Lecture 02 : From dynamics to Kinematics

[Music] previously we have seen and actually have somehow informally introduced the aspects of classical mechanics and I also said that how Newton's law actually helps the first two laws of Newton actually helps in understanding classical mechanics as a dynamical system system because if you see now these are the equations where I can have X and P as two fields and I also have their evolution equation where this dot basically means that when I say a DOT or this is just DDT of this Vector okay same thing for a scalar if I write a Five Dot this is DDT of five so this gives me the equation of the evolution and in understanding the evolution of X I need p I need m i for understanding the evolution of P I need F which is a function of x p and t now you can see that what about this m m actually can be written as a fun explicit function of time now okay so if I just write here Mt instead of that is redundant so instead of writing that I just write here T okay on the other end X and P they should be here okay because if can can be an explicit function of x if can be an explicit function of P explicit function of T okay whatever be the nature according to that we have to uh we have to solve or we have to handle the situation now before going to that point I want to clarify some point which is also sometimes which causes at this level some students uh confusion that is x and x dot are they dependent or independent depent so you see that x and x dot they are functionally independent quantities only when you know the form of X as a function of time for example then only you can do that DDT or the complete time derivative okay total time derivative of x and you can get x dot it looks that this is I mean this is something depending on each other so if I know x i I also know x dot this is not true for example if I give you the value of x so for example uh position 2 I + 3j - 5 K at some t t uh T is equal to 5 seconds can you tell me the uh velocity Vector at that time so you can say no because you you have to know the explicit form of X or the X Vector in terms of T now if I say X is actually 3 t² I + T J minus TB k then you can say okay then I you give me the t is equal to 5 6 whatever for that I can also tell you what will be x dot and that's why they're functionally independent another point is so that I will actually come later okay so that there is a there is another point of initial conditions okay so still now till now we have just wrote two equations of evolutions for two variables X and P and we say this is a dynamical system okay now if I correctly remember the Newton Second Law this is p dot is equal to F so according to our definition that should be m v where V is nothing but x dot so I can just write V is x dot for me is equal to F and you can write that M dvdt plus v DM DT okay I can write that F you see many of times in nature a particle has a fixed mass in time the mass is something which does not change and for for that most of the cases we forget this term and we simply say m dvdt is equal to F is Newton Second Law so this we have to understand this is for just a specific case where your dmdt is zero however there can be several uh occasions where you can model the whole system in terms of a particle of changing Mass for example if you just want to model the reti linear motion of a rocket then you just think that it is actually the best modeled as a particle with I mean variable Mass okay so you see and also when some uh for example when some rain drops they gain weight with time okay so actually you can you can do an extended body analysis but you can also model very simply by a point part particle which is gaining mass in time okay for those systems I will give you some Pro I mean maybe one problem or two problems in the assignment this dmdt is not equal to zero and you have to be very careful in writing m dvdt is equal to F now if your system has constant Mass so Mt is just a constant then for that part you can actually write m dvdt is equal to F and this m x double dot which is DDT of x dot which is X double dot will be F okay or you can say I mean is a equivalent saying M vou V Dot is equal to if okay then the simplified equations of classical mechanics will be x dot so for uh for fixed mass m x do is equal to I would say V because now my Mass is constant okay I don't really need to introduce P differently huh so I just write x dot is equal to V and V Dot is equal to some f Force by m which is a constant I can say that is another function which is a which I call acceleration acceleration or the rate of change of velocity and I say this is a function of x x dot and T this is the most General case possible I'm not saying there are the all I mean if is always a function of x x do and T but on the other hand in general they can be okay now again another uh I mean very um like student like confusion is that sir X is a function of T you have just said x dot is a function of t m was also a function of T but you didn't write a as a fun I mean a as a separate variables whereas for x and x do you are writing no that is not the case the point is that most of the cases when we like to express this Vector huh we have to check that whether we have an explicit dependence on X or explicit dependent on x dot okay that means that for example let's say I am just saying that this if let's say is a function like this some variable some some sorry some constant time x Vector itself so here you see here I mean the X itself explicitly appears in this if this is the case then you can actually you have to write this itself and not as a function of T of course the most uh the mathematically correct way of writing is if x t x dot T and T and here for the sake of Simplicity actually whenever we said this mass is a function of T we assumed that we somehow know the explicit functional form of M in terms of T that we assumed if this is not the case then also we have to write another Evolution equation for mass and we can actal system we are not getting into that complexity so what we have done is a simple Construction in the dynamical system we are not getting into that complexity so what we have done is a simple construction where we actually assume that even if when m is not constant in time but changes with time we know exactly before even solving some situation some external information Source gave me the explicit form of M that assumption is there okay clear now I am going to start with very basic level problems and I will then uh go as as I promised in the introductory video that I will mostly treat the classical mechanics and its Vector formulation in this course but I will somehow start by a very simple 1D motion and the equations associated with that before saying we now have equations of classical mechanics huh and this equation which is the equation where you have the explanation of the motion is called the equation of motion or equations of Dynamics equation of Dynamics okay so this actually explains the motion so if you know this equation then your