[Music] this program is brought to you by Stanford University please visit us at stanford.edu now lille's theorem is really at the heart of Quant of classical mechanics it's at the heart of hamiltonian classical mechanics and a quantum mechanical version of it which is called unitarity is very much at the heart of quantum mechanics it will profit us to understand lille's theorem uh because as I said it is the immediate classical analog of one of the most fundamental principles of physics quantum mechanics um it can also be thought of as the idea or generalization of the idea of information conservation we talked about this I think in the first lecture where let me just remind you I think I've done this a couple of times but let me remind you again uh that if you have a phase space which consists of a collection of points by the phase space I simply mean the space of possible uh configurations of a system in the case of a mechanical system it would be positions and velocity these in the case of just a system with a finite number of states it would be any one of those finite number of states and then the laws of Mechanics for this very simple setup would simply be expressed as a rule for going from any uh configuration to any other configuration and um I can draw it in a more complicated way but there's not much point I could draw s of the lines cross but that doesn't add anything to it you could have some disconnected Cycles but generally that's the pattern and you'll notice that each point has one Arrow coming into it and one Arrow going out of it that's the basic idea of information conservation that nothing gets lost there's no let's call it convergence or Divergence of the flow in the phase space this can be thought of as a kind of flow uh what's what's what's the word for what they do in discos where they flash lights at you stroboscope strob you could think of it stroboscopically where each instant uh you flash and the system moves to the next configuration so it's a sort of stroboscopic uh flow of the points in the face space but um you could imagine populating each of these points populating means imagine a system at that point in Phase space undergoing the motion that the system is supposed to have so we would have a collection of systems filling up the phase space one at each point in the phase space and as time evolves the points in the phase space just move have described this I think probably the best way to describe it is in that they move incompressible among other things that means that if you start with a separate point a separate system at each point in the phas space you'll never wind up with two points or two systems at the same point in the phase space it's not like playing Monopoly where the Monopoly men can wind up on the same uh on the same square that never happens they stay distinct from each other uh the points in the phase space they don't diverge they don't converge what would diverge mean diverge would be a situation where an incoming line would split in two now what would that mean that would mean there was some ambiguity in the evolution of the system if it went from here to here it wouldn't know where to go next sometimes it might go this way sometimes it might go that way that doesn't happen so there's no Divergence of the flow and there's also no convergence of the flow convergence would mean that from two different phase points you come to the same phase point then you would always know where to go but you wouldn't know where you came from if you found yourself over here you wouldn't know if you came from here or over here uh those are the Forbidden kinds of flows as I said you can think of a lack of convergence or Divergence on a phase Bas yeah it's quite thinkable it's not that it's not thinkable to have these things they don't occur in classical physics they don't occur except in so far as you ignore things except in so far as you ignore degrees of freedom um as I've said over and over friction is an example where you start a system in any number of ways you start in many ways it comes to the same configuration well not quite but uh you get the idea friction just slows things down brings them to rest no matter how they start they come to rest many many different configurations can wind up exactly the same but that's only because you've ignored things you've ignored the molecular structure of a of a tabletop um you can take this as an experimental fact or you can take it as a consequence of the basic rules of quantum mechanics but all the classical mechanical systems that are of interest in fundamental physics have the property that there's no Divergence or convergence yeah yes it is saying something more than that it is saying something more than that and we'll see we'll see what it says it's what leoville theorem is what it says uh but we'll come to it for example the sort of thing that can't happen is here's phase space okay every point in this phase space is distinct from every other point in some sense now you could take a group of points and follow them and they might all squeeze together into a smaller volume right that's the kind of convergence that doesn't happen instead the flow in Phase space is a kind of incompressible flow both incompressible what's the opposite of compression and non- Divergent well phase space doesn't blow up it doesn't shrink but we we'll come to with a precise definition of it soon enough uh so in some sense the phase space doesn't pinch together or divert it's as though here's the way you can think about it in the Continuum mechanics case you can imagine populating the phase space let's call it P and q q and P and of course there may be many p's and q's I simply can't draw them on the Blackboard and we might imagine in the same way that we populated each point here with a system we can imagine populating the phase space with a dust with a the dust of points and follow that dust of points let's assume that the phase space is populated uniformly with a constant density a uniform density of points of dust uh and we follow each point as it moves through the phase space it defines a kind of flow but it defines a special kind of flow which is called incompressible and it basically has the property that each point in the face bace can be thought of as having a little volume around it each one occupies its own little volume if we take the density to be uniform then each one of these volumes is the same and as we follow it when we see it later each phase point is separated from each other phase Point by the same amount of volume in the phase space or or you can say it another way if you take a group of points there are three ways to say it all of them equivalent the first is to follow individual points with little volumes around them and watch them move okay uh the volume of each little point or the region occupied by that little little Point stays the same but we can generalize that we can say take any region of the Subspace of the phase space to begin with take any region and follow its points follow its points with time after time that phase space region will change it'll change its shape some points move faster in another points uh it may spread one way it may stretch some direction it may compress in another Direction but the one thing that's true is that the volume of that region will stay the same in other words the volume of a fixed set of points in the phase space will not change connect and it will maintain connectedness and it will maintain connectedness absolutely they don't disconnect the connectedness is simply the predictability if it became disconnected how could it become disconnected a phase point would split somehow which would simply mean it didn't know where to go the equations broke down at this point and didn't tell a phase point where to go properly so yeah regions maintain their connectivity in other words a region like this can't break up into two disconnected region and a hole cannot appear so it's a very continuous smooth uh no holes the the connectivity is conserved but the type of connectivity simple simply connectedness multiple connectedness whatever it is the topology of the chunk of phase space is maintained as is its volume what is not maintained in general uh is the distance in the phase space between points take a pair of points in the phase space and follow them if we follow them we may discover later that they're closer together now you say how is that consistent how is it possible that the volume uh of a phase region doesn't change but the distance between points can change well what can happen for example is a volume can stretch in One Direction and compress in the other direction volume being conserved the volume of all of this collection of points in here is unchanged but if you follow each point you find mind that the shape changes but the shape changes in a way that leaves the volume unchanged but in particular this point and this point are now found here and here so they're much further from each other than they were on the other hand this point and this point are found here and here so they're much closer together than they were H points switch no no everything is extremely continuous conserves orientation uh what is that mean Oh you mean can one well you can certainly have a situation where there's a point in Phase space moving to the right and another point in the phas space moving to the right and they start like this and this one is moving faster than this one and uh passes it yes that can happen no question that uh uh that that can happen for example you can have a situation where if you started with a square in Phase space that square could evolve into a what is