this program is brought to you by stanford university please visit us at stanford.edu mechanics is the basis for all of physics it's the basis of all of physics not only because it describes the motion of objects like particles and mechanical systems and so forth but because the basic framework the basic structure of all of physics is based on the principles of classical mechanics the conservation of energy the conservation of momentum uh the principles by which all systems evolve in nature is the same set of rules essentially exactly the same set of rules in a more abstract and more general setting than the rules which govern how a simple particle moves for example under the influence of gravity but in order to understand it we have to understand the principles in a fairly general context let's begin with the very very simplest kinds of systems that we can think of systems that are so simple that in fact they're simpler than any real systems in nature laws of nature let's imagine laws of nature which are of the most primitive and simple kind acting on the most primitive and simple systems that we can imagine i want you to first of all suppose that time which evolves continuously under the my watch i see the second hand goes around and around and around it goes around continuously the time can evolve and be any real number t i want you to imagine it only occurs in beats a stroboscopic world in which you only look at the world at intervals of time which we could be a thousandth of a second or it could be a millionth of a second let's just take it to be one second intervals and ask how in the stroboscopic world systems change with time let's also imagine a very very simple set of systems systems which are so simple that they only have a handful of configurations configurations mean everything we need to know about the system to characterize it completely so the simplest system i can think of would be a system that has only two configurations heads or tails a coin lay a coin down on the table i don't have a coin so we'll take uh this coffee top here i could put it down heads or i could put it down tails i doubt if you can tell the difference from where you're sitting but i can tell the difference this is this is heads that's tails all right so we have a system then that's characterized by two states two states of being and we want to add to those two states a law a law of evolution in going from one instant of time to the next instant of time from one beat of the stroboscopic light to the next one what kind of laws are allowable what kind of laws do the basic principles of physics allow and what kind of laws don't they allow that's going to be our first kind of question for tonight so our first concept is the space of states in this case just heads and tails it's just two points heads and tails two points in an abstract space heads and tails it's called the phase space it's called the phase space of a system the space of possible states of a system and what do i mean by a state of a system i mean everything that you need to know in order to predict what happens next everything you need to know in order to be able to say with certainty what the next state of the system will be that's called the phase space of the system in this case just heads and tails what kind of laws can you imagine what kind of laws of nature can you imagine for this extremely simple world it is the simplest well i suppose you could imagine a slightly simpler world you can imagine a world with only one state heads not much can happen in that world there's only one law of physics heads goes to heads and heads goes to heads goes to heads it's extremely boring nothing ever happens because there's only one state so how could anything happen but with two states has entails you can have a richer variety of possible laws of physics one law let's take the various varieties of laws of physics we could have one law might just say you stay the same if the state of the system is heads at one instant of time then the next instant of time it will be heads if it starts tails it will stay tails that's a very boring law and let's graph that law by drawing an arrow if it starts heads then it stays heads let's just represent that by drawing an arrow from head to head and from tail to tail the meaning of this arrow you start at the tail end of the arrow and you follow it and you see that it comes around to the same point that stands for the rule that says that a heads stays a heads and in this case a tail stays a tail so there's a very dull law of nature this stays that way for endless amounts of time and this stays that way for endless amounts of time another possible law of nature would be slightly more interesting situation if you look at it at one instance of time the next instance it's the opposite heads goes to tails goes to heads goes to tails these are deterministic laws deterministic laws mean that if you know what is happening at one instant you know forever after you know everything about the system infinitely into the future completely deterministic classical mechanics has that nature to it that is completely deterministic and in fact it is in a certain sense always reversible but uh we're coming to that all right so to draw the graph representing heads goes to tails and tails goes to heads we draw an arrow from heads to tails and from tails to heads and we read that as saying that if you started heads you go to tales if you start at pales you go to heads what is the evolution of a system under this law of nature if you start with heads it's heads tails heads tails heads tails heads tails forever after these are two consistent laws of uh of physics in a world of only two states whether we can think of more laws yes we can but uh but for the moment