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.


Welcome back. So we started off John chapter 18. We looked up to verse 11. The arrest is now completed and Jesus is being now taken to Annas, to the home of Annas and all this is still happening at night. So when you have a court trial being made when accusations are being brought against the accused, it is all done in public with enough witnesses watching so that everything is done in a just and fair and correct manner according to the legal procedures which have been laid down by the law of Moses and by all the leaders over the various centuries. So that everything is done in an orderly way in a correct way. So that's basically how any trial should take place. But over here everything is being done so crookedly. It's being done in the nighttime when the public will not be aware of what's going on and so they are going to fabricate charges and then they're going to present those fabricated charges to the public and you know stir them up into a kind of riot and turn them all against Jesus. So the actual facts, what is really taking place behind the scenes, the public is not being allowed to view that. In fact not when the entire Sanhedrin, you know the governing body which looked after Jewish matters, the entire Sanhedrin is also not being allowed to watch all of this. It's only some of the officials, some of the chief priests and Pharisees who are party to this which basically means, you know it's basically the conspirators and those who are willing to support them who are being allowed to watch all these secret proceedings and so then later when they go in front of the entire Sanhedrin and when they go in front of the public they're just going to give a summary of whatever they have fabricated and cooked up where reality is not going to come out. So we see here that right now Jesus is first taken to the home of Annas. Annas is not even the High Priest. He is the father-in-law of the High Priest. So what we get to know is that Annas used to be the High Priest between AD 6 and 15 which we know learn through historical records. So you know just as a reminder none of this is fiction. These are all historical events which actually took place. The names given here in these accounts we find these names in historical records. So this is not a fabricated story that we are looking at. These are all things which rarely took place in history. So Annas was the High Priest over the nation of Israel between AD 6 and AD 15 and then for some reason the Romans depose him. They remove him from his position and they place his son-in-law instead in the position of High Priest. So we don't really know what happened. Maybe the Romans had political reasons for removing him or maybe the Annas the High Priest did something crooked that he's not supposed to. We don't really know why he was deposed but the son-in-law was made the new High Priest. However Annas continued to hold power. So even though Caiaphas was the one now sitting in the chair of power it's actually his father-in-law who was still running the show and so Jesus is first taken to Annas home. When I say home it's not just one little home you know in case you're picturing a little house. This is going to be like a big mansion with a compound around it and all of that. So this is basically where Jesus is now taken and you know as we see in the other Gospel Sando over here not everyone can enter into the compound. So only those with influence, only those who are able to have some known person on the inside will be allowed to go inside and watch the secret proceedings that are taking place. So we are all very familiar with Peter denying Jesus so we will kind of skip that at the moment. We will look at some of the other details which we generally don't you know tend to reflect upon. So here you basically have Caiaphas and Annas and all the others questioning Jesus. If we could have someone read out for us John chapter 18 verses 19 to 23 24 yeah 19 to 24 if someone could read out please. I spoke openly to the word. Why do you ask me? Ask those people that I said to them. Do they know what I said? And then he had said these things. So if you look at the setting over here, this is the home of Annas. So probably the top conspirators, the high leaders are you know are the audience which is watching these proceedings. The rest of the people are not around and so they would have carefully prepared a list of questions which they're going to ask Jesus to trip him up to somehow get him to say the wrong things so that they can use that as charges. And so here in verse 19 we see that now they start asking him questions about his doctrine about his teachings and so Jesus humiliates them. You know he says why are you asking all of these things like as if you know you're getting to hear my teachings for the first time all along out in the open in the synagogue every day I've been you know teaching these things. There's nothing new over here. So he says in verse 21 why question me. Ask those who heard me. Surely they know what I said. So it's not like as if I've been you know teaching any anything false in secret openly out in the synagogue in front of the entire public. I have spoken out what I believe in. So all my teachings are open for everyone and everyone is acquainted with them. So now why are you questioning me like as if you know I have some hidden teachings which are wrong which are in some way against the law of Moses and so when Jesus says this mockingly you know one of the officials slaps him in the face and Jesus replies this he says you know you're slapping me have I said anything wrong. If I have said anything wrong then finally you know you testify about it and he says if I spoke the truth why did you strike me.  