 So, for our discussion about the accelerators was mostly qualitative. We talked about cyclotron, microtron and some basic principles, however, equation of motion was not developed means if a charged particle's initial speed and position is given to us and a magnetic field distribution is given to us, how charged particle move or its motion will be evolved. We did not see that so far. Now in this lecture, we will develop the equation of motion of a charged particle under the electromagnetic field and this will be helpful to trace out the trajectories when the charged particle passes through a given magnetic field. Means how the charged particle motion will evolve if a magnetic field has dipolar component as well as some gradient in it like we have weak for the weak focusing in cyclotron and etc. So, that kind of trajectories can be traced out using this equation of motion. So, in this chapter we will look into that. Now, so far we also considered only a single particle that is nicely going on the trajectory only sometimes we saw that if a particle is deviated in vertical plane how it can be focused. So, actually the charged particle beam consists of many many particles means it is a large collection of charged particle beam. So, what kind of collection we can say that it is a beam means what are the properties of charged particle collections which we see it is a beam in the accelerator. So, there are certain properties and first of all these particles must of the same species and same charge means if we are talking about the electron beam means all the particles are in the beam is electrons or all the particles are protons if we talk about the proton means there is no intermixing of the particles in an accelerator like we have in plasma there are certain ions and certain electrons that is also a collection of charged particles, but that is not a beam. Then all particles which we consider here have much higher energy than the thermal energy. Thermal energy means the particles kinetic energy due to room temperature and here we are talking about the particles energy much much much higher than the room temperature. So, room temperature kinetic energy you can calculate using the kt is equal to half and v square and in accelerators you will see that the energies are much higher than this kt. And now all particles must travel with this high velocity in one direction means there should be an average beam direction around which along which the particles are going. Some particles may have some deviation in the angles with respect to this central trajectory, but those deviations are very very small. So, almost all the particles you consider that almost all the particles not exactly almost all the particles are moving in the same direction and the transverse size means if a beam is coming like this this is the beam propagation axis and the beam size in the transverse to this propagating direction means this is the A you can say this is the B. So, this A and B are much much smaller than the length. So, transverse extension of beam is very very small it may be in the microns level and in some cases it can go up to centimeters. The spread of energy of the all particles are almost same means if we talk about there is such say 2GV proton beam means all the particles are nearly having 2GV energy means spread or you can say deviation from the 2GV is much much smaller in the case of beam in accelerators. Again here is the word almost means not exactly all the particles are having 2GV, but the spread is much less and similarly here we also use the word almost and these two almost words actually decides how the optics has to be designed for the charged particles. So, now consider that there is a design axis we can say that this is a design axis or optic axis or the central axis or the orbit like that. So, some particle is going in these directions. Now at a particular location of the design axis say this is the location here we can characterize this particle trajectory by two parameters that what is the distance of this trajectory with respect to this design trajectory means how far this particle's location is with respect to this design trajectory and what is the angle made by this design trajectory with respect to this design orbit or a particle trajectory is going in which direction with respect to this design trajectory. This distance we can say this is the x and this angle theta these should be very small means all the particles should move almost in the same direction. Now there may be such various different trajectories of different particles. So, beam may be a very complicated collections of different x different theta each for each user particle and you can say that number of particles in a beam can as high as 10 to 8 to 10 to 15. So, suppose if you plot the location or distance of the particle from the design axis as well as its trajectory is angled with respect to design trajectory then we can get such kind of distribution here on the x axis the distance of each particle from the design axis is drawn and here in the y axis the angle of its trajectories with respect to design trajectory is drawn. Each dot represents a particle and its coordinates are b. So, now you can see that there is a distribution of the particles in location and angles and if you plot the distribution in location only means at certain location what is the number of particles then we can get a certain kind of distribution of the beam and same kind of distribution of the beam will have also in the vertical direction. Now instead of angle if we see that what is the extent of beam from the design axis in the x