initial duty is to try to solve those equations and finally to find X as a function of T and x dot as a function of T because whenever you have a dynamical system and you try to uh study the motion what is the goal the goal is that you know the value of those filled variables let's say here x and x dot at some given instant and using that you want to determine their values at some uh posterior instant or some uh anterior instant or I mean some I mean earlier times or some times later okay so if you want to do that you have to solve those equations so for solving of course you have to solve if you just see it carefully once again I am going to this this equation is nothing but this equation okay and so if you try to characterize this equation this equation is actually a second order ordinary differential equation in X with respect to time so here I am just trying to start with a very simple case of 1D motion so in the classical mechanics as we just have seen that the basic challenge is to solve a number of second order ordinary differential equations you can say well cannot we do every time no it depends on the explicit form of the force F or the acceleration here in this reduced case Okay so ond motion I can simply write like what is the equation of I mean the dynamical equations so dxdt is equal to V I just have 1 D and then dvdt is equal to I have if which you can also write F by m whatever you want okay I am not giving the vector signs because this is only 1D okay so if I now say that how to get so here here in this case I just give you for example this force that is the point where I can I mean I can start solving my equations that I have to know if I have to know or somehow I have to guess if I have to model if that is the whole challenge if somehow in somehow in some means you can write some forms of if and that is just I mean I am writing for this variable that will be easier F is X can be a function of x v and T maximum if B are explicit function of t as well where X and V is not changing but only T is just changing and that changes F if that is possible then we can say this is an explicit fun explicit variation of if with respect to T okay now if T is changing and due to that X and we are changing and those are actually changing if huh then it's an implicit dependence I hope all of you have a little bit uh I mean a little bit you are more or less familiar to uh the concept of explicit and implicit dependence okay I will come back to that when I will talk about the partial differentiations okay so if they are the functions of this and this is a complicated function uhuh then sometimes you know that uh it's not guaranteed that maybe it's can be solved but maybe you do not know how to solve that okay so that's why we have to go Case by case and not to claim that this actually guaranteed if I know this I know what is x t and X do T that I don't know but in general what I can do I can Case by case I can make some strategy so that I can solve the system the first one is is the easiest one so my force or the acceleration I mean just here acceleration is just a function of T only it is just changing with time explicitly but not with X and V explicitly okay if this is the case then you actually have dvdt is equal to Ft and if you can solve that so that will basically give you V is equal to integration ft DT plus some constant of integration C1 and that you know DX DT V is DX DT because of the first equation you just put that and that is now this integration ftdt plus C1 okay now you integrate once again and you get have have X is equal to integration of integration if since this is already integrated I can use a dami variable for this T and this whole thing is now again integrated over time t plus C1 t plus C2 which is another constant of integration okay in this way you can find X and also I mean once you have the whole form of X you can write dxdt you can calculate dxdt and you can have your V okay so you see in this way you can also solve the problems of classical mechanics in more than one dimension now let's say you have a force which is a function of X for example then what to do then we will actually write another form which is much more helpful is this instead of writing that we can write dvdt as DV DX DX DT okay since this is just a problem in one dimension I can write always this thing this thing is not exactly uh possible this type of writing in more than one dimension in more than one dimension we have to do something which we called partial differentiations okay we will do like that that is also another problem I will discuss in the next lecture May maybe okay so if you write that in FX and then you see this is your V so you then have V DV is equal to FX DX because V DV so this is V so this will be actually V DV DX that will be FX and you have D v² / 2 is equal to FX DX and if you integrate it will give you V sare by 2 is equal to integration of FX DX okay of course plus some C1 constant then you can write this as V is equal to root over I mean plus minus that is also another Point 2 integration FX DX + C1 and then what you have here in this in this part what you have you have something you see you have something which is already that was a function of X which is already like now integrated over X okay now you have something which can be atmost a function a constant or a function of T okay that is the possibility now uh let me just think since since our initial assumption is if is just a function of X then t also will not come here so here you are actually saved so if you somehow know this thing then actually your I mean if it is integrated then this will be another just a function of if actually okay just a function sorry just a function of X here yeah because this is indefinite integral this will be a function of X let's say it's a function of let's say GX okay so then what you have to do this V will now be plus minus < TK 2 G of x + C1 now you have another function of X only now you can write this v as DX DT and take that one in this side with DX so finally your thing will be actually DX over < TK 2 GX + C1 is equal to DT and then you integrate both sides and 2 uh GX + C1 is equal to DT and then either you use uh I mean the proper integration like that or if you have some formula known you can use that you can finally have actually some function of some function uh I would say uh I'm just getting mathematically correct some function h of X which is some function some some t+ C Prime another constant of integration which comes here okay and then you can invert if the Inver inversion is possible you can invert and finally you can get X as a function of T okay all these possibilities are here and you see that even if you follow a specific possibility it is not guaranted that always you can come out with a solution as an elegant solution like X and XT all these conditions are also to be satisfied okay and if for example again the last one if f is a function of v then what happens I'm just telling you the general strategy then the specific problems we will do later if