see you know it gets kind of heavy after a while but the essence of no--there's result can really be distilled into a fairly simple formulation which of course can be generalized but I like on as you know.
[02:03] I'd like to pair things down in this series to the essence of the mathematical idea because the generalizations can look far more complicated than than the initial incarnation that for instance I show you but those details while important for the general application of many of these ideas are not so important to get the gist the essence of what the contribution is all about so let's get into then the subject here and let me swap over to my iPad good so we're talking about no there's theorem and I know that I am pronouncing her name a little bit wrong but it's kind of the best that I can do maybe it's new to as neuters theorem something like that but it's gonna sound kind of bizarre if I try to do that so let me just use the standard American pronunciation of no
[03:03] this theorem and the idea is this up.
[03:07] yeah it is up on the screen okay so so.
[03:10] as I mentioned it's all about the relationship between what are known as conservation laws and symmetries and so.
[03:22] let me spend just a moment on conservation laws first then I'll get to symmetries and then I'll show her beautiful theorem that connects the two.
[03:36] all right so what is a conservation law?
[03:38] conservation law is any law that tells us that a quantity doesn't change over time.
[03:45] I mean the most familiar conservation law I think that we're all familiar with is this notion of conservation of energy.
[03:52] you know the energy that you begin with is equal to the energy that you end with if you add up all sources of energy and these.
[04:04] Conservation laws are incredibly powerful.
[04:06] I mean, let me just give you one simple example that no doubt you've encountered various versions of this in your own studies, but it's kind of good just to see it spelled out once, especially in this context.
[04:20] Let's imagine that we have some block that's on a hill, and maybe it's a really complicated looking hill, and the block starts over here and it slides down this hill.
[04:37] Imagine there's no friction, so things are really nice, and imagine that you are challenged with working out what the speed of the block is at the bottom of the hill.
[04:47] Now, of course, one way of going about this, um, the most straightforward, naively, is you simply start with Newton's laws.
[04:55] You've got F equals MA equals M d2x dt squared, and you know the various.
[05:05] forces that are acting on this block.
[05:09] you've got the force of gravity if you got thee the normal force that's the force exerted by the slide along which this block is moving.
[05:15] so you could try to spell these out in detail with the vector components plug them into the differential equation above.
[05:27] from that if you solve that differential equation you could get the position as a function of the time T given that you can take the derivative and you evaluate it at the time when the block gets to this location.
[05:41] so all very doable but a little bit involved a far simpler way to get at the answer is to use conservation of energy.
[05:50] what do I mean by that in this particular case well at the beginning of the blocks journey it is say at rest and
[06:06] therefore all of the initial energy that it starts with comes from gravitational potential energy which we know now I'll just write down here is the mass of the block G the acceleration due to gravity times H imagine that the height of the block above the ground surface is H now
[06:31] by conservation of energy this must also be equal to the energy that the block has at the end of its when it has reached this location over here the cauldron plaque won't change but you know what I mean
[06:47] now at that location the blog doesn't have any gravitational potential because H is equal to zero but it does have kinetic energy which is one-half M times the speed squared and then we therefore have MGH equals one-half MV squared which
[07:07] means we can solve for V multiplied to the m/s go away and V therefore is equal to the square root of 2gh and there we have the answer in you know one line of work no differential equation to solve no F equals MA so it's a real powerful shortcut to getting the answer to the problem that we were looking at so conservation laws the message of this little example really brings home are quite powerful let me give you one other example just for the fun of it let's look at conservation of momentum so let's imagine a circumstance where momentum is conserved and the circumstances under which it is conserved we are going to come to quite shortly but I'll give an example where it is conserved imagine I have say a
[08:08] star in space and it goes supernova
[08:14] imagine that this star just explodes
[08:18] I can't really not good artists but you know imagine that all these little pieces of the star get blown outward
[08:26] this one this way this one that way each one has its own little velocity vector
[08:33] and just draw a few of them so you can get a feel for the issue I'm about to raise which is what if I were to ask you to do what appears to be a gargantuan calculation
[08:49] what if I said to you what is the total final momentum of all those little particles that the star blasts outward into space
[08:58] I want you to sum over all of those little particles the momentum of the ice particle that is flung outward
[09:08] by the star well how do you how do you do that?