these are two interesting ones now how can we generalize this we can generalize this first of all to systems with more states two states is not very many we could have a die die as in dice die has six states one two three four five and six and we could represent them as points six points now we have a large variety of different laws of physics that we could have for example we could simply have we could label these one two three four five and six which one is one and which one is six is not very important but uh but there are six of them we could have a law of physics which says one goes to two two goes to three three goes to four four goes to five five goes to six and six goes back to one this would be a complicated motion when thought of in terms of a die you start with a one up and then it goes to the two up three up and so forth but it's a relatively simple picture when drawn in this manner here you can imagine more complicated laws of physics with six states well whether they're more complicated or not is in the eye of the beholder but uh first of all we could have a similar law where instead of one going to two one could go to three three could go to two two could go to five one two three four something like that that's really not very different than this each one goes to a neighboring one well not to a neighboring one each one goes to a next one and they cycle around and one characteristic of such systems is they will just cycle around forever and ever and ever okay now you can have more complicated i don't know if they're more complicated different laws for example you could have a law which goes this way if you start at one you go to two if you go to two then you go to three and then you go back to one if you happen to start at three nope this is three this is five right five six six six six one two three four five if you start at three you go to four if you start at four you go to five back to three in this case well that's somewhat similar to this you have two disconnected cycles two different more complicated things here's another one in fact i think it would take a long time to draw all the possibilities one goes back to one two goes to three three goes to four four goes back to two uh five goes to six and six goes back to five right they're all acceptable laws of physics wherever you happen to be you know exactly where to go next so it's deterministic it's deterministic into the future meaning to say wherever you start you know where you will be arbitrarily into the future if you start here and you go a hundred thousand times you'll just wind up i don't know somewhere ever wherever if you start here you stay there so these are completely deterministic into the future but they're also completely deterministic into the past which means if you know where you are you know where you were before if you find yourself over here then you know in the previous cycle or the previous instant you were over here and so forth so you can trace your motion either into the future or into the past with complete confidence about where you'll be no matter how far you go that's the character in principle in principle if not in practice in practice things get jumbled up and you can't see them clearly enough and you and you miss detail but if you see the infinitely small detail in a system and get every single bit of physics absolutely right for any classical system they are in this sense exactly deterministic both into the past and into the future now what kind of laws of physics do we not allow the kind of laws of physics that we don't allow i can best illustrate i think by drawing some some possibilities and explaining to you why they are not allowed here's a law of physics that is not allowed by the principles of classical physics three states three states is perfectly all right nothing wrong with that but let's draw some arrows two three two goes to three and three goes back to two that would be perfectly all right by itself what about one well i could have one goes to one but i don't want one to go to one i want one to go to two that's completely deterministic into the future if i start at one i go to two i go to two i go to three i go three back to two two back to three i always know where to go in the next step i just follow the arrow wherever it happens to be but what does it fail it fails to be deterministic into the past supposing i know that i'm at two then where did i come from i could have come from one or i could have come from three so i cannot work my way backward with uniqueness i can work my way forward i cannot work my way backward that's a this is a law of physics which is irreversible it would not allow me to run the laws of physics backward it would lead to an ambiguity every time i were a two another law which is not allowed by the principles of classical mechanics or principles of classical physics would be basically the same thing but with the arrows turned in the opposite direction all arrows reversed here i have a problem not going into the past but i have a problem going into the future let's see yeah supposing now i find myself a two and i want to go into the future i don't know whether to go into the future by following this arrow to one or this arrow to three there are two arrows leading out of two over here one of them goes to three one of them goes to one nothing tells me which i wrote to follow so it's not deterministic into the future i might uh randomly decide to go from two to three or randomly decide to go from two to one these are rules each one of which is deterministic in one direction but not the other these are the sorts of things which are forbidden by the principles of classical mechanics why are they forbidden by the principles which one yeah just one of those two