 So, now solving the Hill's equation by this method, we introduce three new parameters alpha, beta and gamma. We will see shortly what are the physical significance these parameters have. These parameters are known as quiz parameters and some literature says optics parameters for like this, invariant of motion or he has been. Now we see the expression of this invariant of motion and subject. Now you can see that this is the equation in X and X prime means they are two coordinates. One is X and another is X. Expand shows the angle and explain X shows the displacement. So these are two coordinates or two variables. So this is the second order algebraic equation in two ways means it may show a conic section. So for conic section, we see that gamma beta is alpha is fair plus one means gamma beta is greater than this is by the definition of the gamma which we have provided in the slide and the particular determinant which we can calculate using this second order algebraic equation which defines that this second order differential equation is not degenerate means it doesn't show a straight lines. So this shows that really this is a conic section and this shows that this only section must be an ellipse. So this equation shows any as in the case of simple harmonic oscillator, we obtained an ellipse in the X and X dot here in X and X prime we are getting some area of this ellipse is pi A square and A square is invariant means area of this ellipse is a constant quantity means throughout the propagation of the particle in entire optics. Due to variation in gamma alpha beta, the orientation and elongation of this ellipse may change over the area we will mix. So this is a remarkable result in the beam physics. Now we come on the physical meaning of this parameter. The physical meaning of beta is very simple because in the solution we get X is equal to A root beta and value of phase part varies between plus minus 1 at any place possible may be between the plus minus 1. So maximum value of this X is A root beta. Maximum value of A means maximum value of displacement at a particular location S due during the exhibition of beta tron oscillation by means if suppose here we do something suppose we take different combination of X and X prime because these are two variables so we may have various combination and these combination each combination is the same A means these many initial conditions are on the perry of the ellipse decided by this A and definitely at the location where we are getting this the quiz parameter of that location. So this may be the ellipse say S is equal to S location at S is equal to S we have certain this is this may be the one combination of X X prime this is another this is third this is fourth like this so this will define an ellipse. Now these particles propagate through the optics so this ellipse may get rotated may get elongated but the area will remain safe now here you can see the maximum distance from the design trajectory is this and this is A root beta. So by knowing the beta for a particular invariant of motion we can find out what is the maximum possible amplitude of beta tron oscillation corresponding to that invariant of motion at a particular location S is equal to S. At another location because beta changes means say at S is equal to S 2 beta changes because beta is the function of S that location amplitude will also change and in the case of synchrotron in the case of synchrotron suppose this is the orbit of the synchrotron and you are looking at S is equal to X means you had a detector which can record the position of the particle which is passing through this line then you can record it. So suppose in first term you will you record X 1 after first complete turn it will come again this particle and you record X 2 its position and in third term its position with respect to percentage it is X 3 and in fourth term it is X 4 like this you have recorded X 1, X 2 and up to X n. So what will be the maximum value X you will record it is given by A and if you will plot these X 1 to X n suppose this is the number of turn or here you plot the X so you will get such kind of motion which is harmonic motion and these are the beta tron oscillations and the amplitude is represented by A so beta actually shows the amplitude of beta tron oscillation so beta has that significance and definitely we have seen that beta is also related with the phase advance means local wave length. Now for obtaining the physical meaning of gamma that was the third parameter which we used the second parameter is just the derivative of beta so second parameter shows you if beta increases means amplitude increases means beam is diverging if beta decreases so beam decreases so beam is diverging. So d beta by ds for a diverging beam will be positive while we are defining alpha as minus half beta prime so alpha is negative so negative sign of alpha shows the diverging beam and positive sign of alpha shows the converging means focusing and defocus. Now for obtaining the physical meaning of gamma we again write down the expression of constant or invariant of the motion and we differentiate this equation with respect to X and when we do this we get dx prime by dx if we make it 0 that will give you the maximum of X prime what is that maximum of X prime we see it this is simple calculation so gamma X plus alpha X prime will be 0 when we will equate to 0 dx prime is 0 it will give gamma X plus alpha X prime this