Lecture 43 : Generalized Coordinates and Generalized Velocities

Hello and welcome to a new lecture of Classical Motion of a Single Particle. We simply continue our discussion from the previous week on the Constrained Motion. So, in the previous week we have already discussed the concept of different type of constraint and also how to get rid of the force of constraints. So, if you remember just to give you a quick recapitulation, we have discussed the principle of virtual work which simply said that the work done due to the constraint forces associated with the virtual displacement which we can also called virtual work is always 0. We also said that there is a principle called the principle of d'Alembert. Sometimes we say d'Alembert's principle as well, which simply said that the equation of dynamics finally, for a particle can be simply written like this in this form where F i is the force acting on the particle, but excludes any type of force of constraint. Then we already said that equation of Lagrange of first kind, where we said that minus F i plus lambda gradient of phi should be equal to 0 for a specific form of this undetermined multipliers of Lagrange. This phi is nothing but the function associated with the holonomic constraint which should be 0 and this is in general is a function of the position coordinates and time that is 0 and that gives you the constraint actually. So, this is the constraint. So, if you do this type of equation if you use you do not have to introduce or even mention anything about the force of constraint in a system and you can simply find the equation of motion for that system. We have discussed the example of simple harmonic pendulum or simple pendulum. So, simple harmonic pendulum simply means that theta is tending to 0. The example of simple pendulum and also a particle moving in a circle. For this simple pendulum I did the case for fixed length of the string ok. Then finally, I mentioned two important points. The first one is that the Lagrange's equation of first kind can actually associate more than one constraints. So, you just have to remember that corresponding to each holonomic constraint you should introduce one undetermined multiplier of Lagrange. So, if you have two such for example, two such holonomic constraints you have to introduce two undetermined multiplier corresponding to each constraint lambda 1 lambda 2 for example, ok. Then we also said that although this is done. So, Lagrange's equation of first kind we showed only for holonomic constraints, but some non holonomic constraints can also be included in such type of equation of Lagrange of first kind. And those equations are actually not 100 percent holonomic or 100 percent non holonomic class, they are called semi holonomic constraints or you can sometimes say that they have a specific form called Fafian form. I mean of course, semi holonomic constraint it is not always 100 percent compulsory that they should have a Fafian form, but Fafian form is one of those forms which can be a semi holonomic form, ok. Today I will try to introduce another concept, but before that I want to remind you that first you have to do the problem of simple pendulum with variable length where for example, length is L 0 1 plus sin omega t and also a particle moving down an inclined plane with fixed theta and theta is a function of both cases. This is a problem I want to leave for you to try, ok. In today's lecture we will try something else. So, till now we have said that we do not need the forces of constraints to take into account in order to find the equation of motion that is the equation of dynamics, ok. And we successfully actually showed that using our tricks actually the usual equations of dynamics the correct equations of dynamics actually are obtained. So, there is no problem with this, ok. Here at this point I want to emphasize on that on that like aspect that maybe at the maybe at some step you can have this idea that it is a universal truth that all the constraint forces are always perpendicular to the virtual displacements. This is not in general true. This is just for your for the sake of your knowledge I am mentioning here today that we are actually saying that we take some examples of some very usual and familiar examples of classical mechanics which are very simple and for that somehow we could show that the forces of constraints are always perpendicular to the virtual displacement. If it is not true then the trick which we have already used the principle of virtual work, the d'Alembert's principle and also Lagrange's equation of the first kind. So, you will see that we have to take some caution. So, for example, you can easily see that the trick related to the principle of virtual work and d'Alembert's principle would not directly apply. So, rather we have to be happy by just saying or admitting the fact that we will we will not claim that principle of virtual work or d'Alembert's principle is universally true. Rather we will say for those system where the principle of virtual work is true we can actually have d'Alembert's principle to be true as well and for that we can use our trick. Still now we have no alternative on the choice of the coordinates. Always we start with our very basic usual coordinates that is the Cartesian coordinates. Now of course, all of us know that one very crucial advantage of Cartesian coordinate is that the unit vectors of Cartesian coordinate they are always time independent. So, they are constant always this directions whereas, if you are thinking of a spherical polar system or a cylindrical polar system not all the unit vectors are time independent for those non-Cartesian systems. On the other hand in Cartesian system it is also true that several type of problems which have a different type of symmetry is not appropriate to be dealt with Cartesian coordinates. So, for example, if I now tell you that you just I mean do a volume integration or a surface integral to find me the volume of a solid sphere or the surface area of a hollow sphere all of you know that it is the easiest way to use the spherical