lines here that wouldn't come i wouldn't know where to go from three you get stuck at three yeah yeah no good can you ever have see a dangly one over here well you can't have a dangling one you gotta know where to go next no you could let's see we could try to put something like that in yeah okay so i would say we go from one to two from two to three to three to three to three now the problem would be in going backward i think right yeah yeah yeah right so one way or the other you get stuck uh with these rules all right how do we spot what's allowable and what's not allowable well it's very simple at every configuration two three we should have one in and one out we should have a unique one in a unique arrow in and a unique arrow out one arrow in to tell us where we came from and one arrow out to tell us where we're going that's the character of classical physics uniqueness into the future uniqueness into the past when represented in terms of this very simple and analog world analog digital world of finite number of states then the rules of physics as we know them would say every configuration has one in arrow and one out arrow now with this rule of course life is very boring because if you only have a finite number of states all that can happen to you is you cycle around endlessly among those number of states always in the same order you can have slightly more interesting situations there's no reason why the number of states has to be finite you could have a situation where there's an infinite number of possible states a state corresponding to every integer positive and negative a particle on a position on a line where the position could be any integer value and then a simple law of physics would be you go from one to the next wherever you are you go to the next one each point has one in and one out i can't draw them endlessly it'll take forever but each mark one in and one out this of course would also be a rather boring law of nature you just hop from one to the next to the next to the next to the next forever and ever and ever but at least you wouldn't be cycling around endlessly again we can have add some more states onto this we could add this on if we start on the line we simply move off and keep going forever and ever and ever but if we start over here we cycle around so we could have mixtures of both kinds of things some states we cycle around in other states we move off to infinity now notice that some of these laws of nature the phase space breaks up into different pieces which are connected among themselves but not connected to each other for example even with just two states we had two possibilities one like this and one like this in this case where breaks up into more than one more than one piece we have something called a conservation law a conservation law is simply a memory of where we started a conservation law means that something is kept intact for all time some piece of knowledge is kept intact for all time and doesn't change in this case we could label this we could label a configuration over here with a plus one and a configuration over here with a minus one and then we could call we could invent a variable plus one over here and minus one over here it never changes if it's plus one it stays plus one if it's minus one it stays minus one that's a conservation law something which doesn't change with time on the other hand if we hop from plus one to minus one to plus one to minus one we don't have a conservation law conservation laws are always associated with these kind of closed families of different trajectories in the phase space which don't mix with each other which remember um something about the system which might otherwise get mixed up if everything got all mixed together so there's all kinds of possibilities that are inherent in these deterministic laws but always the condition is one in line and one outline for every point that can also be called information conservation it's information conservation in exactly the sense that that you never lose memory of where you started either into the past or into the future if you know where you are at any instant you know where you came from and you know where you'll be information about where you are is conserved never changes into the past and the future whereas if you have one of these bad laws laws which are forbidden by the rules of classical mechanics then you do lose information for example if you find yourself over here you don't know whether you came from here or from well no that's not quite right find yourself over here you don't know whether you came from here or over here so you lose information information conservation is perhaps the most fundamental law of basic classical physics that you don't lose information about now why that why is that so um it's not written into the laws of physics why they are what they are maybe someday we'll understand hyper laws of physics or meta laws or physics or deeper laws of physics which will tell us why the laws of physics are what they are at the moment it's more or less an experimental fact that all the known laws of physics fit into this class of information conserving laws even those which are quantum mechanical but for our subject this quarter we're only interested in classical physics all right so that's the basic setup if we were interested only in this stroboscopic world of discrete time intervals um the real world of course is more continuous than that supposing okay so let's make the law something like this if the last two states were heads heads then it stays heads if the last two states were head's tails then it goes to heads if the last two were ted's tails heads then it goes to tails and if the last two were tails tails then it goes to tails i think that's a possible possible