So because Jesus is completely in the right they can't find anything wrong in the things that he is saying always if you know I mean notice Jesus always used the Old Testament scriptures to back up what he was saying. He never just said you know anything that they can charge us being false anything that you know can be portrayed as going against what they had been taught in the Old Testament because Jesus showed again and again that he is the fulfillment of the Old Testament. So they actually could not find any defect in him so Anas finally gives up and he sends him off to his son-in-law's place. He sends him off to Caiaphas the High Priest. So that's one point that we can draw from this entire you know trial procedure. Moving on maybe to another passage if we could look at verses 31 to 36 maybe. Yeah someone could read out verses 31 to 36. This will be the you know questioning that is being done by Pilate okay. So verses 31 to 36 if someone could read out please. I'm saying you know you can change the answers of a particular problem. And for the Jews said it is not lawful for us to do it. And the same of these points which are filled which is you can try to die or you can try to enter the dogma again. Oh Jesus and so you can. Are you thinking of Jews? Jesus answered me. I'm just speaking for you to talk about this or you can do this concerning me. And I said am I a Jew? Your own nation and Jesus please have been for you. What have you done? Jesus answered. It's not a disability. It's not a disability. I should not be able to do this. Yes. Yeah I mean so sorry your voice kind of faded out. Did you read out verse 36? Okay yeah. Thank you. Yeah so we see that now after failing to find any false charges they bring him to pilot hoping that maybe pilot will just simply you know issue an order saying that he should be crucified. So pilot is not interested in this entire process simply because it's a religious matter and he never really had any interest in the Jewish religious traditions. He learned from history that on various occasions pilot was rather indifferent to the concerns of the Jews. I know whenever they try to bring up religious matters he in fact spoke in a rather derogatory manner sometimes regarding the Jewish people and their religion and their beliefs. So he really does not care about such things at all. So for him this is just an additional unnecessary headache.  Why are you bringing him over here? So that is when they say you know we want him executed we want him killed and we don't have the authority to do that because we are under Roman subjugation.  He goes and asks Jesus are you the king of the Jews and then Jesus does not answer him straight away. He just says are you inquiring whether I am the king of the Jews because you are interested or is it just something that you are repeating see here right now Jesus is giving this man an invitation to believe if he wishes to this is Pilate's golden chance this is his chance to you know have eternal life if he chooses so Jesus starts introduces this thought he says you know are you making this inquiry because you are personally interested or are you just simply saying what others have said and Pilate you know he just kind of casually replies am I at you your own people have brought you over here so what have you done so his basic concern is are you claiming to be a king are you now going to try to you know get onto the throne will this create political issues for me so that's all you know his interest is along those lines because Pilate right now is living under the rule of of Tiberius Tiberius is the Roman emperor at this time and Tiberius was a very fearful Tiberius was kind of you know scared he was very paranoid always wondering whether someone is trying to you know take over his power whether some king in one of his you know wassel countries is trying to rebel against him so Tiberius was very strict with the officials and governors whom he had appointed in his different territories and he expected them to keep all kinds of rebellions and any kind of treason under control so that his you know Roman emperor ship should not be harmed in any way so if Pilate is kind of lax regarding the serious situation if Pilate allows a king to start creating trouble in one of the territories then Pilate can actually get beheaded so now you know that is the that is why the minute he hears that this person may be a king he is worried and so he comes and asks are you the king and so Jesus assures him in verse 36 and says kingdom is not of this world if it were my servants would fight to prevent my arrest and you know very specifically we see that Jesus did not allow his servants to fight for him he did not allow prevent the arrest from taking place why because he has come over here to establish a spiritual kingdom right now yes in the end times the Lord will come to establish a physical kingdom as well but right now when he has come over here he has come with the purpose of establishing a spiritual kingdom and so he tells Pilate very openly my kingdom is not of this world okay so once he says that Pilate is no longer very concerned because this is not going to be a political issue okay so his only concern was that there should be no unrest in his immediate territory and so he has Jesus flogged even though he realizes now there is no political danger involved he hasn't flogged just to please these chief priests and Pharisees who have brought him so even though he understands that Jesus is innocent even though he knows that Jesus has not done a crime he is willing to have an innocent man flogged just to maintain peace with these Jewish leaders so all that matters to Pilate is staying in power holding on to his position he doesn't want anything to come in between that I know and his position so so he has Jesus flogged and then in maybe we could know in chapter 19 we can look at verses 7 to 12 yeah in chapter 19 if we can have someone read out for us verses 7 to 12 please I am so sorry you know your voice is not audible you know if we could have someone else read out but please I will try again wow it's so clear now thank you thank you the Jews answered him we have a law and according to our law he ought to die because he made himself the son of God therefore when Pilate heard that saying he was the more he was the more afraid and went to Gabriel and said to Jesus where are you from but Jesus gave him no answer then Pilate said to him are you not speaking to me do you not know that I have power to crucify you and power to release you Jesus answered you could have no power at all against me unless it had been given to you from above therefore the one who delivered me to you has the greater sin and 