that called parallel sare a parallelogram of the same area that would happen for example if the top was moving faster than the bottom but uh the width of the whole thing that's a sort of sheer motion where the top shears relative to the bottom that's perfectly allowed that uh that conserves the volume if the height and the width stay the same conserves a volume so that's possible and that would certainly allow a point back here to get ahead of a point over here yeah so uh well go ahead ask the question yeah what's that yeah the double pendulum and a gravitation yeah neighborh absolutely absolutely but but we yeah yeah we'll talk about that case uh that is not a counter example to this right it's um an example of it but one which is less smooth and less uh predictable but for reasons we'll come through okay well let me just say it very quickly I say the space maintains a certain degree of connectivity it maintains its connectivity all it happens is shape changes but the shape can change wildly for example this sphere can evolve into something with I'm trying to keep the volume the same as it was before but I'm probably not know horrible uh so the there's no rule that says that the shape can't evolve into some wildly extravagantly different uh structure which eventually starts to look more and more fractal now it never really forms a fractal because continuity tells you that things which are close to each other will stay close to each other but in time it'll start to get more and more fragmented more and more fractured uh but always maintaining it volume okay maintaining its volume so that distinguishable points remain distinguishable and a measure of distinguishability is the volume surrounding a point but uh we will we will return to this particular picture I ought to leave it there because I'll probably wind up drawing it again but uh not right now that incidentally is the um basic case of a chaotic system chaotic systems do exactly this kind of thing non-chaotic systems tend to maintain a greater degree of shape coherence we can call it say it again no they don't they don't stay in close proximity no no no the statement is the volume of the region in Phase space is maintained so what that tells you is that if two points do diverge or let's say that if the phase volume stretches in One Direction it must compress in the other direction okay so it's not completely without rules uh but the only only real rule is that the phase volume is maintained as time evolves volume volume means the same thing that it does in calculus it means the integral it means break up the region into a lot of extremely small cells Let's uh let's talk about area for the moment let's not try to get to higher dimensions for a minute let's just take the case of one Q 1 P then what I mean by volume of phase space is just the area of the region regions have area irregardless of whether they're squares they don't have to be squares or rectangles to have area uh you have to use a little bit of calculus to add all of these up but there's a well- defined notion of area and as you follow it the shape may change but the area doesn't now if it's a higher dimensional system with more than one p and Q incidentally as always there's one Q for HP they come in they come in pairs but we simply have to think about the higher dimensional version of volume where we break up the volume into lots of little cubes count the cubes and add them up yes Michael yes absolutely that's right nearby means close by in position and momentum but it doesn't mean that they stay close by okay and in particular the chaotic case is the situation where there's a great deal of tendency for points which are close to to each other to diverge from each other but always at the expense of Co of compression in some other direction yeah yeah right where having which happen now oh oh oh oh you said well yeah that's that's basically friction you say you say there are many ways to start and you can wind up with a six on the table yeah right if you didn't have friction it would just bounce off the table and uh and uh uh rebound and it wouldn't stop it wouldn't stop yeah yeah right friction is incredibly common right yes yes and when you're doing a a mechanical system in practice if you're an engineer you don't want to forget friction if you're a physicist you often want to forget friction you want to pretend it didn't exist in fact in many situations microscopic physics doesn't have friction if you really follow every degree of freedom of a system then there's no concept of friction in the equations friction is a collective effect of many variables some of which you just decide I'm going to ignore uh so it's a thermodynamic idea ra rather than a um an idea of basic mechanics of course they have to be consistent with each other when you decide not to look at a collection of degrees of freedom the behavior of it may nevertheless reflect the fact that in some bigger sense um the phase Space volume is conserved uh energy conservation is one which uh you might not realize if you were sliding around things on the table so energy is lost kinetic energy there's no potential energy it doesn't go up or down kinetic kinetic energy is just lost looks like energy conservation is wrong um those who would believe in energy conservation would make a prediction and the prediction would be that if you very minutely and carefully measure the temperature of the table the temperature of the table will go up a little bit and if you know the specific heat of the table then you can find out how much energy was absorbed by the table etc etc and uh and conservation of energy will tell you will tell you that there must have been degrees of freedom that you ignored the same is true of this conservation of information if you discover a system which looks like the phase space is Contracting it means you've left things out of the system now that's a that's an empirical fact in the sense that it has always proven to be the case uh but by now it is uh a fundamental fact derived from quantum mechanics but let's take it as a uh well we're not going to take lille's theorem as a given we're going to prove it but we're going to prove it from something we're going to prove it from Hamilton's equations all right so the beauty of the hamiltonian form of mechanics is this flow picture a flow in pH space a very predictable flow in Phase space where the points move in a very characteristic kind of way and that characteristic way is Hamilton's equations the entire flow is determined by a single function of all of the PS and q's if you know that function then you know the flow on the phase space and if you started a little boat sailing on the flow which means you start a system at a particular point in the phase space that hamiltonian flow will tell you where it is at any future time okay and that flow is incompressible but let's uh let's um uh first Define the flow so this is Dynamics as a flow in Phase space as a kind of fluid flow all right so every Point has associated with it a motion that depends only on where you are and a motion means means a Time derivative this is p this is Q It Means A P Dot and a q dot in other words a velocity but not a usual velocity a velocity in Phase space consisting of the ordinary velocities which are the Q dots plus the P dots which are the time derivatives of the PS okay so we just write down Hamilton's equations P sub I dot is equal to minus d h by DQ subi notice they come in pairs is plus DH by DP d p ofi that's it that's all of mechanics in a nutshell in the hamiltonian form okay all right let's consider what it means for a flow to be incompressible what we're ultimately going to prove is that this flow is incompressible what exactly does it mean let's start with a one-dimensional flow and see what it says so a one-dimensional flow just means you have a line and let's populate that line with a uniform distribution of points for Simplicity I'm taking a uniform distribution that means that the density of points is the same everywhere as along the line or equivalently the little tiny separation between points is uniform along the line what kind of motion constitutes incompressible flow well basically there's only one kind and that's just all the points move together with exactly the same velocity what would happen if the velocity varied from point to point along here for example suppose it went faster over here than over here then there would be a clumping up of points in between okay there would be a clumping up and a increase in their density all right but let's uh let's uh consider the mathematics of it what's that yes yes but let's just talk about flows for a minute for a minute let's forget phase space and just talk about the general concept of an incompressible flow uh and we'll come back to phase space as a special case that's right you're no you're right uh but uh nevertheless it's possible to think of incompressible flows in one dimension they're very trivial all you can have any velocity you like but the velocity has to be completely uniform all across the line otherwise there's a clumping now you can think of it in two ways you can either say take a set of points and follow them let's take a connected set of points like this and follow them one way of saying incompressibility is just that the volume of that particular set of points stays the same all right that's a view of it following the points there's another view of it which stays at a particular place in space let's say we take stay at a particular space in place in space in other words a little volume we're not going to move that volume but points are going to come into it and points are going to