[09:11] well you know you could try to work out the detail force that blew the star apart and you could try to figure out how that force exerted and and resulted in the motion of each little tiny constituent that gets blown out where you could try to do that but man that would be tough but using conservation of momentum there is a much simpler way to go about it the total momentum final must be equal to the total momentum initial and what was the total momentum initial well the star was just kind of sitting there not doing anything just imagine that the star to be a little more precise just a solid rock with no motion whatsoever evil just to keep our selves simple in this description and that initial momentum
[10:10] was equal to zero overall the thing was not moving at all and therefore the answer to your question well not your question my question that I asked you that some must add up to zero and we got that without having to do any difference equations no F equals MA nothing complicated at all so again it's just to show that conservation laws are are really powerful they're really important okay
[10:38] so then that takes us to what Emmy Noether the profound contribution that she gave us in 1918 and that profound contribution as I mentioned is this connection between symmetry on the one hand and this important idea that we just talking of conservation laws okay
[11:14] So I've said already a few words about conservation laws.
[11:16] Let me say a few words therefore about symmetry.
[11:20] And there are two important classes of symmetry that are good to bear in mind.
[11:29] There are so-called continuous symmetries and there are also what are known as discrete symmetries.
[11:42] So let me just give you an example of each just to make sure that we are all on exactly the same page.
[11:48] And I should should have some example.
[11:52] I should have set this up.
[11:56] What can I use?
[11:59] Um, well, yeah, sorry, all right.
[12:08] So I do have a ping-pong ball here and it does have some some writing on it, but ignore.
[12:15] the writing imagine this is just a perfectly orange ping-pong ball with no writing on it whatsoever.
[12:21] if that were the case then as I rotate it regardless of how I rotate it it would look the same to you again forget the writing.
[12:32] so the writing does break the symmetry but if there was no writing at all however I rotate this I'm trying to keep the writing on my so I any rotation leaves it looking the same.
[12:41] and by that we mean that the ping-pong ball has a symmetry.
[12:46] a symmetry is a transformation of an object or an equation or the ingredients that make up a physical system.
[12:54] it's a transformation of the system which leaves the system looking the same or we often say it leaves the system invariant.
[13:05] so this is a continuous symmetry because I can rotate it at arbitrary angles however little however big I have a continuum of symmetry transformations.
[13:16] that I can apply to the ping-pong ball.
[13:18] that leave it looking the same.
[13:20] for for a discrete symmetry well any any.
[13:24] symmetric looking object can do let me.
[13:27] let me use my this mug that people.
[13:31] comment on my my dirty muddy water some.
[13:34] this Earl Grey with soy milk.
[13:35] unfortunately has tea leaves and I do.
[13:37] not want to turn it on the side without.
[13:39] drinking it so excuse me I will for you.
[13:42] know for science for pedagogy.
[13:45] let me just drink all the junk that's in.
[13:47] here I do not like to drink tea leaves.
[13:51] but whatever okay so now this is more or.
[13:55] less clean sorry if you found that a.
[13:57] little gross but so this is more or less.
[13:59] clean now and and you'll note that if I.
[14:02] look at the opening the top of this mug.
[14:05] I have a circle here just look at that.
[14:08] circle as I rotate this mug as I rotate.
[14:12] that circle the circle looks the same.
[14:14] the rest of the mug does not look the.
[14:17] same under that continuous transformation of course because I mean forget about the handle but the rest of the mug it has a discrete symmetry by that I mean if I rotate it by 90 degrees it looks the same forget the handle if I rotate it by 90 it all looks the same because of the discrete symmetry respected by the bulk the body of the mug and so this is an interesting example where I have about the continuous symmetry part of it and I have a discrete 90 degree four-fold symmetry for the the rest of the mug you know you can do this with a book you know and why not plug my own book right so you know here here forget about the title of the book it's the shape I'm after you know if I wrote today this way it doesn't look the same the shape but if I rotate it this way it does so I have to rotate it by 180 degrees for the shape to return to the form that I began with those are discrete transformations whoops just knock some stuff over there.
[15:17] you know a more pretty example let me bring that up on the screen here.
[15:23] so here is a you know a prettier example that is the canonical one that people often use.
[15:27] it's a snowflake so you see the snowflake here has a six-fold symmetry.
[15:33] so if you rotate it by sixty degrees each of those points moves over to the location occupied by the previous point.
[15:41] and therefore it looks the same under that discrete six volt transformation.