12 yes 12 from then on Pilate sought to release him but the Jews cried out saying if you let this man go you are not Caesar's friend whoever makes himself a king speaks against Caesar yes so we see that initially when Jesus says no my kingdom is not of this world now Pilate kind of is relieved he you know figures that there's no really political threat involved over here so he will in no way get himself into trouble with emperor Tiberius so he kind of backs off he has Jesus flogged he assumes that this will please the leaders and they will take Jesus away with them and his problem will be over and now the Jewish leaders continue to insist and they say no no no this person must be executed you need to give permission for that because he's claiming that he's the son of God and we have a law which says that anyone who claims to be the Messiah that person should be killed the simple truth is that you know in the in the laws which they drew up you know the Jewish people drew up over the centuries anyone claiming to be a Messiah was not to be given actually a death sentence no I mean this is not part of your law of Moses these are all the legal laws and procedures which people came up with over the centuries so according to that law anyone claiming Messiah ship would not really be given a death sentence but here they are outright lying to Pilate and they're saying in our law according to our law a person who claims Messiah ship should be put to death and this man is saying that he is the son of God now when Pilate hears this in verse 8 it says he was even more afraid because he first of all understands that there's some religious issue going on you know when Jesus says my kingdom is not of this world so he is aware that he has had an innocent man flogged but now this innocent man he realizes is someone who is saying that he is a son of God not a son of God the son of God and so he is afraid and he goes back inside and he says where do you come from because Jesus said my kingdom is not of this world so now Pilate is saying if your kingdom is not of this world where are you from he is kind of beginning to realize that there are greater spiritual implications here and so he is in fact afraid his heart is being stirred the Holy Spirit is convicting him of sin righteousness and judgment you know that's actually what is happening over here and Jesus does not answer him because Jesus answers when someone really wants the answer Pilate is he really open to receive the answer Jesus does not answer then he this Pilate says you refuse to speak to me I have the power to either set you free or have you killed so you better speak to me and then Jesus says you would have no power over me if it were not given to you from above and this really frightens Pilate because he began here I think he has actually in his spirit caught the truth that is not just a human that he is dealing with Jesus said very openly my kingdom is not of this world and now Jesus is saying you have no power over me unless it is given to you from above he understands that something greater bigger is going on over here and now it says in verse 12 from then on Pilate tried to set Jesus free he is actually afraid he is actually feeling the conviction inside his spirit that there is something going on over here this is Jesus personal invitation to this man I mean imagine how many leaders get to have a personal interaction with Jesus ways offering him a chance to repent and turn around and Pilate who is kind of on the fence at the moment wondering what he should do he finally makes his decision based on what the Jewish leaders are saying this is what they say in verse 12 they say if you let this man go you are no friend of Caesar anyone who claims to be a king opposes Caesar so this Jesus is claiming to be a king he is opposed to Tiberius now if you are befriending this Jesus then what's going to happen you are also going to be against Tiberius when Pilate heard this is what it says in the next verse verse 13 when Pilate heard this he brought Jesus out sat down on the judge's seat and gave his judgment it's such a sad thing so he wanted to hold on to his power even though he had sensed in his spirit that something is going on in fact he goes back and says where are you from because he understands begins to realize that this Jesus is not just from this world there's something divine about him and he just you know snuffs out those truths for the sake of this power which he wants to hold on to he goes and sits down on the judge's seat and he gives his judgment you know he agrees to the crucifixion so what did he get out of that maybe he held on to his power for another 10 years for another 20 years after that he ended up in hell for eternity so it's very sad the way people you know prioritized their values was this really worth it I mean holding on to his power maybe for another 20 years 30 years and then for the rest of eternity he has lost out on the invitation given to him so it's really not a good deal at all you know it was such a foolish, foolish deal that he chose and so even though pilot over here in verse 14 he says here is your king he says to the Jews for the Jews shout and they say take him away, take him away crucify him and so finally it says in verse 16 he handed Jesus over to them to be crucified but at least one thing he was sure he wants to do so when he puts up that you know inscription regarding this criminal who is going to be hung on the cross this is the wording which he puts up so maybe we can read out verses 19 to 22 if we could have someone read out for us 19 to 22 if we can have any one of the students yeah we had John Paul reading for a while if anyone else could volunteer please it's just a few verses all you need to do is open your Bible and read it so verses 19 to 22 please yeah go ahead please chapter 19 verse 19 to 22 then the high priest then asked Jesus about his disciples and his doctrine Jesus answered him I spoke openly to the world brother this is chapter 19 you're probably reading from chapter 18 so I read 9 from chapter 19 yes and verse 19 to 22 now the pirate wrote a title and put it on the cross and it was the writing was then many of the Jews read the writing was Jesus of Nazareth the