 We will resume with our discussion of the simple pendulum problem in which recall the Hamiltonian h of theta, p theta was p theta squared over 2 m l squared plus the potential energy which is m g l into 1 minus cos theta and we were specifically interested in finding out what the nature of the solution was as a function of time on the separate ricks which took you for instance from the saddle point at minus pi to the saddle point at plus pi in this fashion and back in this fashion and of course you had small oscillations larger amplitude oscillations inside and this picture got repeated as you went along we were interested in finding out what the explicit solution was on this trajectory here on that particular trajectory the energy had its value twice m g l which is the maximum value of the potential energy and we also saw that theta dot is proportional to cos theta over 2 on that trajectory we saw this last time and all you need to do is to integrate this. So this gives sec theta over 2 d theta is proportional to d t and what is the integral of this this leads to something like log tan theta plus pi over 4 and that is proportional to t on this side apart from some constants therefore theta can be written down explicitly in terms of tan inverse of the exponential of some time if you sketch that solution as a function of t then you discover that on that solution theta of 0 we took this to be equal to 0 theta of infinity was equal to pi theta of minus infinity is equal to minus pi starts at this point and ends at that point and we assume that it passes through theta equal to 0 at t equal to 0 and the solution looks like t here versus theta it started at minus pi pass through 0 and hit plus pi so it behave like a kink as if it had a kink and if you compute the angular velocity corresponding to the same solution here is t here is theta dot the angular velocity this quantity is by and large 0 except it takes a sharp peak up here and then back to saturation value therefore looks a little bit like that and this is like a lump this quantity this solution is related to what is called the soliton solution of the pendulum equation of the sign Gordon equation and this kind of lump solution something which is essentially quiescent nothing happens and then suddenly there is a little blip up there and then back down there to quiescence this kind of thing is a special kind of soliton it is called an instant on since it happens in time but we are not going to get into this business right now except to mention that phenomena like solitons are very characteristic of non-linear equations and the sign Gordon equation as I mentioned is a prototypical non-linear equation which has a very large number of applications and this is a very well known and famous solution of that equation the point I want to make here is that in as simple a problem as a simple pendulum problem it already makes its presence okay now let us get back to our primary discussion which was to understand what kind of phase-space structure does an integrable Hamiltonian system have and for this we go back all the way to a very simple 2 degree of freedom problem so far we have discussed a few 1 degree of freedom problems now let us do something in 2 degrees of freedom and we begin to see how things can get a little more complicated and the model I have in mind is the 2 dimensional harmonic oscillator you could imagine this to be 2 oscillators simple harmonic oscillators at right angles to each other which you are familiar with from elementary physics I write down all the Hamiltonian and since we decided to call q and p the coordinates and momentum so let me call this q 1 q 2 p 1 and p 2 this is equal to p 1 squared over 2 m that is the kinetic energy of the first oscillator plus one half m omega 1 squared q 1 squared assuming the oscillator to have a mass m and a frequency omega 1 plus an identical expression over 2 m plus one half m omega 2 squared q 2 squared for the second oscillator this then is my Hamiltonian and I have the characteristic the typical Poisson bracket relations namely q 1 with q 2 is 0 p 1 with p 2 q 1 with p 1 is 1 as is q 2 with p 2 so the one variables q 1 and p 1 have nothing to do with the two variables the completely independent dynamical variables it is two uncoupled simple harmonic oscillators with in general different frequencies added together to form a two degree of freedom system in four dimensional phase space the phase space is the space of the four variables q 1 q 2 p 1 p 2 each of which ranges from minus infinity to infinity so that is our set of the Hamiltonian of the system what is the set of equations well it is quite evident that they just two decoupled oscillators so we can write down the equations instantaneously this is p 1 over m and p 1 dot to the minus m omega 1 squared q 1 and similarly for 2 with a subscript 2 replacing 1 so q 2 dot is p 2 over m p 2 dot is minus m and of course each of these oscillators if you imagine them to be at right angles to each other oscillates may be in a plane the q 1 q 2 plane completely orthogonal to each other so the problem as it stands is completely integrable it is solvable now let us apply the general results that we know to this case and ask whether we can say something about the nature of the phase space and what kind of motion we really have we need for its integrability since in this problem little n is equal to 2 the number of degrees of freedom we need to functionally independent analytic constants of the