depart from it if the fluid is moving to the right of course it could be moving to the right in some places in the left and other places but let's count all velocities as positive to the right if um if the fluid is moving then some points leave this region and some points enter the region to say it's incompressible says that the number of points that enters is the same as the number of points that leaves in any given uh time interval what is the condition for that well it's more or less obvious that that if the velocity let's imagine that the velocity is fast over here and slow over here all right then it's quite clear that the number of points that are exiting the region is proportional to the velocity at this end right let's call this point 2 and point one the number of points exiting will be proportional to the velocity of the points to the right and that we can just call V2 that's the number of points exiting the number of points entering is the velocity at one so the net increase shall we take increase or decrease let's talk about the net decrease the net decrease in points in this volume here is going to be proportional to V2 minus V1 actually it's V2 minus V1 time the density of the points but as long as the density is uniform then uh then we can just write that the yeah velocity High theity got to be changing that's the point that's the point that's exactly the point wait if the velocity yeah if the velocity is higher at this end or whichever way if the velocity is bigger at one end than the other then the density inside has to be changing incompressible means that the density is not allowed to change okay okay so what it says is that V2 minus V1 must be equal to zero that it cannot be a gradient or a variation of the Velocity this must be equal to zero now if I take a small interval let's let's imagine that this is a small interval then the difference of the Velocity at two and one assuming everything is nice and smooth and differentiable the calculus applies for a small interval we can say that the difference of the velocities is just proportional to the derivative of the Velocity with respect to x times some Delta X where Delta X is the uh is the interval here and then we would write that this has to be equal to zero Delta X is just a small interval it's not zero it's just a number so it follows then that the derivative of the Velocity with respect to X must be equal to zero for an incompressible fluid which which is what we would intuitively imagine that the whole fluid has to move together since all of the motion is along the x- axis I'm going to write this as the derivative of the X component of the Velocity there is only an X component of the Velocity but uh just uh to use a notation which will generalize I'll write that this is dvx by DX is equal to zero that's the condition in one dimension for an incompressible fluid and it is very trivial questions good all right now let's move on to two Dimensions I'm going to call the two directions X and Y why not X and Y now let's ask in a small interval of time what is the increase or the decrease I guess uh V2 minus V1 would be the decrease in the number of points what is the decrease in the number of points in this little square here let's take the square to be of size Delta X and Delta y it's not a square it's a rectangle how many points are entering and leaving from on the various sides of this uh uh rectangle all right let's start with a number of points coming in here again we're imagining that the density is completely uniform so the number of points per unit area to start with to start with is completely uniform if it's incompressible then the number of points per unit area must stay the same must be exactly the same after afterwards so okay so how many points come in from this side here well the number of points coming in this way per unit time is clearly going to be proportional to the X component of velocity the Y component of velocity a velocity in the vertical direction will not contribute to points Crossing this boundary here so so the X component of motion here VX at the left hand end here will constitute the incoming points coming in from the left vertical side of the square what about the number going out oh there's another Factor there's another Factor anybody know what the other factor is Delta y yeah the size of the Y interval here obviously the bigger the Y interval for a given velocity the more points will enter there so there's another factor which is just Delta y now how about the number of points going out on this side that's also proportional to VX but VX at the displaced Point all right so we have to move ahead a little bit and subtract the points coming out on the side exactly what we did over here the answer is going to be the D derivative of VX with respect to X time Delta y times Delta X where does this Delta X come from the difference of the Velocity here and here is the derivative of V with respect to X time Delta X right the difference between the X component of velocity on the right hand side of the square and on the left hand side of the square is the derivative of V with respect to X time Delta X we have to multiply that by Delta y to find the NIT number of points coming through this coming into on this side minus out on this side and that's this okay so this is the net number of points coming into the rectangle through the vertical edges now what about the horizontal edges the horizontal edges things can also move in but now the X component of velocity is not relevant to particles coming in in this Direction only the Y component of the motion is relevant on the horizontal boundaries so how many particles come in from the lower end well that's going to be V suby at the lower end times Delta X but then we have to subtract off V suby coming out of the top end here in order to find the difference of the incoming points and the outgoing points on the bottom and top so that will give us the derivative of Vy with respect to what with respect to Y because we're going from bottom to top times Delta y y the Whole Net change in the number of points in this rectangle is simply proportional to Delta X Delta y that's just the area of the little region times a quantity which is called the Divergence of the Velocity field this is the velocity field it can vary from place to place but it's definite and it's fixed this is derivative of VX with respect to X plus the derivative of Vy with respect to Y if this is positive it corresponds to a decrease in the density what if it's an incompressible fluid if it's an incompressible fluid then the net number of points coming in and going out must add up to zero all right we can forget the Delta x * Delta y y that's just a number for a little area and so the condition of incompressibility is that the Divergence of the Velocity field is equal to zero let's drop this Factor here it doesn't do anything for us dvx by DX plus dvy by Dy is equal to zero this is not a general rule about all possible flows this is a rule about incompressible flows that start with uniform density and maintain the uniformity of the density notice that it does not say that the velocity can't vary from place to place but it does say that if the velocity is increas if the X component of velocity is increasing in One Direction then the Y component must be decreasing in the other direction so it does not say that the velocity can't change from place to place it doesn't require the motion to be just rigid motion uh more more complicated kinds of things can satisfy this kind of flow but uh it is constrained in this way okay questions yes yeah but we we take an instantaneous photograph of it and we ask at this instant what's the time rate of change of the number of points in here so at any instant some something which is coming in here yes it could it could turn around and go up here that's perfectly acceptable but that will be at a later time that it'll be found found going out here so we take imagine a snapshot instantaneously right now what is the time rate of change of the number of points in this volume okay so if a point is coming through here it's not going out there it applies to any number of dimensions so in any number of Dimensions we could write there aren't enough letters in the alphabet but uh I some X Y and Z here just refer to a general space having nothing to do with mechanics just a general flow in any number of Dimensions the general rule for an incompressible flow would be that the derivative of the I component of the Velocity with respect to the i x is equal to zero right that's also written in the notation some of I thank you that's also written in the notation that the Divergence of V is equal to zero Divergence simply means this act of differentiating the I component with respect to the I coordinate and summing them up up and it's called the Divergence yeah I've assumed uniformity of the density yes yeah um if I hadn't assumed uniformity of the density I just did that to make it simpler had I not assumed uniformity of the density what you would have to do is invent a new quantity which is the density at any point times the velocity and then that would be the thing which would have zero Divergence but if the density is uniform then that factors out so just for Simplicity I imagined that the uh and after all if it's an incompressible fluid it stands to reason that its density is everywhere as the same since you can't change its density so um let's come back to phase base what are the X's the X's are are just the coordinates in some space in the case of phase space the coordinates are the p's and q's right the p's and q's are the coordinates so we should identify the X's not