f is a I mean function of v only then what to do then I can simply say that DV DT is equal to FV this case is also very easy you just take this DV by F of v and DT then you integrate both sides and finally first you get v as a function of T and actually what you get is not exactly that you get t as a function of v if that function is invertible then you get first this and then once you get this you can write DX DT as a a function of T and you can integrate to get X as a function of t Okay so our objective is to find equations of motion to I just change the color a little bit to XT and X do T okay and if for example if is a function of neither V or x uh singly but both if x and V or let's say x and t or V and T or all these three things then you have to check Case by case where you can solve this problem by something called separation of variables by simplifying by doing some other methods by reducing this to some uh easy variables or something so there is no General prescription for that even in 1D you see that the classical mechanics which sometimes sounds very uh mundan very ordinary to us uh very um I mean non-interesting to us actually is extremely complicated Simplicity just arises because we try to make them simple most of of the time okay so if we just want to do some small problems and after that we will wrap up this lecture so the first one is this one so f is equal to e the minus t I have to solve and I have to find I have to find find XT and x dot so that is the goal of classical mechanics so then dvdt is equal to e to the minus t if I integrate once I should get minus E to the minus t plus some C1 then I can write this is DX DT right so I mean I should write like here so this is DX DT actually so then X should be I can save one line by writing this X will be again e to the minus t because another minus will come here plus C1 t plus C2 this is the general solution of this equation now you see there is a very interesting point that is if I just say that I give you the value of x then what will be x dot you can say Okay X do is minus E to the minus t + C1 that's exactly what I got here but can I given a value of this I may I say something on x dot at that point let's say if I say at T is equal to 3 I give you the value of x if I don't provide you the value of T and the value of x at that t if I simply say that from this one can you tell me what are the values of x and x dot at T is equal to 3 the answer is no so just knowing the general solution can not help us to know the motion of the particle you need to know the initial condition since this is a second order ordinary differential equation this one finally you are what you are solving is V Dot which is second order in X d2x dt2 is equal to something so actually you need two initial conditions or equivalent equivalently if you think that this is a combination of two first order one I mean first order ordinary differential equation then also you need two 1 + 1 two initial conditions okay to completely specify that and then you can actually see that if I tell you now at T is equal to 0 x is equal to 0 and V is equal to minus1 then immediately you can say at T is equal to 0 this is 1 and this is 0er so X is also given zero so that's why C2 will be minus one all right and if C2 is minus one then you can actually uh from this one also you can say something about C1 easily at T is equal to 0 this one is one so this is minus one and and V is minus1 so C1 will be zero then the solution the particular solution for this system is specific solution for this system is X is equal to e to Theus t uh minus sorry -1 now if you give me any T I can give you X X and v as well sorry and v as well because V is nothing but minus e ^ minus t okay so you see you get rid of the arbitrary constants of integration by using the initial conditions so classical mechanics is all about knowing the position and velocity for constant Mass system position and velocity of a particle at a given time okay it's all about to predict or to know accurately the position and the velocity of the same particle at a later time both by using that information of position and uh velocity at the I mean some given time and the Sol and solving the equations of motion so there are two parts you have to know the system of equations you have to solve them and also you have to know in addition the information of position and velocity at any any single time instant if you give me that I can trace out the whole trajectory for you now before I mean wrapping up this lecture I would like to comment on something if you just check some I mean what we discussed already in the first lecture that you have here two equations this already and I say that this somehow completes here if if for Simplicity I can just think that if if can be at best a function of X PT or for uh constant Mass this is uh just a minute let me go to the thing yeah for constant Mass this is just x dot is equal to V and V do is equal to uh some f which is a function of x x do and T huh so it somehow gives us that we have a Clos system at this x dot level X double dot level okay so because if you go to to X to x dot you have velocity but if you go to x dot to X double dot you have the reason of that motion now whenever you have the reason or the explanation for the motion your dynamical system is closed okay because here actually what you are doing when you are saying that xou dot is equal to F but that f is actually something external if you if you read carefully Newton's law this is an external agent so that AP should be external so Newton's law basically says something very courageous something very bold huh it says that the second law or the first law whatever I mean you can combine both actually both this first and second law actually says that you can relate and that is the beauty or Marvel of the classical mechanic you can think which is totally some intrinsic motion property of a particle you can relate that exact L to something and actually in some units that is exactly equal to something external which you call Force now since this is external you know that so you have this information okay and if you know that that's why you are just using this qu this information to know the X double dot and you back back back tress and you obtain X so this is the approach of classical mechanics okay so that's why Newton's law is so beautiful and so great because it helps in simple cases to close a system in just I mean uh the motion in just up to second order derivative now this is not the end of the story in nature you can actually find several systems and several uh instances where you need more than second order derivative higher than second order derivative the for example third order derivative so third order derivative uh many of you may know it's called third order derivative or position is called a jar okay you I I invite all of you to have some reading on that on internet under which circumstances that can actually be important to understand the mechanics of a particle thank you very much [Music] [Music] [Music]