motion and of course it is not very hard to see that this combination here which represents the energy of the first oscillator is a constant of the motion as is this combination here and since the ones and twos have nothing to do with each other this and that are in involution with each other so you could in fact call this f 1 and this f 2 and f 1, f 2 the Poisson bracket vanishes and each of these is a nice analytic function of the arguments and therefore by the Liouville-Arnold theorem this system is integrable completely we can write its solutions down explicitly so let us formalize this we have n equal to 2 and we could take f 1 to be equal to p 1 squared over 2 m plus 1 half m omega 1 squared q 1 squared in fact we could write f i is p i i i and i equal to 1 or 2 we guaranteed that f 1, f 2 is 0 hence the system is integrable completely now we ask what are the constants what other constants of the motion could the system have it is evident that the phase trajectories cannot be drawn by me since it requires four dimensional space but we could draw the trajectories as projections of the trajectories into various subspaces so for instance if I try to draw the projection of the phase trajectory in the q 1 p 1 plane for example since it has nothing to do with this oscillator it would just correspond to f 1 equal to constant which are ellipses as we know therefore you get phase trajectories of this kind and similarly in the q 2 p 2 space you get ellipses of this kind the periodicity here is 2 pi over omega 1 and 2 pi over omega 2 there what would happen if you try to project a trajectory on to the q 1 q 2 plane what would you get what would you get in general well it is evident that q 1 for given initial conditions it is clear that q 1 stays within this range and q 2 stays within that range so the motion is restricted to some kind of rectangle this being the amplitude in the first oscillator and this being the amplitude of the second oscillator but is the representative point is the projection of this trajectory is it going to be a closed curve or is it going to be an open curve it certainly bounded it is not going to go out of this region but is it going to be closed or is it going to be open it depends absolutely right it depends on the ratio of omega 1 to omega 2 if the ratio is rational this means that the two periods are commensurate with each other one of them is a rational multiple of the other then it is clear the overall motion is periodic and the system given enough time will come back to its original point yes in general you get Lisa juice figures now the question is is the Lisa juice figure going to close on itself or is it going to be completely open you are familiar from from elementary physics with for instance depending on what the initial phase difference is between the oscillators you would perhaps get pictures like this you could get something like this or you would get something like this or you would get a pattern which fills up the space and never comes back to itself depending what on what the frequency ratio is so it is clear we know this from elementary physics that if omega 1 over omega 2 is rational implies periodic motion in other words if we have two integers say L and M I should do not want to use M because I have already used that for the mass say K and L such that K times omega 1 plus L times omega 2 equal to 0 it implies that the ratio is rational where K and L are integers and then it is quite clear that the overall motion is periodic but if no such relation exists with K and L being integers then this ratio is irrational in general and the motion is not periodic although it continues to be periodic for the first oscillator and for this subsystem separately we are now interested in whether the motion is periodic for the entire system or not in other words in the four dimensional phase space does the point representative point come back to its original point position and the answer is no not if the frequency ratio is incommensurate if it is irrational the motion is not periodic such motion where the actual motion is made up of two or more mutually incommensurate time periods is called quasi periodic and the motion in general is quasi periodic in this case let me write this down what can we say about the motion in this case well here is where the general theorem I talked about helps us the motion has to occur on a surface on which H is constant since the total energy is constant so Hamiltonian autonomous system it also has to appear on a surface on which F1 is constant and on a surface on which F2 is constant simultaneously and of course the Hamiltonian in this trivial case is just the sum of F1 and F2 it is not functionally independent of F1 and F2 we have just two functionally independent constants of the motion which are isolating integrals namely F1 and F2 if we now go to action angle variables and there is a standard method for finding the action variable in this case it turns out that this Hamiltonian which we have written down here could be written in terms of action angle variables as follows this Hamiltonian becomes I1 omega 1 plus I2 omega 2 where the two action variables I1 for example is an integral over P1 dq1 over a complete cycle if the oscillator starts at the point say a and moves to be in the first oscillator then it is an integral twice the integral from a to b of Pdq the corresponding Pdq so it is an integral over the orbit that I mentioned the projection in the q1 q2 plane and similarly