polar coordinate system. If you try to use the Cartesian coordinate system for doing that you know that you will have great problem because then the integration limits will be actually coupled with each other and you have to really pay attention to do long integrations. So, if you want to stick with Cartesian coordinates. So, it simply said that the nature of the symmetry of several problems actually sometimes compels us to choose a different coordinate system than that of a Cartesian system. And if we have such systems we can sometimes actually see that some problems are actually simplified that even sometimes one or two coordinates are also reduced directly. What is the meaning of that I am telling you? So, let us say I am just taking the very example of a particle moving in a circle in x z plane. Now, this is O and I am thinking that some point P is over here which is moving in this circle and this is R I just assume this is fixed in time. So, now the point is when I am trying to analyze the motion of this particle I will simply say that for this particle. So, R is basically constant and that means, that x square plus y square is actually equal to R square is constant. So, if you want to stay in Cartesian systems and then I just say that you just tell me for example, a line integration x d y on this circle. Then you can actually have great trouble because then you have to actually write this x in the in function of y which can be given by simply R square minus y square d y and then you have to of course, integrate it within proper integration limits and sometime it is also very difficult to find those limits ok. Now, on the other hand if I just try to tell you that you just go and switch to r theta coordinate system and using this r theta coordinate system if you just want to analyze the motion of the particle you will see the first and foremost thing what you will have is a constant ok and then although this is constant. Now, in order to just do any analysis for example, when I say that what is the particles position vector or something and anything related to the particles motion for example, if I say what will be the velocity what will be the kinetic energy always you have to actually take those both of these coordinates into consideration and then according to the necessity you have to use this constraint. So, whenever you see x you should replace this by root over R square minus y square and whenever you see y you can replace root over by root over R square minus x square you can do any of the two actually, but the point is that finally, you have to keep both the directions or both the variables, but on the other hand if you first of all I am just trying to do this thing right from the beginning if you just say the I will just take r theta coordinate ok. Then what happens that you can say in this problem I know that on this circle R is a constant. So, I will not at all count this as an interesting coordinate from the motion of the particle ok this is just a constant of the system. So, one coordinate directly becomes a constant of the system and that is why the only coordinate with which you can work is the theta the angle ok. So, now, see this is an angular coordinate here which is the surviving coordinate and so here in this system you will have x and y both the variables x and z actually sorry one second I am just making this mistake yeah x and z they are actually two variables, but related by this constraint whereas, in this coordinate system you just have only one meaningful coordinate this is theta. So, you see that due to this constraint actually somehow in one possible coordinate system there exist a coordinate system where you can actually reduce the degrees of freedom of the problem by one that is exactly what I said last week that every single holonomic constraint can actually or rather should reduce the degree of freedom by one, but not in the same coordinate system you actually I mean see the effect in another coordinate system which is the coordinate system of symmetry for that problem. So, here this plane polar coordinate system plane polar coordinate system is a system of symmetry for the problem. Now, you see that this type of coordinate system which finally, makes I mean which finally, gives the number of effective coordinates which are actually necessary to describe the motion of the particle is known as the system of generalized coordinates. So, for example, for this problem Cartesian coordinate system is not a generalized coordinate system. However, the plane polar coordinate system is the system with generalized coordinates or the generalized coordinate system and what is the corresponding generalized coordinate neither x nor y it is the theta. So, in a way I will write that generalized coordinates are the minimum number of independent coordinates in a in a coordinate system I would say in a specific coordinate system specific to that problem. Generalized coordinates are the minimum number of independent coordinates which can describe the configuration of the system complying with all the constraints or rather you know polynomial constraints imposed. So, this is a very important definition the generalized coordinates are the minimum number of independent coordinates. So, that is very important. So, here again you see that I am just in order to just make you realize the meaning of this independent coordinates is that when you are using Cartesian coordinate system you have two variables, but these two variables actually are not independent of each other. However, here let us say I am now thinking instead of a circle I am now thinking the motion of a particle on a sphere surface ok, surface of the sphere and the radius of the sphere is constant let us say small r as well ok. Then we know that x square plus y square plus z square is equal to r square. So, again in 3D Cartesian coordinate system it has one constraint. So, we know that the system will actually effectively have the normal number of degrees of freedom which is 3 minus 1 constraint equation. So, it will have two effective degrees of