just with q but with a p and a q so the number of x's is twice the number of Q's One One X for each P and Q that's the space we're dealing with if there's only one Q then it's a two-dimensional space one p and one q and so forth right what are these quantities over here Pabi Dot and qabi Dot those are the local velocities in the phase space they are the v subis i now goes from 1 to 2 N it runs over all of the coordinates in the momenta but these are we could call this here we could call it the P sub I the velocity along the axis P sub I this we could call V Q subi so the P dots and the Q dots are the things that are being called the um uh the velocity now let's calculate the Divergence of the flow we're going to calculate the Divergence of the flow and of course what we're going to use is Hamilton's equations to see that the Divergence of the flow is exactly zero all right so the Divergence of the flow we take v p subi and we differentiate it with respect to P subi so let's uh let's write the Divergence of the flow over here first of all we have derivative of the P I'm not going to bother writing the I well maybe I should no let's let's let's just do it for two Dimensions first then we'll come back and talk about it for any number derivative of VP with respect to p with only one p and 1 Q what is that well that is equal to the derivative with respect to P times VP but VP is minus the derivative of H with respect to Q all right that's the first term what's the second term in the Divergence the second term is just DV Q by DQ all right what is VQ that's over here plus DH by DP so this becomes d by DQ of plus DH by DP if we have several coordinates in momenta then we would put an i here and some over I everywhere also in the lower equation here I won't bother with it though well now you can see what happens partial derivatives can be interchanged the derivative with respect to p of the derivative with respect to Q is the same as the derivative with respect to Q of the derivative with respect to P the P the Q is the same as DQ DP it doesn't matter which order you differentiate something in right this one has a minus sign this one has a plus sign so these cancel and pairs they cancel and pairs in fact it's not just that the Divergence is equal to zero it's equal to Z Z in a special way where they cancel in pairs but that's okay the Divergence the sum of these the P that's exactly equal to zero okay so the sum of these add them together that's exactly equal to zero and that proves that the flow in Phase space is incompressible that is a quite a profound theorem about uh about mechanics the incompressibility of the flow the fact that uh the phase points can't get squeezed into a smaller volume that they maintain sort of they have elbows and they keep uh from running into each other that is probably the most important uh fact about hamiltonian mechanics that it is this incompressible kind of flow uh let's concentrate for a little while on two Dimensions one p and one Q remember this does not say that the distance between points stays the same as I said what it says is that the volume of a region of phase space says same so a region like this could flow into a region that looks like this as long as the volume stays the same and in particular two points which are fairly close can get fairly far so there's no conservation of the distance between points in the phase space only the volume only the volume of a of a patch of phase space um this idea that let's take the two-dimensional case that area is the important thing and not distance you can see in another way let's take the world's simplest mechanical system a particle moving along one axis with an energy which is x do^ s a kinetic energy which is x do^ 2 over 2 I have even gone so far as to set the mass equal to one to make it simple right there's a canonical momentum P sub X which is equal to the derivative this is the L grangian lran is x do^ 2 over 2 the momentum canonically conjugate to X is the derivative of the lran with respect to x dot and that's just x dot now I could have used other core another coordinate I didn't have to use x to describe the system what would have happened had I used 2x suppose I defined a new variable let me call it y it does not stand for another axis it just stands for another representation of exactly the same physics uh a new coordinate I'll Define and let's call it Alpha time x Alpha is just any number fixed could be two it could be a half it could be seven whatever it is this is just another representation of the same uh system where it's been labeled by a different Axis or an axis with a either closer or further points points uh different units it really comes down to a unit transformation okay uh what is x dot X first of all is equal to Y over Alpha and x dot is just equal to Y dot over Alpha let's rewrite the lran the LAN doesn't change lran is the same numerical quantity no matter what coordinates we use lran will now become X it's still x do^ 2 over two but that becomes now y dot 2ar over twice Alpha squ I've just written that x do^ 2 is y^2 alpha s that's the Newan it's just another description of exactly the same system what is the canonical momentum conjugate to Y well it's DL by Dy dot P sub y is equal to Y dot over Alpha squar okay let me now write right over here we have y = Alpha * X we have py = y do over Alpha squared but that's also equal to x dot over Alpha I've used that y dot is equal to x dot let's see what if I written yes y dot is Alpha * x dot so wherever you see y dot here just stick Alpha * x dot that cancels one factor of alpha all right but this is also equal to P subx over Alpha all right so let me combine that with this equation over here P sub y is equal to P subx / Alpha notice what's happened when you make a coordinate transformation that stretches the xais it shrinks the P axis sorry P yeah if you make a transformation which stretches the x-axis remember now X and Y are not different directions of space they're just different coordinates describing the same thing this is a coordinate transformation Y is equal to alpha x along with it goes a transformation of the canonical momentum but in the opposite direction if you stretch the x-axis you shrink the P AIS okay that means for example if you take the phase bace let's we can represent it either in terms of X or Y and we take a little region let's call this X and PX right suppose we represent that same system in terms of y and p sub y well the um the width of this box as measured in y units is Alpha times as big as it was in X units but on the other hand the height of the region is 1 over Alpha times what it was in X units so whether the the height of this box and the width of this box as measured in either X or Y are not the same but the area of the box as measured in X and Y are the same this is a characteristic feature of classical mechanics that the various coordinate Transformations that you would do if combined you do some coordinate transformation but you also calculate what happens to the canonical momenta you what you'll find mind is that areas in the phase space always stay the same so the idea of an element of area in Phase space is very fundamental it of course comes up in quantum mechanics in quantum mechanics it's basically roughly speaking the statement that there's a smallest area in Phase space a Quantum of area in Phase space we'll come to that but not I'll just mention it now that things cannot be localized in the phase space better than that Quantum of area how big is the Quantum of area plks constant no no not squared not squared plunks constant is not a measure of Delta X or Delta P it's a measure of area in the phase space okay and it says that there's a fundamental unit of area in the phase space and that you cannot think of anything smaller than that that's the uncertainty principle Yeah question absolutely absolutely the mathematical statement is that the time evolution of a system is a canonical transformation yeah that did or did not what say it again why does the uh why does it not conserve energy no it loses it loses potential energy gains kinetic energy right um good okay so let's all right the statement the statement that the flow in Phase space flow in Phase space is incompressible that's called Le ail's theorem so you ago thees where where where here no this wasn't representing the time dependence of anything this was just two different sets of coordinates that's all it could well uh that call this is a canonical transformation okay yeah yeah right and we haven't we haven't discussed that at the moment but but the answer is yes that is a canonical transformation all the interesting transformations of mechanics are called canonical but we haven't defined them yet so let's uh all right let's talk a little bit since it's been raised about what chaos means that's right it means the area in quantum mechanics it's of course connected with the uncertainty principle uh that's right a p Dot and a q dot P Dot and a q dot you want to multiply P do times a q dot okay let's multiply a p Dot and a q dot what doesn't change is Delta P * Delta Q the area of a little region of face of face space it doesn't change you have you have X and p p subx and you have some other coordinates which you can call y and p sub y and if you calculate the take a little region of area here break it up and calculate its area by adding up the squares and giving them and assigning them an area Delta P sorry Delta x * Delta P subx or add them up by assigning them Delta y * Delta P sub y you'll get the same answer so it's the area element the size of an area which will be the same in whatever coordinates you use whatever