direction or in the y direction means what is the extent we say a and beam last figure then we can see that this beam may look like this. So, we have to consider the stable motion of all these particles and the particles which are deviated from the central trajectory. So, we will develop now equation of motion of these deviated particle. Now first of all whenever we try to solve any dynamical system means we try to formulate the equation of motion of that dynamical system first of all we have to choose a suitable coordinate system. Here also in accelerator physics we will choose a suitable coordinate system. Cartesian coordinate system which we know very usual power use generally x, y, z is not very useful in this case. In accelerator basically we make the design path itself as a reference. We calculate what is the distance of a particle with respect to design axis and how the particle trajectory is making angle with respect to its design axis means design axis itself can serve the purpose of the coordinate system. Now consider a particle which is nicely following the design trajectory. Design trajectory is basically decided by arrangement of the dipole magnets means if we arrange some dipole magnets and from the starting point which we want that this should be the starting point of the trajectory if we launch a particle in the correct direction then it will trace out the path due to those dipole magnets which we have arranged. So, design trajectory is decided by dipole magnets and a particle which follows exactly the design path or ideal path with the correct energy and it reaches in the RF cavity at the correct phase. We say this is a synchronous particle because it is synchronous with respect to RF phase and it is exactly going on to the design trajectory. We will take this synchronous particle as the origin itself and with respect to this synchronous particle we study the motion of other particles. So, suppose this red curve shows the design trajectory so our coordinate system will also be on this and now suppose at one moment the synchronous particle at this location. So, this is the origin at that moment and the beam propagating direction is generally taken as s and here you can see that trajectory is making some curved path. So, our origin also moving on this curved path. So, here you can see that local little coordinate system is having some curvature. So, this is a curvilinear coordinate system and because the origin is moving now suppose synchronous particle is at that location the origin is also here. So, origin is moving and it is a curvilinear coordinate system so that this is a moving curvilinear coordinate system in which we develop the equation of motion. Now here because the radius of curvature is in this direction so opposite to that we take the x axis. So, radius of curvature and axis is in the same plane. So, the plane formed by x and s this is known as horizontal plane. Generally because in accelerators bending of trajectory takes place in the horizontal plane so we say this is a horizontal plane rather it may be a bending plane also if it is not in completely the horizontal. And vertical to this x and s axis there will be a third axis which is the vertical axis. Now if synchronous particle is moved here then origin is also here. So, our coordinate system is moving now as you have said that this x s form the horizontal plane and this y and s similarly this y and s this will be the vertical plane and x and y both these make the transverse plane because this is the transverse in the direction to the beam propagation direction beam propagation direction is designated by s. So, these are three planes and our equation of motion presently we will develop that is in the transverse plane. Means we will consider x and y not the s itself. So, presently we will study the transverse plane dynamics. Now considering this coordinate system particle which is deviated from the design axis. So, this red curve shows the design path it is it has certain curvature the radius of curvature of the design path is designated by rho 0 and some particle is situated at p. So, our coordinate system local coordinate system will have origin at here. If we draw a projection from p to the horizontal plane this distance will be the x which is the distance in the horizontal plane from the design axis. This distance which is the vertical distance from this horizontal plane is y and center of curvature is here. This is this means making some circular path. So, center of curvature of this design trajectory is here. So, the distance rho 0 plus x is the small r in the figure and with respect to this center of curvature the radius vector of this deviated particle is capital R. So, capital R which is the radius vector with respect to center can be written down as r x cap this is the r and x cap is in this direction this is in the escape direction and then y y cap y is in this direction. So, y cap is also in this direction. So, this distance is y and this distance is r r is equal to rho 0 plus x. Now here if synchronous particles comes here then our x will be in this direction. So, you can say that the unit vector x changes this direction with course of motion. So, our unit vector in this coordinate system is not constant it changes with time. So, now in this coordinate system we derive the velocity what is the velocity of this particle because we have the position vector r capital R. So, velocity will be dr by dt and because we have r is equal to r x cap plus y y cap. So, we will differentiate it. So, first dr by dt x cap which is written here now because x cap unit vector is not a constant. So, it