freedom. So, degrees of freedom actually if you just lost with this vocabulary they are simply one degree of freedom is one independent variable or one independent information to designate the motion ok. If the particle needs two independent informations to get designated or get identified then the motion will have two degrees of freedom ok. Of course, whenever we say that the independent information it is the independent information about the position of the particle. Of course, you know that for every position coordinate there will be a corresponding velocity coordinate as well. For example, if you have x you have x dot ok. So, just for Cartesian you see ok, but that we will not considered as a degree of freedom. So, degree of freedom specifically. So, if you just in an in an n dimensional space the number of degrees of freedom of a free particle is just n ok. Because you need n coordinates n numbers ok and these n coordinates are actually totally independent of each other n coordinates to actually locate the particle in that space ok that is the perfect word to locate the particle yeah. So, here you see if now instead of Cartesian system. So, that was Cartesian if you take a spherical polar coordinate system and that is r theta and phi ok then basically again you have r square is equal to constant. So, already one variable becomes constant. So, it will be not taking place in the analysis of the motion anymore in a significant way because this is just nothing, but a constant. So, the whole problem will now be considered to be described by theta and phi which are the pole attitude and the azimuthal angle ok and that is why you see that since it has 2 degrees of freedom actually I find a coordinate system which is not this one, but an different one ok where basically the system can be described by exactly 2 variables and here for this problem you actually have these to be the generalized coordinates. Now, finally, what I was trying to say is that here I said the number of independent coordinates here you see that x y and z they are actually not independent and they are actually bound by this constraint equation here theta and phi they are independent of each other because the only constraint is actually totally taken care of by the constancy of this r ok and that is why this is the generalized coordinate for the motion of a particle while moving on the surface of a sphere ok. Similarly, you can now think of a simple pendulum. So, again we take our standard example of a simple pendulum with mass m and length l let us say the length l is fixed in time and then I just have this is x direction and this is z direction of course, now you know that any arbitrary position can be written as x and z where x is equal to. So, if this is theta we just saw that last week that x is equal to l if it is also l ok this is just an arc of a circle this is theta. So, I would say x will be equal to this this is x and this is then this is z. So, x will be simply equal to l sin theta and z will be minus l cos theta that is because I am taking according to my convention z to be vertically upward ok. So, l sin theta and z will be minus l cos theta ok. Now you see in this equation once again you can again see that x square plus z square is equal to l square is the constraint. So, if you now say that this is a 2 D problem, but there is a constraint. So, x and z they are no longer independent of each other, but if you now go to just l theta or r theta system you simply can have that l is constant and only theta which is the effective variable. So, for this problem once again you see just like the motion in a circle this ok you will have theta to be the only generalized coordinate. Now, why we use the word generalized? Because this although here in this course of classical mechanics I am introducing this concepts of generalized coordinates, they are of superb importance in many domains of science and social science as well where this coordinates and velocities finally, they are becoming immaterial ok. You will see that some way what we actually call in a sense generalized coordinate they are actually can be variables according to our necessity from system to system, they can be angles, they can be energy, they can be momentum, they can be angular momentum. So, you can be surprised that how can momentum be generalized coordinate this is also possible, how can be energy a generalized coordinate it is also possible. So, you you can see sometimes in a in the amplitude in a Fourier series for example, that can be actually considered in some problems as generalized coordinate ok. Exactly what is the proper purpose of making the definition of generalized coordinate to be so generalized that may not be apparent at this moment, but it will be actually clear if you just go through some books of dynamical systems where they use this somehow this type of techniques using this generalized variables and something. Some for example, I am just to draw your attention for example, in economics, in social science, in epidemiology sometimes you can actually encounter systems where you need to designate generalized coordinates and which have actually nothing to do with the meaning of coordinates in physics ok. So, just before wrapping this lecture up I would like to say that whenever there is a generalized coordinate ok, d dt of that generalized coordinate is defined as generalized velocity ok. In general generalized coordinates are designated by the symbols q and generalized velocity are designated by q dot ok. So, for example, in in a system the effective degree of freedom is 2 actually you need 2 generalized coordinates q 1 and q 2 and there will be 2 corresponding generalized velocities q 1 dot and q 2 dot ok. I want to I I want to give you a small exercise can you also give me the number of the generalized coordinates for the problem of this simple pendulum when the length is changing ok with time and also the number of generalized coordinate for a particle which is moving along an inclined plane with some fixed angle theta making with the horizontal ok. Thank you very much.