coordinates and momenta if the coordinates and momenta come from the same lran okay so yes so area area has an invariant meaning in the phase space uh it takes on its sharpest physical meaning when we get to Quantum Mechanics where it really corresponds to the uncertainty principle well I don't know I I um what does it mean the fact that it's conserved means that it's useful uh that uh that yeah yeah yeah yeah yeah yeah it it it tells you what the right probability measure is on the phase space that an invariant probability measure on the phase space and that's useful in statistical mechanics yeah yeah and uh it uh for example for example if you have some chaotic system there a phase bace so that means the phase Point moves around and gets you know really moves around in some wild way uh and you ask how often or what is the probability that is found in a little region of phas space then the answer will depend only on the area of the region of phas space uh so area would be the thing which would tell you the relative probab if you are interested in the relative probability probability could mean the amount of time spent in this little region here the phase Point moves around goes crazy we stand there and watch it and ask how much time does it spend in that little volume that's a measure of the probability that you would find it in that volume versus this volume all right the answer is is that the relative probability of being in two different volumes of phase space is simply the ratio of the areas right that's for a chaotic system for a chaotic system yeah for a chaotic system yes right well here I'm defining probabilities not by expectations your your naive expectations I'm defining it by the amount of time integrated time that it spends in a region so that's a that's a stronger yeah right the the yeah the um the amount of time that it's in here divided by the total time that you sample the system tends to a finite limit for chaotic systems yeah right no yeah yes no it probably does but I have to think about it no I'm not sure what the word stochastic means frankly I think it just means an element of Randomness uh what does stochastic what does the word stochastic mean non non- deterministic yeah well of course chaotic systems are secretly deterministic but you can't follow them very long in time whe unless you have um a degree of precision Which is far beyond what uh what's reasonable I don't want to I don't want to get into chaos too much right now I will just tell you very quickly one other thing about chaotic systems or the difference between chaotic and not chaotic both of them have the feature that the phase space volume stays the same but that seems a little odd because you would think that for systems which are chaotic you know just um systems banging into each other in random collisions not random really but governed by the equations of motion but nevertheless complicated enough that they seem to scatter all over the place you would think that the phase Space volume increases increases because you get because after a while you know less and less about where the system is well that's not exactly true here's the way it works supposing you take all right so the correct statement is that this little sphere in Phase space will evolve and it will evolve into something which may look more or less like a sphere but it may also look uh you know grow all kinds of complicated fibers and become very fractal I don't know is fractal the word probably not you know start to grow uh okay now here's what you do you I want to put some more in okay Etc all right now if I literally mathematically follow the points incidentally this will be just as continuous as this if you look at it under a microscope you will discover that this phase space is completely continuous nothing uh discontinuous is broken off it or anything like that but as time goes on it tends to get more and more fractured like this let's just call it fractured if you take each point each point goes to a definite point if you follow the volume you'll find the volume is the same but now you can do something which is called course graining coar graining means a purposeful decision to draw little circles or spheres around phase points and roughly speaking what it corresponds to is saying I don't have enough Precision to be able to distinguish points of phase space strictly speaking I can only distinguish this little volume from that little volume from that little volume and that little volume so what we do is draw little spheres around each phase point and follow the little spheres rather than the individual phase points follow the Spheres and what will happen is this the sphere may go over here the sphere may go way over here this one may go way over here this one may go way over here but if we were to take the total area occupied by these spheres I think it's pretty obvious that the total area will be much larger than the area that you started with if this thing branches out and fractalized and it is so the coar grained following of the phase space allows the area to get bigger and bigger and it does get bigger yeah yeah because of the course graining it's effectively random in other words if you take the course grain phase space the nasty thing that happens is no matter how small a course graining is if you wait long enough points within one coarse grain here will separate and get far outside a course grain right if the co grain is defined by little spheres so what's that no no no then not the mapping from the first no the mapping from the no no no they're not the mapping from the first they're just spheres in the face Bas again spheres that you can distinguish by the coar grained apparatus that you happen to have so the coar grained apparatus that you happen to have defines a course graining on here and it's not the same as the course graining that would inherit by following this course graining it's just what you can measure what you can distinguish uh and of course what you can distinguish depends on the Precision of your uh of your measuring apparatuses and so forth but no matter how precise your apparatuses and your description is no matter how small a course graining here eventually points within a single cell here of this Co graining will diverge and get outside the uh the corresponding Co grain description of what happens later that's what happens uh a chaotic system most systems are chaotic of course uh the uh the non-chaotic systems which sort of maintain their their integrity without in the course grain sense those are rare and exceptional yeah say it again yeah change in time yeah the area of the of the cor SC you you take the original region of phas space you figure out exactly what it does and then in the original region of face space you draw spheres if the original region is a nice simple sphere or something like a sphere then when you add up all of the uh these coarse grain spheres you'll get the volume of the original sphere back pretty much but now you take this tree this very very um finely fractalized tree that it grows into which may fill up this very big piece of this Blackboard with very fine filaments very very close to each other okay so for example if there are filaments extremely close to each other so close to each other that the coarse graining uh sort of slops over or slops over the distinction between these different filaments then the total volume of these coar grained green spheres will in general exceed the volume that you started with why yeah it's basically just going to cover this whole uh the whole tree there yeah yeah you just yeah yeah yeah so you just look at this picture with fuzzy eyes and you see how big uh how big this region is this region can be much much bigger yeah lots of ways the what yeah you're counting everything within this little sphere here even though the fiber in here may be much thinner than the sphere but the fibers are so thin that your apparatuses cannot t tell exactly where you are in the sphere all you know is you're somewhere in the sphere so at the end of the day if you ask where am I if I start in here the answer to the question is you're somewhere in here in a much bigger region if you had infinite precise control over the either the mathematics or the measuring measurement then the volume of this fiber um ultra fine filamentary structure would be exactly the same as the original ball here but you don't follow things that way that's not the way you follow things in an experiment you say Okay I I I detect the I detect a a particle somewhere is in this phase region over here I can't tell exactly where it is because it requires too much Precision so in that sense the phase Space volume does expand okay it's called course graining and it is of course the origin of the second law of Thermodynamics okay there's one other point that I should emphasize uh I don't know when when and if we'll come to it in a more thorough way but the entropy of a system is a measure of the volume in Phase space that the system occupies uh the det the detectable volume in Phase space meaning to say if you know about a system that it's somewhere in here you take that volume of phase space that volume of phase space which corresponds to the boundaries of the region that uh that your knowledge tells you that the system I'm getting I'm getting tied up you have some knowledge about the system the knowledge of the system is that it's somewhere is in here in this volume of phase BAS the entropy is a direct measure of that volume in fact it's the logarithm of that volume okay essentially that's now the system evolves and what you know afterwards is that it's somewhere is in here you're not either