for I2 and if you use those as your variables and they can be shown to be canonical variables then the Hamiltonian this is what I call K of I as promised does not depend on the angle variables but only on the action variables and it is linear in the two action variables you can easily check that the dimensionality is right because action has the dimensions of energy multiplied by time and the frequency is inverse time so the product of the two is the free has the dimensionality of energy no angle variables are present here and indeed you can see that in this simple case delta K over delta I some I is in fact omega I and the frequencies are independent of the amplitude of the motion because the Hamiltonian expressed in the action variables is linear in the action variables therefore when you differentiate it there is no further dependence on the energy we could therefore now go to a representation of the phase space in terms of the action angle variables themselves and then what does it look like well I know I1 is constant and I know I2 is constant and this is the combination I1 this is like an ellipse in the Q1 P1 plane and then ellipse apart from units is topologically equivalent to a circle of some kind and similarly in the Q2 P2 plane this is an ellipse and it is this combination that remains constant that is again like an ellipse which is again topologically equivalent to a circle so I could in fact now identify what the angle variables are going back to P1 Q1 once I give you F1 in other words once I give you this ellipse then the state of the system of the first oscillator is specified completely by specifying for instance what this angle is as the oscillator moves around and similarly for the second oscillator therefore the state of the full system once you give me F1 and F2 which depends on the initial conditions is specified completely by specifying two angle variables which in this problem can actually be identified with the phase of the oscillator in the Q1 P1 and Q2 P2 planes what kind of space is spanned by two angles each of which runs from 0 to 2 pi two independent angles it's a two dimensional space what would that space be well a rectangle but with the ends identified because 2 pi should be identified with 0 and the other 2 pi should be identified with 0 so it's completely right it's like taking a square of length to of side 2 pi this is a square of side 2 pi so here 0 to 2 pi in the first angle variable theta 1 and 0 to 2 pi in the second angle theta 2 but I must identify that point with this and this point with this I must further identify this point with that and this point with this so I must take this piece of paper and roll it around and stick this glue this onto that and do that onto this what happens if I just glue these two together preserving the orientation I get a cylinder I take the cylinder and I preserve the orientation of the two ends and I glue them on and I get a two torus so this is how the torus structure appears in the action angle space so this phase space is nothing but the surface of a two torus in which I could measure theta 1 as the angle in this direction and theta 2 as the angle in the transverse direction in the cross section of this torus and any point on the surface of this two torus represents a state of this oscillator for prescribed initial conditions in other words the action variables the size of the action variables is prescribed once and for all periodic motion would correspond to the following situation as the representative point traverses in this direction the complete loop it should do so an integer number of times in the other direction that would be 1 is to 2 or 1 is to 3 or even 1 is to 1 more generally K times this period plus L times the other period that they should be commensurate with each other K times 1 period should be L times the other period in other words the frequency ratio should be rational in that case this representative point which wanders on the surface of the torus returns to its original point and the motion is periodic if the motion is 1 is to 2 or 1 is to 3 this simply means it takes if it goes around here once maybe it goes around in the other direction twice or thrice it's hard for me to draw this and comes back to the original point the trajectory therefore would not fill this torus up it would just be a curve on this torus folding around a few times like this and then it does this and then it does this etc and eventually comes back to the original point on the other hand the frequency ratio is irrational then we are guaranteed that the representative point never comes back to its original value precisely but does so arbitrarily close to the original value and infinite number of times the motion becomes quasi periodic one way to see this would be to ask let's imagine this frequency to be unity and then we ask how often does this representative point come and hit a cross section in this direction so I take a cross section of this torus and I ask here's the circle which represents the cross section I start at some point here so I'm here meanwhile it's going around the other direction when it comes back it comes back somewhere else and hits this cross section at some other point it comes back a second time it hits somewhere else and so on the question is does it come back to the original value ever or does it keep filling up this circle completely it again depends on the frequency ratio and we can prove the following theorem quite rigorously given a circle