king of the Jews then many of the Jews read this title for the place where Jesus was crucified was near the city and it was written in Hebrew, Greek and Latin therefore the chief priests of the Jews said to Pilate do not write the king of the Jews but he said I'm the king of the Jews Pilate answered what have I what I have written I have written here actually is proof that Pilate has sensed in his heart the truth which Jesus presented before him so when he is putting the inscription over there it does not they generally put over there an inscription telling the crime which the person has committed so in case someone has committed a certain number of murders that's basically what would be there on the inscription so that everyone would know why that man is being crucified because he's a very terrible serial killer or a murderer or whatever so over here what is the crime that Jesus has committed he is the king so he does not say that this man is claiming to be the king he very openly just simply writes over there Jesus of Nazareth king of the Jews and he writes it in three different languages that nobody misses out on that so this man who rejected the offer of eternal salvation which was being given to him at least he does this because he has caught a glimpse of the truth and he has understood it at some level and now he puts that in writing and so chief priests are very upset and they say you should have written that this man is claiming to be a king why are you writing over there that he is the king and Pilate says I have written what I have written yes so it is a small concession that Pilate gave but it was actually not enough he could have gone all the way and said yes this is the king and I choose to submit to him I choose to accept him that would have just changed his entire eternity but he chose not to do that and so yeah he loses out on salvation but you know God allows circumstances to be orchestrated in this way so that what is finally put over there on that inscription is not just a proclamation of the crime which this person has committed but rather a declaration of who this person is that he is the king of the Jews so in that capacity as their king Jesus chooses to die for this nation of Israel and then not only for the nation of Israel but also for the entire world maybe we can move into some final passages I mean simply because we have so much to cover if we could have someone read out for us verses 28 to 30 so this is chapter 19 and if you could have someone read out for us 28 29 30 after this Jesus knowing that all things were not accomplished the descriptor might be fulfilled said I thirst now a vessel full of sour wine was sitting there and they filled a sponge with sour wine put it on to his mouth now when Jesus had received the sour wine he said it is finished and when he said he gave up his spirit so if you look at the other gospel accounts we see that earlier on during the crucifixion process he was given another liquid to drink and he refuses to drink that because that is something that will dull the pain the crucifixion is a highly painful process literally hammering nails into your wrists and into your feet so they tend to give you something that will dull the pain a little bit and at that point Jesus says no he refuses to drink that because he is over here to bear the pain to become the sacrifice on behalf of the entire humanity and so he wants his mind to stay alert he wants to go through that entire process fully experience the entire thing because this is the punishment that he is taking upon himself on behalf of the world so that the world will not have to face the wrath of God so he does not drink that earlier liquid that was offered but now it says knowing that everything had now been finished now he is willing to drink what they are offering so now Jesus says I am thirsty and then in verse 29 it says the version which sister read out it talks about the wine now this is something that was cheaper kind of wine which the ordinary people would drink you know not that kind of expensive wine which is drunk by those who are more wealthy this is just something which maybe the laborers and the poorer people would drink so that is basically given to Jesus now and he is very thirsty he drinks that when you look at verse 22 that is basically where we are told what he is feeling in his body when this whole thing is going on because over there it talks about how his mouth is so parched you know that his tongue is sticking to the roof of the mouth that is because on the cross he would literally be gasping for breath because you know you would have heard sermons about this because of the positioning on that cross you cannot breathe until you lift yourself up on your feet each time you lift yourself up on your feet you can feel the you know the nails which are there on your feet poking in and so he would be gasping for breath his mouth is completely dry but he does not drink until he has finished the entire you know process of atonement and so in verse 28 it says later knowing that everything had now been finished and so that the scripture would be fulfilled Jesus said I am thirsty and now he you know they handed up to him with a sponge and he sucks on the sponge he drinks that and then it says he says in verse 30 when he had received the drink Jesus said it is finished and with that he bowed his head and gave up his spirit he did not just die he gave up his spirit he chose the timing when he would give up his spirit so this was all throughout this entire you know description that we have of the arrest and of the entire trial proceedings and the crucifixion we see that Jesus is very much in charge nowhere at no point is he helpless he is completely in charge of all that is taking place he is volunteering to subject himself to the different things which are being done at any point if he had wished to walk away he could have walked away so now he even chooses the moment at which point he is going to give up his spirit because generally when people were crucified they would be hanging over there for days you know for 2-3 days because it is a very slow death but in Jesus case he has not just come over there to hang on a cross he has come over there to atone for