[15:48] so you have these discrete symmetries and you have continuous symmetries and what no they're found is that if you let me go back over here.
[16:01] if you use continuous symmetries so everything we're going to talk about now is focusing on the continuous symmetry case and which is a different color.
[16:09] so these continuous symmetries are related to me related to conservation laws and that's it's that.
[16:20] connection that I now want to spell out for you.
[16:25] and it's actually in the pared down version that I'm going to focus my attention on.
[16:28] it's actually a pretty straightforward argument.
[16:30] again as I showed you know this paper does get pretty involved.
[16:32] you can take these ideas to a great level of generalization abstraction.
[16:37] but to get the gist of it you don't really need all of that.
[16:41] so what am I gonna do I'm going to work in the context of Lagrangian mechanics.
[16:47] and if you don't know what that is then you're gonna have to just kind of follow along at a 30,000 foot level.
[16:55] but if you want to have little background on that you can look at the episode that we had earlier on the least action principle.
[17:08] where I described the basic idea of Lagrangian mechanics.
[17:11] I'm not gonna go all the way back to review that here let
[17:21] me just well that is a serious em that I have there can I just erase this little part of it yeah it's gonna get in my way let me just do that like that so I'm going to start with a Lagrangian which you will recall either from your own studies or from our previous episode say depends on coordinates their derivative so a dot means DX DT I could also have a T in here but I'm going to pair it back to to keep things simple and imagine that the Lagrangian is a function say of X and X dot and then B momentum is defining to be DL DX dot and what we showed in an earlier episode and again it's standard so many of you have encountered it elsewhere is that the equation of motion that the coordinate X will follow will satisfy as a function of T can be gotten by looking at D by DT DL DX dot and said in that equal to DL
[18:24] DX in essence these are or I could say.
[18:29] this is in this one-dimensional case and.
[18:31] I'm writing down this is Newton's equation.
[18:33] this is just Newton's law.
[18:34] this is F equals M a.
[18:34] DL DX is basically F and.
[18:39] D by DT of DL DX dot is d by DT of P.
[18:43] which for constant mass is just mass times second derivative of X with respect to time.
[18:48] which is MA okay so it's a fancy way of writing F equals MA and.
[18:54] now what we're going to do is we are going to imagine that there is a transformation.
[18:59] a continuous transformation in which we are going to imagine taking X and replacing it by some transformed value of x which I'm going to call X of lambda.
[19:17] so lambda will be our transformation parameter so for.
[19:25] instance when I was rotating the ping-pong ball lambda could be the angle through which I am rotating this ball.
[19:34] say in three-dimensional space so I'm going to take X to be X of lambda and I'm going to imagine that this is a symmetry of my Lagrangian.
[19:46] which means that when I plug X of lambda in place of X in my Lagrangian imagine I substitute X of lambda for X the Lagrangian won't change at all.
[19:58] which means if I now frame this in just the language of calculus it means if I look at D by D lambda of L of X of lambda X dot of lambda that that derivative will be equal to zero.
[20:14] doesn't change at all as a function of land that doesn't change at all as a function of this continuous symmetry okay so with
[20:26] that as a starting point a Lagrangian
[20:28] together with the continuous symmetry
[20:30] what no other is able to prove is that
[20:32] there is a quantity and I'm going to
[20:34] give that quantity a name I don't know
[20:37] maybe pink is phone a call it AI for
[20:41] invariant or you call it C for conserved
[20:43] it doesn't matter it's just a definition
[20:45] of a letter I'm going to define this
[20:48] invariant quantity to be DL DX dot
[20:53] multiplied by the derivative of X with
[20:56] respect to lambda and the claim is that
[21:00] this quantity does not change in time it
[21:04] is a conserved quantity so
[21:06] mathematically then what we need to
[21:08] prove to establish this is that if I
[21:12] look at di DT the claim is no this claim
[21:17] is that this will be identically equal
[21:20] to zero it will not change in time so
[21:25] that's the connection between a symmetry
[21:29] as described by this continuous
[21:32] transformation and this conservation law