of unit circumference if I add at each time step I add a rational number you're guaranteed that wherever you start that point will be returned to in a finite amount of time after finite number of iterations every point on the circle whatever be the initial condition is going to be revisited over and over again all points are periodic points and the motion is periodic on the other hand if the number you add is irrational modulo one because you are on a circle then one can show rigorously that no starting point is going to be revisited there are no periodic orbits at all in fact you can go further you can show that the iterates of any starting point are going to eventually densely fill up this entire circle and uniformly so so it's not only important that it fills it up densely but uniformly there's no bias there's no particular part of phase space which is preferred over other parts of phase space on this torus so the iterates the trajectory any trajectory which starts here will wind around this tightly till eventually it densely fills up like a space filling curve it densely fills up the torus but never comes back and closes on itself comes arbitrarily close to its initial point many many times an infinite number of times but never recurs and the motion is quasi periodic so this is characteristic of quasi periodic motion that the system densely fills up a certain subspace of the phase space and it's said to be ergodic on this is on this torus so quasi periodicity implies to use a technical term goddess on the torus I repeat by ergodic I mean that any initial set of any initial conditions I take a little patch of initial conditions here as time goes on each of the points in this patch evolves in time wanders around this torus this little patch visits every neighborhood of this on this torus an infinite number of times comes arbitrarily close to every starting point and eventually fills up this entire torus uniformly and densely with no exceptions it's said to be ergodic on the torus what that implies is if you want to compute long time averages of any physical quantity you can replace the long time average by a statistical average with a uniform measure on this torus a uniform probability distribution on this torus and that's the implication of what this word ergodicity means and in fact you know that in statistical mechanics you normally replace time averages by ensemble averages by statistical averages and the distribution function has to be prescribed to you by the conditions under which the system is kept and that implies a certain kind of ergodicity that assumes that time averages can be replaced by statistical averages this is exactly what happens here in this dynamical system which displays quasi periodicity we can go further our phase space is four dimensional any motion for given initial conditions is restricted to a three dimensional subspace of this four dimensional space namely the energy surface now in a three dimensional space the motion is further restricted in each case to a two dimensional surface namely some kind of torus and it's evident if I try to draw this for instance perhaps there is a torus here and if I draw a cross section of it it's like this a different initial condition would correspond to a different torus of this kind like that for instance schematically so the whole space the energy surface is simply filled up by these successive nested tori and moreover since phase trajectories cannot cross themselves it's evident that if you start on one torus you're never going to escape you're going to be on that torus if you start on a different torus you remain on that torus the gaps between these tori are also filled up completely by other tori and that's the structure of the phase space for an integrable system like a two dimensional harmonic oscillator but the matter is more general than that even more generally than this if you have a two degree of freedom system which is completely integrable then the motion in general is restricted always to some two dimensional torus what would be the major difference between an arbitrary two degree of freedom integrable system with bounded motion and a harmonic oscillator what special about the harmonic oscillator what's the feature that's absolutely special about the harmonic oscillator you can't jump between tori but that is true even for a nonlinear system even in general this would be true and that's simply a consequence of the fact that once you are on a torus that represents a constant of the motion and you are on that surface what's very special about the harmonic oscillator what the fact that the cross section is a circle we're doing we're talking about topological aspects here so the fact that the cross section is a circle instead of an ellipse and so on this makes no difference that's simply choice of units what's particularly special about the harmonic oscillator the equations of motion are linear that's certainly true it's not a nonlinear system but what's very special about it go back and recall that omega i was in general equal to delta k of i over delta i sub i what happened in the case of a harmonic oscillator it was constant the frequencies were constant independent of the initial conditions independent of the energy of the oscillator independent of the amplitude of oscillation this feature is not true in general in general this is some function this is equal to some function of all the action variables and that would mean that the frequencies on the