law then you would say i can't tell from the fact that it's heads where it goes next and indeed i can't but that would just be another way of saying that a specification of a heads by itself is not what you would call a state what you would call a state would be the specification of the previous last two entries because you need two entries to tell you what happens next now that raises the question that that that's very very important in classical mechanics how much and what exactly do you need to know to say what happens next if the phase space is the space of things space of possibilities but always in such a way that they tell you exactly what happens next what is it that you do have to know next so that brings us to continuous physics let's take the motion of a particle let's take the motion of a particle is it enough to know where a particle is to say what happens next let's hypothesize that the generalization to continuous time is that we need to know the exact location of a particle along a line we've run out of ink i'm afraid now there it is all right so let's imagine the motion of a particle along a line then you might think that the analog of a state is just the location of a particle where is it but is it enough to know where a particle is in order to to say what happens next no what else do you need to know its velocity in order to know where a particle will be next you need to know not only where it is but how fast it's moving you need to know its velocity so that means the state in the same sense that i used it that which you need to know in order to know what happens next does not just consist of the location of a particle but you can say it two ways you need to know not just the location of the particle but you need to know also the previous location or better yet what causes what is equivalent to knowing the previous location the velocity right the velocity that tells you that the phase space the space of states the space of configurations is two-dimensional not one-dimensional it's not just a line it's a line that represents the position of the particle and a second line which represents its velocity either to the left or to the right positive velocity means moving to the right negative velocity means moving to the left supposing you're over here where do you move next you stay the same place why you're at the origin and you have no velocity what if you're over so we could just we could draw a little loop here to say that you come back to the same place what if you're over here you stay the same place because you're moving with zero velocity vertical axis is velocity so you come back to the same place what if you're over here where are you one second later let's chop time up into one second intervals where are you next somewhere to the right huh what if you're over here now what if you're over here move twice as far or roughly twice as far to the right what if you're down here you move to the left so we could fill up this space here with little tiny arrows to show where you move next but notice to know where you move next you have to know not only where you are in the sense of what x is but you also have to know the vertical component namely what the velocity is so that means the analog the analog of a point in the phase space is a point in the space not only of positions but also velocities yeah so if there's a conceptual issue here that i understand that is you would think that if you were able to discretize time finally enough yes the two should be the same and yet why a velocity that came in which isn't readily apparent where that would take place if we had this infinitely small discrete no it means exactly what the gentleman asked me before what if you had a law of nature which tells you in order to know where to move next you had to know your previous two entries right now one way of saying it is all right in that case i need to know the previous two entries and it just doesn't fall into this class of things or you could say that the space of configurations doesn't just consist of a heads or a tails but it consists of a pair of entries a pair that's right we might we model that that's right we need two axes instead that's right we need an axis for the present configuration and the past one yeah that's exactly right also in classical physics well of course not because in classical physics the position of a particle and so forth is a real number a real number is uh something that you can never determine exactly uh and so there's always imprecision and that imprecision always represents a degree of uncertainty in where you will be next now it gets worse and worse in general it's likely to get worse and worse as you try to take larger and larger time intervals a given degree of imprecision in what you actually know can get magnified and get magnified into worse and worse in precision as time goes on so in practice in practice classical systems don't really have the property that you can predict endlessly where they're going to be and exactly what they're going to do but you can always say given a time interval i want to be able to predict exactly for the next 30 seconds where every molecule in this room will be let's forget quantum mechanics now uh i want to be able to predict exactly where every molecule will be then there's a certain degree of precision in the present information at exactly one instant of time which will permit me to be able to extrapolate for 30 seconds okay it will not permit me to extrapolate for 40 seconds if i try to extrapolate for 40 seconds i will find the errors get magnified out of control if i want to predict correctly for 40 seconds i will have to do even better in my initial conditions and my knowledge of exactly what the the state of the system is so