[21:35] given by this di DT identically equal to
[21:40] zero statement so now what I want to do
[21:43] is just give a little proof of that fact
[21:48] and the proof is in this very simple
[21:52] pared down version not hard to come by
[21:56] at all we just really need to follow our
[21:59] nose as people say so we're going to
[22:02] just calculate di DT so we're going to
[22:06] calculate D by DT of DL DX dot DX D
[22:14] lambda
[22:15] and now I'm just going to use the
[22:19] product rule to work this out so this is
[22:21] the same as the by DT of DL DX dot times
[22:27] DX D lambda and now let me add in the
[22:31] second term DL DX dot now I can put my D
[22:36] by DT acting on DX by D lambda but I'm
[22:41] going to assume that the symmetry
[22:42] transformation does not depend upon time
[22:44] again one of the little simplifications
[22:46] you can more be more general as I've
[22:48] said before but just to get the gist of
[22:50] it we'll look at that simple case which
[22:52] means I can interchange the lambda
[22:55] derivative and the T derivative and I'm
[22:57] going to do that I'll write this as d by
[23:00] the Atlanta of D X DT okay so far so
[23:06] good now what I'm going to do is oh
[23:09] sorry I didn't mean to bring that full
[23:12] frame sorry let me do this again so what
[23:14] I'm going to do now is I'm going to
[23:16] simplify or perhaps a better way of
[23:18] saying it is I'm going to make use of my
[23:21] understanding of the equations of motion
[23:22] to substitute a different expression for
[23:25] D by DT DL BX dot how do I do that well
[23:28] look I'll scroll right up over here
[23:30] notice that what we had before are these
[23:32] equations of motion that emerge from
[23:36] this least action principle this is
[23:38] nothing but F equals MA in slightly
[23:44] fancier language but the point for us
[23:47] now is the first term D by DT D L DX dot
[23:51] we can now substitute from the equations
[23:53] of motion set that equal to DL DX this
[23:58] guy right over here so let me do that so
[24:01] I'll write this now as equal to say DL
[24:05] DX times DX D lambda and then for the
[24:11] second term I'll write it as d L DX dot
[24:14] and using DX DT is nothing but this guy
[24:19] over here is nothing but X dot this is
[24:21] DX dot d lambda okay that's kind of cool
[24:27] because
[24:29] you will recognize this is nothing but D
[24:31] by DT of L of X and X dot because by the
[24:37] chain rule right the first term that I
[24:40] will get and carrying at that time
[24:42] derivative is d LD x times the XD whoa I
[24:47] said it wrong I'm sorry I didn't mean a
[24:49] T there I meant a lambda that's what I
[24:53] mean good I can even call it out with a
[24:55] different color right so when I do my D
[24:58] by D lambda derivative using the chain
[25:00] rule the first term is 2 L DX DX D
[25:02] lambda the guy I have here but I have
[25:05] the second dependence in the Lagrangian
[25:07] the X dot and when I carry out that
[25:10] derivative it's the L DX dot DX dot d
[25:14] lambda why do I like that I like that
[25:16] because notice that D by D lambda of the
[25:22] Lagrangian is equal to zero in the case
[25:25] that we are talking about asymmetry and
[25:28] my assumption is that the transformation
[25:30] X goes to X of lambda is indeed a
[25:35] symmetry so what we now have then is our
[25:39] proof that this quantity called I does
[25:43] not change in time its derivative with
[25:45] respect to T is equal to zero that's
[25:48] what we mean by having a conserved
[25:50] quantity and invariant quantity and we
[25:54] have gotten this conserved quantity by
[25:57] starting with a transformation that
[26:00] leaves the lagrangian invariant which is
[26:02] what we mean by symmetry and that gives
[26:06] us via the proof that we've just looked
[26:08] at a conserved quantity called i dl DX
[26:13] dot DX d lambda and that's the proof
[26:18] that's how we go from a symmetry
[26:20] continuous symmetry transformation to a
[26:23] conserved quantity now again this is in
[26:26] a very special simplified setting no
[26:29] this theorem applies regardless of the
[26:32] number of particles and number of
[26:34] coordinates it applies to
[26:36] lagrangians that are not for particles
[26:38] but are for field so it's a very
[26:40] powerful general result
[26:42] which I've now shown Yuna the simplest
[26:44] possible case that if you have a
[26:46] continuous symmetry that yields a
[26:49] conserved or invariant quantity now
[26:52] beautiful beautiful result but let me
[26:55] finish up by just giving you a couple of
[26:58] examples often people in the comments