the sense of the word so once he is done he gives up his spirit because his work is completed so even death is under his control he gets to decide when he wants to give up his spirit and so we have the famous words which Jesus speaks where he says it is finished and as most of us know the Greek word that is used over there is tetelestai which basically is talking about complete total completion I mean as in if it is a project that you are working on you can say when you would say tetelestai when it is totally fully finished there is nothing no component of it is left it has now been closed finalized in every way so you know that is the kind of term that is used so having understood in verse 28 that everything had now been finished he now declares it out verbally and he says it is finished and having done the task for which he has been sent he gives up his spirit so Isaiah 53.5 you know explains to us what exactly was finished so we learn in Isaiah 53 you know verses 3 to 5 in the last portion where it says he was bruised for our inequities it says he was wounded for our transgressions and then it says the chastisement for our peace was upon him and in the last portion it says by his stripes we are healed so these are the things which were finished we see that the transgressions the inequities which we have committed the price for that has been paid for by him the chastisement the correction the judgment which should have come upon us was put upon him by for our peace so that God can be at peace with us so that God would no longer be angry and brothful against us so that he can have peace with this almighty God he chooses to have the judgment poured out upon him rather than upon us so now peace has been established between God and us and not just peace but also healing has been imparted so there are two things over here that have been completely totally finished first peace has been established between the almighty one and us anyone who comes to this almighty God in the name of Jesus Christ they will be granted peace with God God is not at against them is not angry with them and the second thing is healing physical healing emotional healing has been imparted to everyone who comes to the father in the name of Jesus now first Thessalonians 213 if we practice what's given in first Thessalonians 213 then what Jesus did on the cross you know becomes effective in our lives if we don't really practice first Thessalonians 213 we don't enjoy even though Jesus declared and said it is finished we will not enjoy that finished work unless we practice first Thessalonians 213 let's turn in our Bibles you know if we can have one person read out for us first Thessalonians 213 very important scripture for this reason we also thank God without seizing because when you receive you received the word of God which you had from us you welcomed it not as the word of man but as it is in truth the word of God which also effectively work in you who believe amen exactly these Thessalonians when they heard the words which Paul and his team were preaching they didn't accept them as human teachings being presented by a team no they accepted them as the word of God and because they accepted those words as the word of God that word began to work in them actively and that is what can happen for us if we can take the scripture where Jesus declared and said it is finished are we willing to believe it as the word of God to us that it is not just a statement made by John the writer but it was the divine word of God being released when he said it is finished when he said that what he had prophesied in Isaiah 53 verse 25 is now being fulfilled it is being completed because first in Peter understood that because when Peter wrote down first Peter 224 he says uses the past tense he says by his wounds you have been healed it is a finished work it has been completed it is not something that is going to be done in the future it is already done in Isaiah 53 which was written before the crucifixion you just have the open present being used over there but when first Peter 224 refers to the same verse he says by his wounds you have been healed because Peter understands that the process was finished completed just as Jesus declared you know when he said it is finished so those of us who choose to accept this as the divine word of God we will discover that this word starts working in our lives it will accomplish what the word has been released to accomplish so nothing can stop that so it all comes down to how we respond to the truths which are presented over here whether we choose to believe them as the divine fulfilled word if we do that that word begins to work inside us in all of its power and it will accomplish the purpose which it was released from Jesus mouth on the cross so it is a very mighty thing that is being offered to us as a gift we need to accept these scriptures as the truth and walk in it and when we do that the power which is there in that word begins to work in our lives to accomplish all that it is meant to alright we really don't have much time left maybe we can look at some very important points that are left maybe verses 32 to 37 if someone could read out John 19 verses 32 to 37 then the soldiers came and brought back of the first and of the other who was crucified with him but when they came to Jesus and saw that he was already dead they did not break his legs but one of the soldiers pierced his side with a spear and immediately blood and water came out and he who has seen has testified and his testimony is true and he knows that he is telling the truth so that you may I am so sorry your voice faded out I am assuming that we could hear up to verse 35 and then I just kind of faded out so it's yeah that's fine so the Jewish leaders they have finished crucifying the Messiah they knew that he is the Messiah he gave them clear signs of who he is so in spite of knowing the truth they chose to crucify him and now they are very deeply concerned about the Sabbath day that the Sabbath will somehow be corrupted if they allow dead bodies to be hanging over there and so they request and say you know if you could kindly tell these people who have been you know who are hanging over there before you know the entire