in practice this idea of determinism is defective it's defective because in order to determine for a given length of time you have to have precision which is so good that it's way way beyond anything anybody can do but in principle given any length of time in classical physics there exists a degree of precision which allows you to extrapolate for that length of time does that answer the question yeah so some systems are very predictable some systems are less predictable and get out of control very quickly they're called chaotic systems but uh the principles are the same that it's just a degree of precision which you need to know in the beginning in order to extrapolate for a given length of time well you could uh i'm not sure what the difference between predictable and the way i use the terms i use them interchangeably so i can't say and they are that there's no way out except that they don't give you an algorithm to predict because prediction requires okay okay yeah right the equations are deterministic if you know the initial conditions or deterministic are predictable infinitely predictable if you know the initial conditions with infinite precision you never do and therefore they're never completely predictable now the fact that you need to know both the position and the velocity in classical physics in order to predict what happens next reflects itself in the structure of the equations of mechanics specifically it tells you that the equations of motion newton's equations in this case are what are called second order equations instead of first order equations let me illustrate it by starting with a first order equation a first order equation what a first order equation means is that it only has quantities of first derivatives with respect to time in other words only contains velocities second order means it contains not only first derivatives but second derivatives means it contains accelerations the equations of motion acceleration is the second derivative of the motion um we could write we know what the real equations of motion of newton are f equals m a it contains acceleration let's write a phony equation let's write f equals mass times velocity force is f velocity is velocity and let's suppose that force just depends on where you are we have a particle that moves along a line it's subject to forces which vary along the all line force may be big here small there and so forth so the force depends on position and let's imagine this fake equation of motion that it's equal to mass times velocity and what is velocity velocity is the time derivative of the position the x by dt and i will use continuously throughout this course the notation that time derivative is just indicated by a dot dot means time derivative all right what does this tell me what do i need to know in order to predict what happens next i say for this equation we only need to know the position of a particle if we know the position of a particle i can tell you what the velocity is just from the equation if i know that the position is a particular position then i know the force on it and from the equation that tells me what the velocity is does it tell me the acceleration how about the acceleration let's see if i can compute the acceleration to compute the acceleration from this equation we just differentiate it once more we write that df by dt is equal to mass mass is just a constant times the second derivative of of the position or just the acceleration so we have over here acceleration what about the f dt the time derivative of the force the force varies with time because the position varies with time so the f by dt is just a reflection of the fact that the position of the particle varies with time and we can write the f by dt using the standard rules of calculus as just you the the change in f with respect to position times the change in position with respect to time in other words the velocity all right we've already figured out what the velocity is knowing the force knowing the position so we know the velocity and we can read off from this equation what the acceleration is we can figure out all of the derivatives of the motion if we know where the particle is by multiply differentiating this equation so what it tells us to make a short story out of it is it tells us that if we know where the position of the particle at any instant of time then we know where it's going to be in the next instant the next two instance the next three instance it completely predicts the motion but this is not the character of newton's equations newton's equations say f is equal to mass times acceleration not mass times velocity so let's look at this equation f equals mass times acceleration can i predict from this what the velocity is no this is no equation for the velocity if i know what the position is i know what the force is that tells me what the acceleration is but there's nothing in these equations which tell me what the velocity is that means i have to add in the velocity as a piece of information to begin with i have no choice i have to tell you in order to predict i have to tell you the position as well as the velocity then if i know the position and the velocity i can then predict the acceleration next to a third derivative the fourth derivative in all of them so that tells me that in order to know where i am and where i'm going to be i have to know the position and the velocity the phase space is a two-dimensional space so we see then that um that classical mechanics does have this character of a configure or a phase space of different configurations with a set of little arrows which tell you where to go next but the phase space itself has a position component to it and a velocity component to it we will go on and study the classical equations