different tori would be different in general it also immediately implies that if you have a nonlinear system of that kind on certain tori depending on what the values of the action variables were because that's what determines the frequencies frequency ratios could become commensurate but you move to another torus it could become incommensurate in general so the motion could be actually periodic on some tori and a periodic quasi periodic on other tori generically of course it would be quasi periodic but there could be accidental cases where you have resonances and the motion reduces to periodic motion on some tori that's indeed what happens what happens further is even more interesting in a general 2 degree of freedom system if the system starts out as integrable and you add to the Hamiltonian a small piece which takes you away from integrability and you lose the integrability property then what happens is even more interesting in general the set of tori no longer exist because it's no longer an integrable system but in general it doesn't happen globally in other words first for certain sets of initial conditions for certain delicately balanced frequency ratios the tori get destroyed the rational tori get destroyed first and gaps form between tori and in those gaps the motion actually becomes irregular it's no longer periodic or quasi periodic becomes chaotic and this happens when you have not an integrable system but to an integrable system you add a small perturbation which destroys the property of integrability and then typically you have chaotic behavior in the gaps between these tori as you jack up the strength of this perturbation and make it go further and further away from an integrable system in other words certain symmetries are destroyed more and more then successive tori go on getting destroyed and still certain tori persist on which the frequency ratio is quote unquote extremely irrational so now we're getting into number theoretic considerations and we have to discuss we have to decide what's meant by a number being more irrational than another number we'll come to that in a second and eventually all the tori get destroyed and the motion becomes completely chaotic and we will talk about chaotic motion in much greater detail later but this is the mechanism by which this is rough idea of what happens in such systems in the same breath let me mention that if you took an n degree of freedom system for example n harmonic oscillators independent of each other what would the motion be on well the individual energies of these oscillators are n constants of the motion in involution with each other so the system is integrable in the same sense what would then decide where this representative point is just the angle variables the individual phase angles of all the oscillators in the corresponding qp spaces what would you represent that kind of phase space by by an n torus the generalization of a two torus to n dimensions in which n angles would specify the state of the system and then you could ask is the motion quasi periodic on this n torus or is it periodic and in general of course the answer is it would be quasi periodic unless you had n-1 relations of the form k1 omega 1 plus l1 omega 2 plus etc up to something times omega n equal to 0 and you had n-1 such relations between the frequencies if you did then all the frequency ratios are commensurate with each other and you have periodic motion but if even one of them is lost you do not have that relation then you have some degree of quasi periodicity and if you have no such relation at all with integer values of these parameters here then the motion is completely quasi periodic on an n torus I cannot draw a picture of an n torus what I have done here is to draw a section of a two torus that is the best I can do on the blackboard but you can easily conceive of the fact that you have motion on an n torus if the system is integrable the phase space dimensionality is 2n the energy surface dimensionality is 2n-1 and the torus is an n dimensional object now in three dimensional space it is clear that two dimensional objects such as these rubber tubes they strike the entire space you can put one inside the other and fill up this entire space if you had instead of three dimensions if you had four dimensional space and you took these two dimensional objects here they clearly do not fill up the space there is no way in which the entire space is laminated by them so it is evident immediately that if you took three or more degrees of freedom n greater than equal to three the phase space is 2n dimension the energy surface is 2n-1 dimension if the system is integrable the motion occurs on an n torus in certain variables and this is n dimensional if n is 3 then this is a three dimensional object this is a five dimensional object the difference in dimensionality is 2 not 1 so it is clear two three dimensional objects cannot separate this five dimensional space or to put it another way in three dimensional space if I have a two dimensional surface of this kind a torus there is an inside and an outside it splits up this whole space into an inside and an outside that is not possible if the dimensionality between the object you are considering and the dimensionality of the space is greater than one is this clear if I take three dimensional space and I draw an infinite plane in it that is a two dimensional object there is an up and a down it splits the space into two parts but if I took a straight line which