clever enough or precise enough to be able to distinguish these fibers so all you know is that the system is somewhere in here the face base volume is bigger afterwards words than it started in the beginning because you haven't followed the details sufficiently carefully to know precisely to distinguish this bit of fiber from that bit of fiber that's why entropy increases basically it's a statement that when the phase space fragments like this you lose information for not because information is really lost but because uh you have a hard time following the system it's a technological thing you can't follow things with a sufficient Precision okay that's uh that's lille's theorem as I said it is um extremely fundamental to both classical mechanics and its Evolution into quantum mechanics what I wanted to do next completely different direction was to work out the equations of a particle we have not studied part very many examples incidentally you'll notice I want to study the example of a particle moving in an electromagnetic field there's one thing that's new in a magnetic field that we haven't encountered yet and it's the existence of velocity dependent forces so far the forces that we've imag iMed depend on where you are but not how you're moving when we said that there's a potential energy function U of X and we differentiate it to find the force well the left hand side is just a function of position so the force on a system just depends on where you are there are forces in nature which depend on velocity magnetic magnetic field acting on a particle is a prime example incidentally friction is also an example friction is an example where the velocity of a where the force on an object does depend on its velocity if it's standing still there's no friction Force if it's moving there is a friction Force typically the faster it moves the larger the force there's a fundamental difference between magnetic forces and friction forces and the simplest way to say it is that forces due to make magnetic fields although the velocity dependent are derivable from a principle of least action they have a lran formulation there's a conservation of energy and there's a hamiltonian formulation it's just that the lran and hamiltonian have a new little twist of them that uh that will that I want to work out now forces due to Electric Fields uh are no different than the forces we've already studied uh I think I'll come back to Electric Fields I want to concentrate on magnetic fields because there's a new phenomenon there that uh that's interesting all right so what is the force of a particle uh if it's in a magnetic field magnetic field is called B it's a vector and it depends on position B of X the magnetic field what is the force of a Charged particle on a Charged particle V cross B well actually Q * V cross B where Q is the charge of the particle velocity cross magnetic field it's a vector the cross product of two vectors both of these are vectors this is theoc velocity of the particle this is the magnetic field and so it says what uh everybody know what a cross product is assume everybody more that means yes or no yeah right right let's see [Music] um right something to do with a Thum I'm going to assume everybody knows what a cross product is although will Define it okay so let's define the cross product right now the cross product can most easily be defined through its components V has three components x y and z b has three components X Y and Z the cross product has three components here's the rule the Z component let's forget Q here and let's just write the rule the Z component of this this What's called the Z component of it V crossb the Z component of it is equal to VX by y minus VY BX I remember one one of them I remember the Z component and then if I want any of the other components I just cycle Z goes to X goes to y so the next one would be x v cross B the X component and I cycle X goes to y v y y goes to z v z minus minus v z b y and the last one V cross B in the y direction is equal to X1 y z y z x minus v x b z did I get that right I think I got it right okay that's the definition of a cross product now we need one other concept we need the concept of the vector potential the vector potential is simply a way of writing magnetic fields but you would think well if it's just a way of writing the magnetic field why not just use the magnetic field and uh the answer is you cannot write the mechanics of a Charged particle just in terms of the magnetic field you need the vector potential uh you can of course write the mechanics fals ma but you cannot write a hamiltonian you cannot write a L grangian you cannot write even an expression for uh for canonical momenta you need an auxilary quantity called the vector potential and the vector potential is just defined by the condition that the magnetic field is the curl of the vector potential let's write down what curl means because we'll need it later the vector potential is also a vector magnetic field is a uh is a vector the vector potential is a vector what's special about things which are are curls of other things they have no Divergence and the magnetic field is a thing which has no Divergence and that's why it's convenient to write it this way but for our purposes let's imagine that the vector potential is something fundamental that we begin with with and that the magnetic field is a derived quantity we're going to see that the mechanics of a Charged particle is best written in terms of the vector potential not the magnetic field okay let's write down the definition of cross product a cross product again can be defined by its components Del cross a the Z component of it now it's basically the same thing except wherever you see a v up there wherever you see a b stick an A and wherever you see a v over there put an upside down uh Delta all right so this just says that this is what is what does delta mean it just means the collection of three derivatives Delta X Delta Y and Delta Z are just derivative with respect to X the derivative with respect to Y and the derivative with respect to Z I assume you all know that kind of thing a little bit of vector calculus but let's just write down exact ly what this means this means derivative with respect to X of a sub y minus the derivative with respect to Y of a subx and then cycle it down Del cross a the X component of it is the Y derivative of a z minus the Z derivative of a y and just cycle it down X goes to y y goes to z z goes back to X X I won't write them all down so we can now write that the force on a Charged particle due to the magnetic field is the charge time V cross B which is also the charge time V cross the curl of a I'm going to bore you a little more with this by writing out the individual components to by writing out the Z component of force here it's a little bit tedious but we'll need it we'll need the expression for the Z component of force explicitly in terms of the vector potential so let's write it out this is just an expression this is just an exercise in substituting all right I want the Z component of the force f subz is equal to the charge now the cross product we have the cross product where is it we have the Z component of V crossb that's VX by minus VY BX but now let's just substitute in for b y where is it um oh brother uh this is Q VX now go to um [Music] H by Y is DZ ax minus DX a z that's the first term minus v y time BX which is dy a z minus DZ a y well this is a fairly complicated expression for the Z component of force but if you know the vector potential you can work out what this is it's just some Vector field you can multiply it by the velocities and you can compute the force the Z component of Force if you want the other components again you just cycle the X X goes to y y goes to Z you just cycle through all right that's the force on a particle and the equation is just Newton's equation f equals ma but now the new thing is that the acceleration doesn't only depend on the position it does depend on the position through the uh through the position dependence of a but it also depends on the velocity so these are called velocity dependent forces of course they depend on velocity we want to formulate the theory of a Charged particle moving in a magnetic field in the lran or hamiltonian form not obvious that we can do it it is not obvious that this is of the form uh that uh that allows us to have a hamiltonian lran and so forth or an action principle the easiest way to confirm that is just to make a guess at what the action is and then follow it through the new thing in the action well first of all we do expect that if the electric charge was Zero that there would be the usual terms all right uh so let's write the action let's make a guess for the action the integral of the lran DT well first of all guess has the usual m v^ 2 over 2 DT where v^2 means the sums of the squares of the X component y component and Z component of the velocities the usual thing this is what we would do for a free particle and if the particle has no electric charge it doesn't feel the magnetic field and then there's another term and the other term is as simple as it could possibly be we take the orbit of the charged particle we break it up into to little segments each segment we call DX also Dy and DZ but let's just abbreviate it by calling it DX each little segment defines a differential displacement in DX Dy and DZ the added term that's first of all proportional to the electric charge if there's no magnetic field then obviously viously there is no additional term even if there is an electric charge the extra term is related to both the charge and the magnetic field but it's related through the