Sabbath takes place so that you know the Sabbath day is not corrupted it's so hypocritical I mean they have participated in a hypocrisy they have brought false charges they have killed an innocent man they have done all that and now they are so deeply concerned that you know the Sabbath day should not be contaminated I mean they have done everything possible to contaminate the Sabbath day and now they are so concerned that the body should not be left hanging and that's the reason why in this particular case they don't leave the you know the criminals hanging over there for 2-3 days which was generally you know what used to happen so they want the people dead so that you know they can remove their bodies so their soldiers come they break the legs of the two criminals because those men are still alive and but when they come to Jesus again it's very clearly mentioned that he was already dead and so it's kind of being you know emphasized that Jesus chose when to give up his spirit he voluntarily decided to give up his spirit and again John's you know stresses and he says in fact one of the soldiers wanted to make sure so he pierces the side of Jesus and you have blood and water flowing out so this is John telling us that you know I actually stood there I saw it happening I saw the blood and water coming out it was proof that Jesus was indeed dead okay because afterwards rumors started saying that he didn't really die and you know all those false rumors which these religious leaders wanted to you know spread so here John is writing for his readers and saying see I was standing over any other cross during the entire process so I know I saw it happening I saw Jesus side being pierced I saw the blood and water coming out this was indeed a definitely dead person and then he says in verse 36 not one of his bones will be broken that scripture was fulfilled and the scripture which says they will look on the one that they have pierced these scriptures were literally fulfilled and John says I was the one who watched all this happening and my testimony is true is what he says and then you know in the few minutes that we have left if we can just look at one more fact which is mentioned that would be verse 41 where it says at the place where Jesus was crucified there was a garden and in the garden a new tomb in which no one had ever been laid so that is the place where Jesus is buried so it is very important for John to write over here for his readers that this was a new tomb no other bodies had ever been placed inside this tomb now in those days the well to do families who could afford their own burial chamber these would basically be caves so they would take a large cave they would buy a large cave and so inside that cave they would have generations of their family being buried so inside the cave they would have many maybe you could use the word shelf like small small shelves being created so when the person is first buried they would place them in one particular portion of the cave allow a few years to go by so that the dead body kind of dries out and all the internal organs the fluids that are there everything comes out so finally the body kind of shrivels up and now you just have skin and you have bones so at that point of time they would very reverentially pick up those bones and put them inside one of those you know the shelf compartments which they have dug into the sides of the cave so in that way they are able to have entire generations of the family being kept inside that particular tomb so they would have a special kind of a box and in that box they would keep the remains of the different family members so over here in this tomb there are no other bones there are no other bodies stored so when the resurrection takes place and the one single person who is supposed to be inside disappears there is no other body over there in case they had been in case it had been an older tomb then someone could have said oh you know Jesus just deteriorated decomposed extra fast and so the body lying over there that actually is Jesus they could have said that but you see this is a brand new tomb which you know Joseph of Arimatia has never ever used so there is no other body in there there was only one person placed in there and that person disappeared he was resurrected and so no false claims could be made about Jesus so this was another important fact which John includes over here so that his readers can be fully assured that Jesus indeed died and Jesus indeed rose up from the dead so this is him affirming that he has witnessed these things and is recording these details for his readers to know alright maybe we can quickly close with the word of prayer Lord we just thank you so much for all the things that we could cover today from your scriptures Lord we thank you for the intercessory prayer that you prayed for all of your disciples including us so Lord who are there today we thank you Lord that because of your prayer even today we are being kept we are being protected by the power of the name of Jesus we thank you Lord that because of your intercessory prayer even today we are going through this process of sanctification where we are getting more and more set apart for you and our relationship with you is growing deeper our relationship with the world is getting weaker thank you so much you are doing this for us and we thank you that because of this one day you are going to proudly present us before the father without blemish without fault I mean what a privilege your Lord that we who were sinners have now are now going to be glorified to the extent where we will be confirmed to the image of Jesus Christ himself what a privilege we have been given a Lord so we pray that we would serve you in a way Lord that honors and glorifies you thank you Lord in Jesus name amen amen thank you hello pastor hello yeah I can hear you you have a question is it sorry I joined the class late maybe you could have already addressed this issue the assignment I cannot access the assignment if you want permission to access okay I will I will inform the IT team you know in case maybe there's some technical issue they will clear it for you I will definitely inform them sure yeah okay thank you thank you