is a one dimensional object it does not split three dimensional space into two parts at all in exactly the same way a two n minus one dimensional energy surface cannot be split up into disjoint parts with by n dimensional objects if n is greater than or equal to three when n is two this happens because this is two and that is three that is exactly the situation we looked at there the implication is deep the implication is these gaps here would all be connected to each other there is no way you can have an inside and an outside unambiguously all such gaps you must imagine would be collected to each other there is enough room to go from one gap to another gap and I already mentioned that once you have a perturbation which destroys integrability then some Torah I get destroyed and part of the motion for some initial conditions becomes chaotic but if the motion is chaotic in this region and this region is connected to all the other chaotic regions this means that the chaos can spread that all random motion all chaotic motion in non integrable Hamiltonian systems with three or more degrees of freedom all such regions of chaotic motion are in general connected to each other and they form what is called a stochastic web and a kind of diffusion will occur over long time periods from one region to another and this has profound implications in dynamical systems especially in applications such as accelerator physics this immediately makes a huge difference because it is in some sense it says that regions where the dynamics is unstable exponentially so are actually spread out all over the phase space not confined to certain regions of it will try to come back when we talk about chaos will come back some implications but the reason I digressed here was simply to point out that very simple arguments based on dimensionality can actually lead us to these conclusions fairly widespread general conclusions in this case so having talked a little bit about quasi periodicity let's now let's now take up a specific example you go back a little bit and take up a one degree of freedom system where I show you explicitly how the nonlinearity plays a role and how time periods could actually depend for instance on the amplitude and the energy let's try to see if you can generalize the harmonic oscillator so let's look once again at a one dimensional 1d nonlinear oscillator I have a single degree of freedom let's call it Q and a conjugate momentum P and the Hamiltonian in this case is H of Q and P and this is equal to one and a half let's write this as P squared over 2m plus a potential energy let's fix the frequency of this oscillator the natural frequency or some constant to be equal to 1 so in suitable dimensions this thing here becomes Q to a power which is not 2 this is 2 would mean a harmonic oscillator a linear harmonic oscillator but let's in general have a power 2R and let's set the mass to be equal to 1 so we just have this as the Hamiltonian divided by 2R to start with let's say R is a positive integer so R equal to 1 implies simple harmonic oscillator the linear harmonic oscillator well we could write the equations of motion down immediately and what would they be as before Q dot is P over m and P dot is minus delta H over delta Q so this would be equal to minus 2R cancels so Q to the 2R if R is greater than 1 it's clear this is a nonlinear system and a badly nonlinear one because if R is 2 for example this is already a Q cube term and you can't linearize about the origin where are the critical points the origin is the only critical point so Cp at 0 0 what is the linear matrix say if you linearize this what would this be this would become yeah this becomes 0 right here and then what would happen here you'd get a matrix a linear matrix if I tried linearizing I'd get L equal to 0 1 over oh let's just take this to be 1 since I have chosen such that the mass is 1 so this is 0 this is 1 and out here I get 0 and 0 because this is already a higher order term for R greater than 1 what would you say are the eigenvalues of that 0 and 0 any triangular matrix the eigenvalues sit on the diagonals so this is in fact a degenerate case we're not able to decide if the origin is a center or if it's some other kind of critical point based on linearization simply because the system is degenerate the critical point is degenerate but let's take recourse to physics and ask what kind of system we have and then I'll point out a method by which we can decide that the origin is in fact a stable center and we can do that very trivially in this problem by simply drawing the phase trajectories Q versus P yeah yes I'm taking a special case by the question is why am I restricting myself to R equal to a positive integer I'm going to relax this condition subsequently let's first take R equal to a positive integer see what happens draw some conclusions and then we relax this condition see what happens what kind of phase trajectories do you get for this system the way I've written it it's clear that whatever phase trajectory I get the closed curves for all positive E because these are all positive terms each of these it's completely symmetric about the Q axis as well as the P axis the motion is evidently restricted to some kind of rectangle so you're going to get some kind of oval this being the amplitude of the motion when R is one of course it becomes an ellipse when R is equal to 2 then you get P squared plus Q 4 equal to a constant and that's the sort of squashed oval by the way what do