vector potential where where's Vector potential did uh oh boy did I erase the most important equation I did okay I assume you all memorized what V crossb looks like in terms of the vector potential and you'll recognize it when I write it down I didn't mean to erase it all right what can you write down it's going to have the vector potential in it so let's call it a sub I the simplest thing you can write down is the integral the line integral of the vector potential dotted into the differential displacement DX in other words what this means is every place in space you have a vector potential as the particle sweeps out through its trajectory there's a contribution to its action from each little differential displacement here and the contribution to the action is just take the dot product of the vector potential with the displacement in other words the component of the vector potential along the direction of the displacement and add them all up very simple prescription how do I know that's the answer well know that's the answer because it was written down in the early parts of the 20th century and I learned it as a student yeah dxy good and or you can just write a. DX where you can think of both a and DX as little vectors one is a little Vector the other is just a vector all right a. DX can represent the dotproduct of the vector potential with the differential displacement it doesn't seem like units are working out right is time how do you know what the units are do you know what the units of a are what the units of Q are a [Music] time but how can you tell me what the units are unless we know what the units of Q are okay or a how do we know what the units of a are well the units of a and the units of Q obviously the combination Q * a is the important thing and the combination of Q * a has units that are required to make this have the same units as that what are the units of an action incidentally or momentum times uh position sorry momentum yeah momentum time x uh no problem you can always give Q the right dimensions this is correct um uh in some appropriate choice of units this is correct and it depends on the on on the uh conventions in some conventions there are pies floating around or speeds of light but you can set them all equal to one and since I'm trying to describe a uh a phenomena rather than a numerical effect let's just set all the numerical constants equal to one or to absorb them into the electric charge here okay now this is yeah yes that's exactly what I was about to do this does not have the usual form of an action which depends on velocities and positions the A's depend on position as a rule but what is this DX here here all we have to do is divide and multiply by DT DX by DT that's just the X component of the Velocity right so we can also write this as uh a dotted into the velocity integral DT or I can write it in terms of components uh a sub I X subi do DT and now we see the usual form the what's it some over I yeah Su over I all right so we see and this of course is also XI do s sum over I I'm not going to write it all right so we have then a typical lran we can write out now what the lran is and it depends on position and velocity the way any respectable lran would L grangian for a particle in a magnetic field is 12 the mass of the particle time x sub I do squared where I goes over X Y and Z sum Plus Q the charge of a particle x subi dot times the vector potential a subi what's the next step the next step is to prove that the equation of motion for this object is f equals ma with the right hand side being where is it QV V cross B F = Q V cross B if we can do that if we can now take the lrange equations here's the lran take the lrange equations and see if we can prove that the force on a particle this is equal to mass time acceleration that this is equal to QV cross P if we can do that then we're finished we found the lran form ulation of particle in a magnetic field so this is now an exercise in just plugging it through I could tell you to go home and do it but let's do it let's just carry it out there is one Curious Thing coming up that we'll see in a second first of all what is the canonical momentum piece of X well it's the derivative of the Lan with respect to x dot so let's take the X component I now mean the X component we could write down Y and Z also that's going to be m x dot from here usual thing normal momentum but then there's another term plus q a subx and of course the same kind of thing for py and pz so we have a system now where the canonical momentum is not just m * V it's m * V plus something proportional to the vector potential um sometimes M * x dot is called the mechanical momentum it's sometimes called the mechanical momentum meaning to say you know it's meaning to say nothing other than that it's mass times velocity and the whole thing is called the canonical momentum all right so that's that's one new thing here we see something new that the there's a discrepancy between the canonical momentum and what we naively think of as momentum okay but now let's work out the equation of motion let's work out the Z component we can know there's three of these but let's get the pz is equal to m Z dot plus q a subz I'm going to work out the Z component of equation of motion what is it it's the time derivative of the momentum I'll just write down over here D by DT of DL by DZ dot is equal to DL by DZ DL by DZ dot that's just pz okay so we have to take the time derivative of this thing here let's write down the time derivative it's m z double dot that's masstimes acceleration and then there's another term in the time derivative of pz it's plus Q time the time derivative of a z now in a moment we'll come back and work out what the time derivative of a z is but let's go to the right hand side of the equation the right hand side of the equation is the derivative of the lran with respect to Z what is it that depends on Z in here x do or Z dot doesn't depend on z a a and all of the A's a x a y and a z all depend on uh on uh on on Z in general so let's go through them Q now the first term here would be x dot time a subx so we would get a term which would be x dot times the derivative of a subx with respect to Z remember we're taking derivative of the lran with respect to Z the only thing that depends on Z are the three components of a let's start with the X component X do ax and differentiate it with respect to Z the next one is plus y dot times the derivative of a y with respect to Z and what's the last one Z dot times the derivative of a subz with respect to Z okay all I did was differentiate all three terms here with respect to Z each one of them depends on Z through the components a what about the left hand side what do I do with a subz dot let's suppose incidentally that it's just a static magnetic field magnetic field not changing with time it's just a fixed magnetic field does that mean that this is zero if it's a fixed magnetic field no no because the particle moves through the magnetic field I don't mean when I say it's a fixed magnetic field I don't mean it's the same everywhere as in space I mean it doesn't depend on time magnetic field varies from place to place particle moves through it and because the particle moves through it there's a Time dependence of the magnetic field of the ve Vector potential at the point of the particle so how do we calculate that well we simply write m z do plus Q now a z depends on x d a z by DX time x dot plus the a y sorry what are we doing here yes sorry the the a z sorry a z a z we're calculating AZ dot which is the derivative of a with respect to x times the time rate of change of X plus the a z by Dy * y dot plus d a z by d z time Z dot that's equal to what's in the bracket over here equals this okay I'll let you stare at it for a minute remember the right hand side came from explicitly differentiating the lran with respect to Z the leftand side came from differentiating with respect to T but notice they have similar form each one here has a velocity time a derivative of a on both sides so maybe they combine together in some nice way well first of all you'll see that this term and this term are the same Z dot time d a z by DZ they're the same so they cancel let's get rid of them now what's left let's look at what multiplies x dot x dot has D ax by DZ and then when we shift to the other side it will have what d a z by DX so it's going to have Let's uh let's move this over now I forgot what it was plus y dot what was it d a y by DZ yeah partial a y with respect to Z right yeah okay now what the term which multiplies x dot when I transpose it it's going to have minus d a z by DX oh that's starting to look familiar and when I transpose the term that has y dot in it that will have minus d a z by Dy I've transposed this term over to the right hand side and I now have MZ dot or mass times the Z component of acceleration is equal to this mess over here well what is this mess this mess is not so complicated it's got linear terms in the velocities and what are these objects the magnetic field the curl of the vector potential in fact this one is um ax1 I think this is b y is it b y or minus b y and this one I think is minus uh BX the X dots and Y dots are the components of the velocity if you go through it or just go back in your notes to where I wrote it down in the beginning this is just the Z component of V cross B velocity H yeah okay right VX time DZ ax minus DX a z and so forth yeah okay so you see that first of all there does exist a lran formulation here it is for particle in a magnetic field you need to know the vector potential first question is the vector potential unique in terms of the magnetic field no