Now, Amidtide security protesters in those days joined Nigerians across the country to seek good governance and sustainable democracy. The protests which commenced at 10 am lasted for more than two hours and disrupted the free flow of human and vehicle movement around King Square, the airport road and its environs. Aggrieved about the current challenges bedeviling the nation, the protesters under the aegis of a coalition of civil society organizations, students and other concerned Nigerians, marched around the city center, chanted pro-democracy songs and submitted a petition to the state government. What we practice is far from a democratic society and I will tell you that we spend more than four days with all relevant security agencies just for how to organize these protests, marching for a true federation where regions can use their resources at their disposal the way they want for the benefit of their region. We are talking about democracy, the first segment to be the true security officer of the state, not your ceremonial security, security head of the state. To draw the attention of the Ramuama Dubuari retired, that Nigeria is bleeding profusely. Nigeria is sitting on the keg of gunpowder. Today security agencies, security agents are key to daily basis. You cannot travel to any more states. We must take, as almost be taken over by terrorists. You cannot travel from Benin to Auchi for fear of being kidnapped. We are protesting for our rights because our rights are alive, so we can have a better change. I mean look at our number one citizen, President Muhammadu Buhari, yesterday in his broadcast. His broadcast was empty and I think his salad means more than even his broadcast for a margin, saying that he's going to remind the evils what happened in the past, he's going to teach them lessons, he's going to flush them out. That's not what the President says in the broadcast. We are all on the streets protesting, protesting for good governance and we have distinguished between the military and the civil rule and we have agreed that what the difference is freedom of speech and democracy is for all of us and we struggle to have democracy and we are saying democracy must continue to exist to the extent that we have freedom of speech. Am I here as of assembly? You know what has happened in the past. As I speak I don't have a representative in my constituency. It's a call with constituency too. I don't have a representative because these guys, characters, when they were voted they decided to follow a Godfather and you know the ingredient of the struggle is to straighten the situation. That is our major concern as activists is to straighten the situation. Today Godfathers are taking over democracy that you and I struggle to have.


This is Keon White, Georgia Tech, defensive line. Great story. All right, so you go to Old Dominion first.   How long were you a tight end? So I was a tight end for my first two years. So started as a red shirt freshman. And then after that, my coaches were like, yeah, I know you're doing really well like a tight end, but we're going to switch you to defense. All right, so they're telling you this. Everybody wants the ball, right? Everybody wants to be George Kettle or whoever in the NFL. And now they're telling you're going to go to defense. What do you think about that? It was fine for me. I kind of sort of like defense anyway. When I was getting recruited out of high school, I played both. I was getting recruited for both. And so when I made the switch to defense, it wasn't like a big fight for me. The coaches told me that we're going to try you out of spring ball.  And so then you end up at Georgia Tech. And in 2022, it all happens. Why did it all happen for you this past season? I feel like it's just the same recipe that everybody does. Like nobody knows. Like nobody, it's not a secret. Hard work. Just hard work and just sticking to the process. And that's kind of sort of what I pride myself on. I'm not always going to be the best of something, but I'll try hard and work hard until I will be. 6-4, 2-86. Yeah, so we're waiting here at 2-80 today. But I've kind of sort of been going through a body reconfiguration process. So leaving the season at Georgia Tech, I was probably about 22, 23% body fat at 2-87. And then now I'm 2-80 at 15% body fat. So definitely going through a body, a recomposition phase. OK, so where do you play on the defensive line in the NFL? The defensive line. You play everywhere. And that's what everybody talks about with you. For sure. Definitely. And I like to be versatile. I like to be able to play all the positions and everything like that for sure. And the fact that you could move inside and still be able to rush the passer, NFL teams are looking for guys like you now. Yeah, definitely. So being versatile does nothing but help you. So I don't see it as men that keep moving me around. They keep, I like that. I like being able to be recognized as, hey, he can do this. He's not just a one-dimensional guy. So from Old Dominion at tight end to Georgia Tech as a defensive lineman with 7 and 1 half sacks and 57 tackles and 11 and 1 half tackles for loss. I mean, this has been a process, which I'm guessing means you're probably enjoying all of this as the process continues. Yeah, definitely. So I kind of sort of have a different perspective than most guys, because when I was at Georgia Tech, they do a good job of setting you up for life. And so I was working in corporate America. And I was working at a company named McKinney's and another company called Certified Finishes. And I was working in the office. So I realized then that office life isn't for me. And so I have a new appreciation for football. Like not to say I didn't love football before, but it kind of makes you say, man, I could be working in a cubicle right now, but I get to do the interviews. I get to be in the senior bowl. I get to go to the combine and things like that. So this is a better job. Yeah, sure. We'll do it as long as you can, trust me. It pays well, too. Yeah, for sure. Keyod White, Georgia Tech with us at the 2023 Riesa Senior Bowl.


This is the SF Productions podcast network It also moves from realistic stories about cancer and drug addiction to silly concepts.