will also be differentiated. So, we will have r dx cap by dt and then y cap this unit vector is constant. Right now we are not considering that bending plane is changing its orientation we always thinking that our bending plane is in the horizontal plane. So, y axis will y unit vector will not change its direction. So, y unit vector will be always constant in the constant direction. So, this will not be differentiated. So, only we have dy by dt and y cap. Now, for equation of motion what is the equation of motion? Equation of motion is f is equal to m means we have to calculate the acceleration also means we have to differentiate this velocity again then we will get the acceleration. So, we will do that. Now, before going into that how we calculate this dx cap by dt means how we will obtain the rate of change of the unit vector this can be done very easily. Suppose at certain time at the location s unit vector is in this direction this is the direction of the unit vector x and after some time when it makes delta theta angle at the center the orientation of x cap become this. This is now the direction of x cap at the location of s plus delta x. So, if we want to calculate what is the change in unit vector we again redraw this here this is the x cap at s and we put the pale of the second vector also here. So, this will be x cap s plus delta s. So, this vector will show you delta x cap. So, this delta x cap and this is the delta theta angle. So, you can say that using this figure you can say delta x cap magnitude wise will be delta theta x cap at s. It is just a simple now here this is a unit vector. So, its magnitude will be 1. So, delta x cap can be written down as delta theta. So, delta x cap by delta t will be delta theta by delta t. Now, we have calculated the magnitude of delta x cap means what is the change of unit vector we have calculated magnitude of that, but what will be the direction because it is a change in the vector. So, as you can see from this figure as we are making delta theta tends to 0 this delta x will be in the s cap direction means delta dx cap by dt can be written down as d theta by dt which is the magnitude and direction will be s cap means dx by dt is equal to d theta by dt s cap. So, our velocity we can write down dr by dt x cap plus r d theta by dt s cap plus d by dt by. Now, we will differentiate this velocity to obtain the acceleration. Now, in acceleration when we will calculate it here x cap will also be differentiate it. So, dx cap by dt we have already calculated here s cap will also be differentiated its direction is also changing with time and in the similar way as we have derived dx by dt we can calculate dx by dt and it will be minus t theta by dt x. So, proving of this result is left as an exercise for often. So, now we have acceleration right we will this is the definition of this d 2 r by dt square x cap and one x cap will be from here r d theta by dt square and then s cap will be one s cap coefficient will come through this. So, that will be 2 dr by dt d theta by dt and one s cap will be through this. So, this is the complete expression for the acceleration. This is the acceleration in defined coordinate system the coordinate system which we have defined it is in there. Now, we will calculate the force also because m into p is equal to f. So, we have calculated the a now we will calculate what is the f force. So, force can be calculated very easily q v cross b is equal to q v s v y this is just the cross product you can obtain it like this this will be x cap this y cap and s cap. So, here will be you can see that this is v x this is v y and this is v s then v x v y and we are considering only transverse magnetic field means no component in the longitudinal or beam propagating direction. So, this coefficient will be here matrix element will be changed. So, using this you can calculate force easily. So, this force will be q v s v y plus v s v x now using this force the equation of motion along x here we can see that along x acceleration is this d 2 r by dt square minus r d theta by dt square. So, in the x direction this is the force. So, equation of motion will be d 2 r by dt square minus r d theta by dt square is equal to q v s v y this is v s v y q upon gamma m again we have inserted gamma due to relative state effect and this is again in the x direction which is perpendicular to the velocity v s that is why we can insert gamma here directly. And this v s v y if we multiply this quantity by in numerator and denominator both by v s we will get q v s square v y and gamma v s and gamma v s will be momentum v s which is written here. And now in accelerator we are saying that beam has almost same direction and in that direction velocities are highest means transverse component of velocities are much much lower than the component of velocity along the s direction along the beam property direction means we have e is equal to p x square plus p y square plus p s square. However, p x and p y both are much much smaller than p s. So, p is approximately equal to p s this approximation is used here. This approximation is known as paraxial approximations means we are not very far from the design axis and particle trajectories are not making much larger angle with respect to design trajectory. So, under this approximation this p s can be written down as now as you know that momentum can be written down as q b into r here r is rho 0 we have taken radius of curvature of the design trajectory as rho 0. So, at the place of p we can write down b rho by q. So, q by p can be written down as b rho 0 in accelerator this term b rho 0 b rho 0 or