you can add things to the vector potential we'll come to that but not right now but if you know the vector potential or you know a particular expression for the V Vector potential then you can write down the lran you can the Lan depends on the vector potential but the equation of motion only depends on the magnetic field so even if there is an ambiguity in the vector potential it doesn't affect the equation of motion itself okay now we have an example of a velocity dependent force and Force which is V crossb it's almost like a friction force it is different the friction Force acts along the direction of the Velocity okay if an object is moving in this direction the friction force is along the same axis what about V cross B does v crossb act along the direction of the Velocity no because V cross B is perpendicular to to both the velocity and to B so this and this is a big difference a big difference uh whether the whether the velocity dependent Force acts along the direction of velocity or perpendicular to it if it acts along the direction of velocity it will obviously have the effect of either slowing it down or speeding it up depending on which way it it's acting all right but if it acts perpendicular to the velocity it'll simply change the direction of the Vel velocity if the force is perpendicular and it won't change the energy so energy conservation will we can work out now we can work out what is the expression for the energy we do that by calculating the hamiltonian and when we've calculated the hamiltonian we then have from previous theorems the theorem that says that energy is conserved right so we can work that out let's work it out I think we're I think we'll just that'll just about take us to the end of the the hour oh excuse me there's something I need to tell you I won't be here for the next uh two lectures now I would like to make up one of them I think uh the question is what is a good time it could be this week yet but it can't be next week and it could be the week after but not on Monday uh yeah we can do it this week does that uh does that work out for people if would do it say Thursday Friday is a bad day for me or we can put it off we can put it off till I get back and uh and uh uh uh just double up the week that I come back I'll be gone for H should we wait until uh to take it Thursday Thursday how many cannot take it Thursday well that's all that's almost half the class uh supposing I were to put it off I'll be back on the 11th now the 11th uh is probably a Monday uh no 11th is a Tuesday wnes H yeah I was thinking about Wednesday the 12th I think it is is it 12th or 13th 12 does that work no get something I know that well we can stay on Monday and uh but uh yeah no we'll continue we'll continue but at some point uh we uh we run into the next quarter I you know playing you know points have to move rigidly uh can't uh well we could do that another thing I have to do as I'm supposed to uh fill in the missing lecture from the first uh the first lecture I was thinking of doing that this week but uh no no no they we didn't miss a lecture but they didn't uh they didn't um film it what's that no doesn't have to be no no I'm happy to this is fun for me I just do it for the fun of it uh no we no I intend to keep going until the uh until sometime near Christmas um but uh we will miss the next two lectures so I thought I would try to make up one what's that yeah two Mondays but then we'll be back on Monday well we will be back on Monday and we'll see if we need if we need to fill in if we need to fill in with another lecture we'll just figure out at that point when to fill in we're already well I don't think we we'll do Christmas Eve uh that's yeah yeah we well look why don't we just decide this when I get back on the uh on the 17th we'll try we'll see when we can fill in an extra lecture and I think that'll pretty much bring us to where we want to be I mean I'm pretty comfortable yeah uh three weeks yes three weeks from today right right right right next lecture is the 17th um okay let's try to just finish up what I was talking about here I thought I think I finished um energy yeah okay let's do energy so all we have to do is calculate the hamiltonian incidentally I'm assuming that the vector potential has no explicit time dependence in other words equivalently that the magnetic field is constant in time in that case the hamiltonian will be time independent and there will be a conserved energy the conserved energy is surprisingly simple when expressed in terms of the Velocity is a little more complicated when expressed in terms of the momenta so let's work it out let's first calculate uh uh p subx as we saw before was MX dot plus q a sub X same thing for Y and Z and I won't bother writing them down separately okay the first thing we have to do when we calculate the hamiltonian actually we don't need to we can just let's let's calculate it not in terms of P but in terms of x dot let's first do it by calculating in terms of velocities to see what the expression for energy is in terms of velocities all right so what do we have to calculate we have to calculate p subx x dot plus p suby y dot plus p subz z dot minus the lran right that's the definition of h take each momenta multiply it by the corresponding velocity and then subtract the lran that's a very general rule that we derived right if I plug in for the P's their expression in terms of velocities I'll have an expression for the energy in terms of velocities that won't be good for Hamilton's equations but it will be good for telling me what the energy is in terms of velocities so let's do that let's do that first p subx x x dot that is m x dot + q a subx time x dot likewise for Y and Z and I won't write them out then we have to subtract off the lran minus l so subtracting off the lran means subtracting off minus M v^2 or MX do2 I'm concentrating on the X term M MX do 2 uh divided by two and then what else minus Q X do a subx well that gives me mx. S over two the usual thing no difference there and look at this Q axx do minus Q axx do they cancel completely so the expression for the energy is exactly just the original kinetic energy just MX do^ 2 over two likewise for y and z the magnetic field does not contribute to the energy when expressed in terms of velocities it's just 12 mv^ squared end of story now why is that why is that that's equivalent to another statement it's equivalent to the statement that magnetic fields do no work what's that yeah that's right it's because the force is perpendicular to the velocity so when a particle moves in a magnetic field the force is perpendicular to the velocity the force being perpendicular to the velocity means that the magnitude of the Velocity doesn't change right a force perpendicular to a velocity deflects an object but doesn't change its velocity uh and so as the particle moves along its direction may change change but its speed doesn't so it's the original expression for its energy in terms of mx. s is just the full energy magnetic fields do no work which is the same statement as a statement that the energy when expressed in terms of velocities and so forth is exactly the same as it was in the first place 1 12 mv^2 is conserved and you know how particles move in a magnetic field for example if there's a uniform magnetic field into the Blackboard particles move in circular orbits with uniform speed so the kinetic energy is conserved that's statement number one but in this form you cannot use the hamiltonian to write down Hamilton's equations you have to express it in terms of um in terms of the momenta but that's easy all we have to do is solve for the X dots in terms of the momenta that's easy X x dot is just P subx - q a subx / M so we can write down immediately what the hamiltonian is in terms of P's instead of in Terms of X dots and it's just equal the P subx minus q a subx squared over M2 over 2 m^2 excuse me I've just substituted for the velocity their expression in terms of the momenta yeah no it does well it it accelerates oh it does it does it does but radiation is a higher order effect that that's only important when the particles uh when yeah it does uh it certainly does radiate right but thus far we have not even tried to think about radiation in order to understand radiation we would have to formulate Maxwell's equations and formulate them in a hamiltonian form and discover that the radiation field itself has energy yeah um right so yeah this is the radiationless approximation mean these yeah sure absolutely plus Z term right so it's a little bit odd and a little bit surprising that when expressed in terms of velocities the energy just doesn't change but when expressed in terms of canonical momenta the expression for the energy does change and now if we applied Hamilton's equations to this expression here we would again get precisely the same uh uh equations of motion what's that yeah yeah all you do is replace the canonical moment the momentum by the momentum minus q a subx in other words this is just a mechanical momentum here the MX do term the mechanical momentum and the canonical momentum differ by Q * a the energy is just mx. s so it's just a mechanical you know just the uh the square of the mechanical momentum but you substitute the canonical momentum if you want to do Hamilton's equations so here's an exercise take this hamiltonian work out Hamilton's equations and check that you get the same equations of motion uh mass time acceleration is equal to V crossb that's a that's a homework assignment [Music] okay the proceeding program is copyrighted by Stanford University please visit us at stanford. 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