Okay, so let's do this problem. It's very similar to the last problem that we did. We're going to take the atomic orbitals from the atoms that make up this polyatomic ion and make the molecule, or the polyatomic ion from it. Okay, so molecular orbitals. So remember what we do. We have to build an atomic orbital valence shell, electron diagram. So in this case it's the 2s and 2p. Let's go ahead and fill these orbitals before we do anything else. Okay? So we just count 1, 2, 3, 4, 5, 6, 7. 1, 2, 3, 4, 5, 6, 7. Then f plus is going to have one glass than that, so it must be 4, 5. Everybody got that? So now let's build the molecular orbital diagram. Remember for each bonding orbital we have an antibonding orbital. This one's going to be the sigma 2s orbital. This one's going to be the sigma star 2s orbital. And then remember for these small atoms the pi and sigma orbitals will repel each other. So we're going to have that normal, what you would think of as the normal electron theory. So the sigma 2p comes before the 2. This is where you're getting confused or whatever, but they label this just 1. And then the antibonding orbitals would be pi star 2p. Bonding sigma orbital and sigma star. So this question asked for both to figure out what the bond order is and the electron configuration. So in order to do both of those we're going to have to fill our molecular orbital diagram. So remember you're filling roles. So 1, 2, 1, 2. So that's all four of those electrons. So how many electrons do we have? 1, 2, 3, 4, 5, 6, 7, 8, 9. So 1, 2, 3, 4, 5, 6, 7, 8, 9. Okay so let's do the electron configuration first since it's right up there shown to us. So what's the electron configuration? So we just start from the bottom and remember put it in those parentheses. So sigma 2 star 2. Sigma 2s, sorry, sigma 2s 2. Of course it would be something I would say. Sigma star 2s 2. Sigma 2p 2. 2p 4. Hopefully you can still see this on the camera. Pi star 2p 3. You can see all that. Well at least you can see it for a second. And then the bond order. One half of the bonding electrons minus the anti-bonding electrons. So 1, 2, 4, 6, 8. Right? The bonding electrons. So anti-bonding 1, 2, 3, 4, 5. So 3 halves. So it's just telling you how strong the bond is. So anything that gets higher and higher and higher. As long as you're getting higher you're getting a stronger bond. A bond order of a half is not as strong as a bond order of 1. So if you want to think of a single bond that's a bond order of 1. A double bond would be a bond order of 2. I mean it's between a single and a double bond. So I guess we've already answered that question. But I guess the next question would be does this molecule exist? Or would this ion exist? Why? Because it's above 0. Even a bond order of a half exists, but it's very weak bond. But stronger than a non-no bond. Is everybody okay with that one? I think we captured it in a bottle there.


Okay folks, Larry Pesaventa with the mid-afternoon update. Stock's under pressure today, following through from yesterday, and that's surprising because if you look at the NASDAQ Banking Regional Index, the KRE, I'll post that a little later. Every single stock, every single one, folks, 50 of them are actually, yeah, 50 of them are all down. I mean that doesn't happen very much, 2.6% but the market's bouncing back, we've bounced back another 20 points, here in the S&P we've bounced back another 150 points in the Dow so there's buyers coming in, and as they should, because believe it or not folks, the low today was a beautiful ABCD in the SPY down there at 40-20. So there's nothing very about the technical picture as of yet, but who knows, Treasury Bonds still moving to the upside, Gold of course is in the area of Obi-Wan-Tin Obi out there in the stratosphere with CP-30 and R2-D2. It hit yesterday folks, 2080, it had such a beautiful, I'll go into that in the next segment here, but had such a beautiful 78% level at 2045, backed off $18 really quick and able to put a little bit of pocket change in the minds of a short-term trader and then at the end of the day, fortunately I was not here and not involved in the market anymore but it took off like a stripe of deer running through the forest with a tiger behind him. Anyway, it looks like we're going up to the 2100 level in gold and $27 in silver that certainly looks like it's going to be moving in that direction. Bitcoin was picking up speed and they had several people on Bloomberg today, actually at UHENRY who's one of the smarter dudes in the world. When it comes to this kind of stuff, he's a former hedge fund dude out of the UK. He said he was buying Bitcoin at $25,000 if given the opportunity and he also thought that gold was going to go up. This biggest fear was the fact that these blockchain cryptos will be taken over by the government and when you try to buy something with their electronic money, they're going to say, well, you can't buy it with that, you have to buy it with something else and then you have to give so much to charity and they're going to know exactly where you are and exactly what you're doing at any particular time and that's not good.<PHONE_NUMBER>, we'll be right back.