you can say b rho is known as magnetic rigidity. It actually is the another way of telling the energy or momentum. So, in accelerator generally we say about the momentum or energy in terms of magnetic rigidity and that magnetic rigidity is b rho 0. Now, we have obtained equation of motion along the x axis. However, we are not interested in the evolution of motion with respect to time. Rather than we are interested in knowing the particle's trajectory evolution with respect to s means as the synchronous particle is going ahead how the particle trajectory of other particles are evolving. So, we are interested more in s rather than t. So, we have to convert this equation from t to s means our independent variable we will make s rather than d. So, we transform this equation into that for that parameter. So, d by dt can be written down as d by ds into ds by dt. Assuming we have constant why we are assuming we have constant with the same approximation here because the energy is not changing by the magnetic field. So, momentum remains constant and momentum is almost equal to the momentum in longitudinal direction. So, velocity is also approximately equal to v s. So, and this is a constant quantity. So, d 2 by dt square will be ds by dt square d by ds square. So, now in this equation instead of d 2 by dt square we can write down d 2 by ds square using this equation. And now what is ds? ds is simply d rho d theta rho 0 is the radius of curvature of the descent so, this can be written down as rho 0 d theta by dt and d theta by dt. The deviated particle what is the angular velocity of that derivative particle is shown by d theta by dt. So, this can be written down as v s by r. r is the distance of particle from center of curvature. So, this gives the velocity angular velocity for the deviated particle that is v s by r. And r from figure we call it r is rho 0 plus x. We again we draw here the figure figure was like this. This is the descent trajectory. Here was the center of curvature and we are taking some deviated particle here. So, this was the y and this is our origin of the accelerator coordinate system. This is x and this is s and this distance was x in the accelerator coordinate system and this is rho 0 the radius of curvature of the descent trajectory and r was this. So, r is equal to rho 0 plus x it is written at the place of r we have written rho 0 plus x. So, ds by dt will be v s rho 0 rho plus x. So, now we have obtained ds by dt also and in the last slide we have obtained d 2 by dt square is equal to ds by dt square d 2 by d s square. So, at the place of ds by dt in this operator we will put the value which we have calculated recently. So, by putting this our d 2 by dt square operator is related with d 2 by d s square using this equation or this expression. And d theta by dt is v s by r which we have calculated this is v s by rho plus x. So, our equation of motion at the place of d 2 x by dt square we have written d 2 by d s square and at the d r we have written down rho 0 plus x. And because we are converting d 2 by d s square so, this term appears. Similarly all the terms can be calculated very easily it is a just simple algebra. Now here the rho 0 is constant so, it will not be differentiated. So, we will have d 2 x by d s square from here. And after a little bit of algebra you can obtain this equation by that equation. So, this is left as an exercise for all of you that you have to derive this expression using the past this term. So, now you can see that d 2 x by d s square this is now x is our accelerator coordinate system it shows the distance from the descent trajectory. We are differentiating it with respect to our path length of the descent trajectory. So, this equation is now in the accelerator coordinate system. Now here b y magnetic field if we will put the magnetic field as a constant magnetic field means a dipolar magnetic field we have seen that dipolar magnetic field means a constant magnetic field over this space. So, if we take b y as a constant it will be the equation of motion under the dipolar magnetic field. And if we take b y as a constant plus some gradient into x means this equation will be for the particle which is passing through a dipolar magnetic field which is having some gradient. And this kind of magnetic field which has some gradient was used for weak focusing in this cyclotron. So, this equation can tell us the motion under such kind of magnetic field. So, we will do that now we will put b y as a constant magnetic field it is which shows the dipolar component and this is the gradient. So, at any distance x from the descent trajectory b y will be b y 0 plus del b y by del x 0. So, on the descent trajectory field will be b y 0 and as it will move it will change according to the gradient. Whether it will increase or decrease it depends whether the del b y by del x has positive sign or negative sign. So, now in equation of motion which we derived here we will put this value of b y and after a little bit of arrangement of mathematical terms you finally get this equation of motion. So, this is differential equation or equation of motion in the accelerator coordinate system under the magnetic field and that magnetic field is the dipolar field having some gradient in it. So, this is the equation of motion which can tell us completely the particle's trajectory evolution in the cyclotron or in other magnets if particle is passing through the magnet. Now, you can see one important thing here that this is d 2 x by d s square plus some quantities into x is equal to g. So, this is the equation of motion similar to simple harmonic oscillator equation.