[27:02] asked me if I could just give some
[27:05] examples so these abstract ideas perhaps
[27:09] can be made a little bit more intuitive
[27:13] and this is a instance where giving
[27:17] examples is particularly easy to do
[27:20] so let me oblige and do so so let me
[27:23] look at example number one consider the
[27:25] following transformation that X goes to
[27:28] X of lambda which is nothing but X plus
[27:33] lambda all we're doing is taking a
[27:36] physical system in this case a particle
[27:38] removing it either to the left or to the
[27:42] right we're translating it by an amount
[27:44] lambda lambda positive it goes one
[27:46] direction lambda negative it goes the
[27:48] other direction so imagine that our
[27:50] physical system is invariant under this
[27:53] transformation this is a continuous
[27:55] symmetry of our theory what is the
[27:58] corresponding conserved quantity well
[28:01] for that we just need to go to know
[28:05] there's general result we take DL DX dot
[28:09] and multiply it by the XD lambda so you
[28:13] look at DL DX dot times the X dot d
[28:18] lambda now the X not the X dot e lambda
[28:23] DX D lambda now the X T lambda is
[28:26] particularly simple here because since X
[28:29] of lambda is nothing but X plus lambda
[28:31] this quantity over here is nothing but 1
[28:36] so our conserved quantity is just the L
[28:39] DX dot what is the L DX dot we'll just
[28:45] scroll with me back over here DL DX dot
[28:49] is the momentum so the conserved
[28:53] quantity in this case is
[28:56] the momentum P now does that make sense
[29:01] to you well yeah it does if the system
[29:04] has no dependence on where it is that
[29:08] means it's not being acted upon by any
[29:11] external forces remember if there is an
[29:15] external force forces come from minus DV
[29:19] of X DX and if there is an external
[29:25] potential that is varying say with
[29:28] respect to X so it has a nonzero
[29:30] derivative that means the system will
[29:33] depend upon X that potential energy
[29:37] function changes depending upon the
[29:40] location along the x axis if it doesn't
[29:44] change along the x axis which means
[29:47] either the potential is constant or take
[29:49] that constant state to be zero there
[29:52] will be no force if there's no force
[29:54] then the momentum doesn't change right
[29:57] Newton's second law F equals the PDT no
[30:01] force no DP DT no DP DT no change in P
[30:06] over time P would therefore be conserved
[30:09] indeed will be our invariant quantity so
[30:13] there's a nice simple example where
[30:15] translation invariance in space
[30:17] yields momentum conservation let me do
[30:21] one more example example number two just
[30:25] to do a little bit more complicated
[30:28] example let's do an example in two
[30:31] dimensions now I only derive things for
[30:36] a single X say in this expression single
[30:40] X dot but the generalization is so
[30:44] straightforward that I'm just gonna
[30:45] write it down and you can recognize
[30:48] immediately that it is the
[30:50] generalization so what I'm going to do
[30:54] here is I'm going to consider a system
[30:56] and let's imagine that the system has
[31:01] coordinates say X 1 and X 2 some
[31:05] physical system and imagine that the
[31:07] system is rotationally
[31:10] in variant right so if I take my
[31:12] coordinates x-one and x-two and say I
[31:14] rotate them right oops that's a little
[31:18] funny so let's say I rotate not a very
[31:20] good drawing over here let's say I
[31:23] rotate my system through some angle
[31:26] theta
[31:26] imagine if the system just doesn't care
[31:29] it's completely insensitive to its
[31:31] angular orientation in this
[31:34] two-dimensional space it's completely
[31:36] invariant under that transformation now
[31:39] what would that transformation look like
[31:42] well you know I could write it down as
[31:45] x1 x2 right as a little column vector
[31:49] imagine this guy is rotated and the
[31:53] two-dimensional rotation matrix you may
[31:57] know what that looks like but let's
[31:58] write it as cosine lambda minus sine
[32:02] lambda sine lambda cosine lambda acting
[32:07] on X 1 X 2 this would be what we mean by
[32:11] the vector X of lambda and I'm going to
[32:16] choose lambda to be small the angle that
[32:19] I call it theta over here I'm just
[32:21] calling lambda in my transformation
[32:23] matrix since ultimately I'm going to
[32:25] take a derivative respect to lambda I
[32:27] can choose lambda to be infinitesimal
[32:30] just to make my life a little bit easier
[32:32] and for very small values of lambda you