of motion and study them in a variety of different formulations but always the connecting link will always be conservation of information the idea that the laws of physics are completely deterministic and described by equations which tell you where you will be next that's the character of classical physics okay that's a result of this bifurcated system bifurcated one system you need two pieces of information it's one system but you need two pieces of information instead of one here i'm not sure what you mean by bifurcating two pieces and tails up there at the very top up here here you only need no no here you need one piece of information to say where you'll be next if you're here you stay here if you're there you stay there all there is is heads and tails that's all there is and if you know where you are you know where you'll be next you know when there were six states then you had six rules you can have more no this this is one rule heads goes to heads tails goes to tails that's a single rule and if you know that you're at heads then you know you'll be next at heads if you know your tails you know you that's so you only need the piece of information heads or tails that does it that tells you everything now we i suggested a different uh well yeah you don't need to know where you were before you only need to know where you are now to know where you'll be next so both of these laws require only knowledge of where you are at one instant to tell you where you'll be next you simply follow the if you start someplace you follow the arrow until you come back or you follow the arrow till you get to the next place and that tells you where you'll be next you don't need any more information than that the only question is what one of these points corresponds to does one of these cut points correspond to how much information does it correspond to a point in the space is it enough to know heads or tails to know what you'll do next or might you need to know the two previous things that's a different that's a different setup um let's stay with the heads and tails for a minute let's suppose that in order to say what happens next you need to know the previous two configurations let's do let's work out that example and let's write it in this form all right let's make up a law i think i had one down before heads heads goes to heads head's tails goes to what tails tails heads goes to heads and tails tails goes to tails okay is there something wrong with that oh this only depends on let's go to the right you're right you're right you're right sorry sorry sorry heads yeah we want this one to go to tails second one goes to heads good yeah that's also true yeah yeah yeah yeah it looks a little hard to work an example isn't it um let's see switch the second one this one make it tails okay i think that's perfectly all right you lose information why well okay here's here's a setup here's a setup if you know the previous two then you know what happens next in fact uh let's let's now see if we can make a table out of this not a table but a set of points a set of points then there are four possibilities heads heads heads tails tails heads and pales tails okay let's see his head's heads it's tails tails heads and tails tails now supposing you go from heads heads to heads this is heads heads heads that means you go from heads heads to heads heads right you go from heads heads to heads heads now supposing you go from let's see what happens if you start with heads tails where do you go to so you go from heads tails to tails tails okay okay what happens if it goes from tails tails oh we're going to run into trouble aren't we tails tails goes to tails so tails tails goes to tails tails bad not allowable by the laws of physics and now tails heads goes where tails tails goes to tails tails wait tails heads tails heads goes it goes to head's tails so that goes here well it seems to me we got we should be able to make a consistent law this one this one oh tails tails goes to uh tails tails goes to heads yeah okay so then tails tails goes to heads and that means that tails tails heads zombies tails tails goes to tails heads ah oh okay good good now we have a workable law now we have a workable law the only thing we had to do was to say that what we originally called a configuration namely a single heads or tails was not a complete specification of information a complete specification of information involved two pieces of information and once we recognize that we were able to uh to to write this down as a law of physics which is deterministic and reversible okay so you don't know offhand you don't know to begin with what pieces of information you need to know in order to know what to do next but that's what a step that's what the configuration space or that's what the phase space is it's the collection of all the things you need to know to know what happens next now of course you could go beyond this you could say i need to know the first three things in order to know what happens next you can do that you'll simply need more states to make a deterministic system what would it mean in classical mechanics to need more information than the positions and velocities suppose you needed a position velocities and accelerations that would mean third order differential equations but what would it say about the phase space it would mean you would need positions velocities and accelerations to represent the face base okay as it happens that is not the case for uh classical mechanics that's an experimental fact but it wouldn't stop us if we did if we did need the accelerations we would just write third order equations and we would make our phase space three dimensional the preceding program is copyrighted by stanford university please visit us at stanford.edu