is the maximum value of X prime because we are equating the first derivative to 0 this will be 0 and using this equation we get this so what is the value of X at the location of X prime max we obtain this now if we use this value of X in the expression we will get the complete expression in the form of X prime so we have converted the whole expression in X prime so we will find the value of X prime what is that value we will see it this we get gamma beta minus alpha square however you know that gamma beta is 1 plus alpha square so this equation become gamma beta minus alpha square is equal to 1 plus alpha square which is the gamma beta minus alpha square so this is the one so numerator becomes 1 and in denominator we have gamma so 1 by gamma X prime max square is equal to a square so X prime max square is gamma a square and X prime max is k so similar to beta function which shows the amplitude of the betaton oscillations gamma shows maximum angle during the betaton oscillation at a particular place so beta shows the displacement maximum displacement at a particular location and at that location gamma shows the maximum angle trajectory will be with respect to descent descent and alpha relates these two parameters so alpha generates a correlation between the displacement and angle so now we can plot and ellipse using gamma X square plus twice alpha X X prime plus beta X prime square is equal to X prime how this ellipse can be plotted we have seen two methods in one methods we generate various combination of X X prime corresponding to same k and that various combination we produce an ellipse and the area of this ellipse will be pi a square which is invariant and second technique is that instead of generating various initial condition take a single condition in a synchrotron and record its position turn by turn definitely you have to record the X prime also for generating the ellipse so on each turn particle crosses the desired location record X and X prime and after many turns when you will connect these points you will see this is an ellipse so there are two methods to generate this ellipse one is combination of various initial conditions another is single initial condition but multiple turns in both cases you will get this ellipse so in this ellipse the maximum amplitude is up to this point so this is represented by a this is this ellipse is shown the betatron oscillation so this is the excursion of betatron oscillation in displacement and this is the excursion of betatron oscillations in the angles so in angle the maximum excursion of betatron oscillation at a particular location s i'm always talking about a particular location because at another location this ellipse will be different because gamma and beta at that location will be different than this because beta and gamma are the function of s however the area of this ellipse at that location will be same as the area of this ellipse here because area is invariant of the motion now you can connect this thing with also rivoli's theorem in classical mechanics if you studied rivoli theorem this shows that under the conservative forces volume in the phase space or density in the phase space remains constant so it is a similar thing if however you can say that rivoli's theorem in the phase space and we are talking about the x and x prime what is the relation between the x x prime and phase space you can see that x prime is basically the p x upon p s and already we have seen in paraxial approximation that p s is approximately equal to p and which is a constant because magnet doesn't change the momentum so x prime is directly proportional to p x so if we say that we are plotting x x prime the similar kind of plotting will be in the x p x also in our case so we say that x x prime also represents some kind of phase space it is strictly known as trace space the space made by x x prime is the trace space however because x prime is proportional to p x we can loosely say that this is the phase plot of phase space plots so this is the plot in the phase space so rivoli theorem also holds here however the rivoli theorem holds only for conservative system and here because we are considering the particle is passing through the conservator magnets it is also conservative system because magnets do not change the energy of the particle however an acceleration takes place the area of the ellipse may change because when acceleration takes place under the time varying electric field the system is no longer considered so in that case we will see what would be the invariant of motion and in some cases when charged particle feels some acceleration it can radiate so that is a kind of dissipative forces in the nature because in the form of radiation particle loses its energy so that again becomes non-conservative system in that case also the area of this ellipse may not remain constant but in this course whatever we are studying we can think that our area of the ellipse remains constant and in this case a root beta shows the maximum amplitude of displacement and a root gamma shows the maximum amplitude in angles so we have known something about this solving this equation and obtaining the twist parameters all these books are same which we will cover actually the same reference will cover our integrals and in next lecture we will study more on these beam parameters and how we can characterize and optics using these parameters we will see that in one next lecture