[32:35] may recall say from our episode on the
[32:40] most beautiful equation the Euler
[32:43] identity that you can do a Taylor
[32:45] expansion and cosines and sines
[32:47] hopefully perhaps even know this without
[32:49] the episode itself but I'll write down
[32:52] the result that you get for small values
[32:54] of lambda to first order
[32:56] cosine lambda is 1 the land that
[32:59] dependence doesn't start till 2nd order
[33:01] and I'm going to be taking the
[33:03] derivative around lambda equal to 0 so
[33:05] that term will drop away sine lambda
[33:08] that just gives me a minus lambda sine
[33:11] lambda also to first order gives me a
[33:13] lambda and again the cosine gives me 1 X
[33:16] 1 X 2 and therefore this is telling us
[33:21] that X
[33:23] one goes to X 1 minus lambda X 2 and X 2
[33:30] goes to X 2 plus lambda X 1 just doing
[33:36] that little matrix multiplication so
[33:39] this is my infinite towel in Venetta
[33:43] simal version of that continuous
[33:46] rotational transformation and now I'm
[33:50] going to assume as I said my system is
[33:52] invariant under this transformation what
[33:54] therefore is the conserved quantity
[33:58] well the conserved quantity we know what
[34:00] that looks like is DL DX dot DX D lambda
[34:06] in the 1d case in the 2d case what is
[34:10] this equal to
[34:11] well 2d I just do DL the X 1 dot DX 1 D
[34:18] lambda plus DL D X 2 dot DX 2 D lambda
[34:25] and now this is something I can easily
[34:29] plug in so the LD X 1 dot that gives me
[34:33] P 1 the momentum in the one direction
[34:37] what is the X 1 D lambda well I get that
[34:41] from right over here it's nothing but a
[34:43] minus X 2 so x minus X to the DL D X 2
[34:50] dot that's the 2 component of a momentum
[34:53] what is the X 2 D lambda well that is
[34:56] nothing but X 1 and therefore I can
[35:00] write this as X 1 P 2 minus X 2 times P
[35:06] 1 that is my quantity that is invariant
[35:10] it's conserved it won't change in time
[35:12] because of the assumed invariance of my
[35:15] system under this transformation under
[35:19] this rotation now what is this quantity
[35:23] called I physically well if you've taken
[35:26] basic physics you will recognize this as
[35:29] nothing but angular momentum
[35:35] and so what we have learned in this
[35:39] simple two-dimensional example if we
[35:41] have a system that's invariant under a
[35:43] rotation then the angular momentum of
[35:47] that system is conserved does that make
[35:50] sense it does because being invariant
[35:53] and not depending on the angle means
[35:55] there must not be any external twisting
[35:58] force right just as in this case for
[36:01] linear momentum the system being
[36:04] invariant under a translation meant that
[36:08] there is no force external force acting
[36:10] on it in this case invariants under a
[36:13] rotation means there's no angular force
[36:17] no twisting force no torque acting on
[36:20] the system and if there is no torque
[36:22] acting on the system no angular force
[36:25] then angular momentum will not change it
[36:29] will be conserved so that's a nice
[36:32] second little example that shows us know
[36:35] there's theorem in action but you know
[36:39] just to finish this up you have this
[36:41] very beautiful result that we now have
[36:45] relating symmetries and conservation so
[36:49] this is really what it is right here
[36:51] that if a Lagrangian is invariant under
[36:55] a transformation come on let me scroll
[36:58] thank you so if a Lagrangian is
[37:01] invariant on this kind of a continuous
[37:03] transformation its symmetric under it
[37:06] then this quantity will not change in
[37:09] time that is the beautiful powerful
[37:12] not--there's theorem that we use in
[37:15] classical mechanics use it in quantum
[37:18] mechanics we use it in quantum field
[37:20] theory you can substitute instead of
[37:22] particle positions you can put down
[37:24] values of fields in a Lagrangian that
[37:26] depends upon fields and in this way
[37:29] we're off and running in terms of
[37:32] getting this powerful notion of
[37:34] conservation laws emerging from now a
[37:38] turn of the crank understanding of the
[37:41] underlying dynamics you give me a
[37:42] Lagrangian has any continuous symmetry I
[37:44] turned the crank and I extract a lot
[37:47] amino there
[37:48] a conserved quantity alright so that is
[37:51] what I wanted to cover today no--there's
[37:54] theorem hope you get the gist of it in
[37:57] these simple examples easy to generalize
[37:59] powerful result but that is your daily
[38:03] equation for today until next time take
[38:06] care