you think will happen if R becomes larger and larger what do you think will happen to this curve you're plotting a curve you're plotting a graph of this quantity equal to some constant E which is positive and what happens to this graph as R becomes larger and larger in which direction is it going to get squashed what's going to happen it's it's evident that this is with this amplitude restricts your the total energy restricts this amplitude this amplitude in fact is given by if this is a then it's quite clear that a to the 2 R over R 2 R is in fact the energy because the kinetic energy is 0 these points so what kind of graph would you get if R becomes larger and larger I fix the energy and then I just change R what happens what would this would this ellipse become fatter like that or would it become thinner in this fashion what would it do it's clear no I fix the amplitude I fix this I fix the energy and then ask what does this graph look like I fix the amplitude and ask what does this look like I'm talking about okay let me be specific I fix the amplitude and now I ask I change this R I increase it I do the same problem for a different R as R increases what happens to this graph do this you start doing this as R increases it would do this eventually as R tends to infinity it would become the rectangle suppose R will take what he said a little seriously now suppose R becomes smaller and smaller if R is equal to 1 what would happen then of course you could say this term changes sign if R becomes less than 1 for example this term changes sign we want to keep everything positive so let's take modulus to do it to R what happens if R is equal to 1 it's just an ellipse what happens if R is equal to a half it's not a parabola so let's go back let's do a little bit of elementary calculus I ask the question X over A to the power R plus Y over B to the power R is equal to 1 and I want to plot this on the XY plane what does this graph look like A and B are some positive numbers since these are positive numbers it's quite clear that the motion is restricted to some kind of rectangle of this kind of this is size A and this is B when R is 2 this is an ellipse so let me draw that what if R is equal to 1 what kind of straight line these are moduli absolutely so it would be some curve like this this is R equal to 1 what if R is equal to 4 it would get fatter and fatter it would start doing this and finally this rectangle is a limiting case when R tends to infinity what happens if R becomes less than 1 yeah absolutely the curve starts getting concave it starts doing this so this would correspond to R less than 1 what happens if R tends to 0 well as R tends to 0 either X is equal to A and you are finished or Y is equal to B and you are finished and the rest the other variable must be 0 so it would be the axes themselves it would be the axes themselves so it would tend to this so it's really a whole family of graphs that we are talking about here the ellipse is just one such member of this family so coming back to what we were doing here is q and p on this side the phase trajectories are some kind of ovals completely symmetrical in this fashion and for every set of initial conditions you have a concentric oval of some kind and these are the phase trajectories a very nonlinear oscillator and of course the system is integrable as it stands simply because there is a constant of the motion just the Hamiltonian itself in one degree of freedom that suffices it some algebraic function of p and q and that's a constant the motion is bounded its periodic motion it's completely clear what can we say about the time period of motion how does it depend on the energy how do we determine this of course in the case R is equal to 1 the simple harmonic oscillator case we could solve for p and q explicitly because the equations of motion were linear and you got trigonometric functions and we know the time period of trigonometric functions we know the periodicity but what happens if it's nonlinear what for instance would happen if R is equal to 2 or 3 or 4 what would happen how would you determine the time period of this oscillator specifically I would like to know how it depends on the energy of oscillation on the total energy what would one do in such a case well you agree that the time period of motion by symmetry is 4 times the time it takes to go from there to there and by symmetry it's just 4 times the time taken to go from here to here or the oscillator to start at the origin and go to the end of its amplitude so suppose the amplitude is a and I look at this portion of the trajectory and I write the time period as equal to over the full orbit dt 0 to t dt but I could write this as equal to 0 to a dq over dq over dt which is q dot but I must make sure about the sign of this quantity which could either be plus or minus it's positive in the upper half part of the diagram and negative here and by symmetry since we said it takes the same time to go from here to there as from there to here this is equal to 4 times on this set to go from here to there to go from 0 to the amplitude where by q dot I mean the positive square root of e minus this quantity apart from some constant factor so this is proportional to apart from some constants numerical constants it's proportional to 0 to a dq divided by q dot but q dot squared p squared by the way is just same as q dot squared since I have set the mass equal to 1 is e minus q to the power 2r so this is square root of e minus q to the power 2r apart from numerical constants 2r over 2r but what's e for a given amplitude a a to the 2r over 2r so it's evident that this whole thing keeping all factors of a present apart from some numerical constants this goes like integral 0 to a dq over square root of a to the 2r minus q to the 2r how are we going to evaluate this for general positive integer r this is not easy to evaluate it's some complicated integral you can't write this in simple form by any trigonometric substitution as you could in the case 1r was equal to 1 then of course you got a sign inverse and you could do the integral but we don't need to do that we need to only extract the dependence on the amplitude a what should I do to extract this dependence I could expand it by normally integrate term by term and then reconstitute the whole thing but there's a much simpler thing method by which you can extract the entire a dependence of t in one shot q is bounded between 0 and a so what does that suggest to you well the only length scale in the problem is a we'd like to know how t depends on a so I should yes I should scale out this quantity a so I put q equal to a u change variable of integration and then this implies that t is proportional to an integral from 0 to 1 numerical integral dq that gives me a factor adu divided by I get an a to the 2r here which comes out of the square root and gives me an a to the power r so a to the power r square root of 1 minus u to the power 2r multiplied by a numerical factor that integral is guaranteed to be some finite number because the only singularity which you have is at u equal to 1 and for instance if you had u to the power 4 here then you have to worry about whether this integral exists or not but you have 0 to 1 square root of du du divided by square root of 1 minus u to the power 4 and that's an integrable singularity let me show you why for instance if you had 0 to 1 du over square root of 1 minus u squared this of course could be written as 1 minus u times 1 plus u and then you immediately see that this is the part that crosses trouble at one not this is quite finite and this is like dx over square root of x when you integrate the square root of x goes on top and the contribution at the upper limit vanishes if you have a higher power like 1 minus u to the power 4 for instance then that could be written as 1 minus u squared times 1 plus u squared and once again it's only square root of 1 minus u that creates a problem and this is true for every positive integer r so the integral is actually finite there is no difficulty at all and it's some gamma function it can be written down in terms of Euler gamma functions etc. But what we are interested in physically is the dependence of the time period on the amplitude and that's like 1 over a to the r minus 1 what happens when r is 1 it's independent of the amplitude that's exactly what the harmonic oscillator is but for all other r there's a dependence on r on this index r in fact if you set r is equal to 2 and we have an oscillator which is a quartic oscillator you have the Hamiltonian as p squared over 2 plus 1 4 to the power 4 the time period of oscillation in this oscillator is proportional to 1 over the amplitude in other words the smaller the amplitude the longer the time period seems counterintuitive but why is it happening why is this happening why is the amplitude increasing as the amplitude decreases and in fact as the amplitude goes to zero the time period is going to infinity what's the reason this is happening what's the shape of this potential energy it's very flat at the origin absolutely right so if I plot V of Q versus Q this potential has a third order minimum here it's extremely flat so as you come smaller and smaller as the total energy decreases the restoring force becomes smaller and smaller because it's extremely flat here if you started with this much as the total energy then by the time it gets here it has quite a lot of kinetic energy and it zips right past but if you start here it doesn't have enough kinetic energy to zip past this it rolls down this extremely slowly because it's not a simple minimum of the potential it's not parabolic at that point you have a diverging time period as a function of the amplitude this is characteristic of non-linear behavior in this case extremely flat higher order minimum is immediately going to lead to such funny kinds of behavior we're actually interested in finding out how the time period or the frequency changes as a function of the total energy of the system how are we going to do that we already have this result T goes like 1 over a to the r minus 1 so how does it depend on the energy exactly a to the power 2r is proportional to the energy which implies that a is proportional to e to the power 1 over 2r and that in turn implies that T goes like a to the power 1 minus r which is a to the power 1 minus r over 2r T e to the power 1 minus r over 2r so that's the dependence on the energy of the time period we could translate this to the frequency and we'll do this and see what the action variable is in this case and we see how the tori actually depend the size depends on the initial conditions in this case so it's a very simple illustration of a non-linear oscillator in which you have almost the same kind of structure as in the simple harmonic oscillator but because it's non-linear you have an actual dependence of the frequency of oscillation on the energy of the system so let's stop here and then we take it from here.