Lecture 03 : Equations of dynamics and constants of motion

[Music] hello and welcome to the course of classical motion of a single particle so in the last time I mean we were discussing different aspects of classical mechanics as a dynamical theory we also said that uh classic mechan In classical mechanics the basic idea is to know the position and the velocity and more broadly if you know the uh position and the velocity at a given instant then using that you have a set of equations of evolutions for position and velocity you can trace the particle and its state of motion at a posterior moment right so this is the basic goal of classical mechanic then we said that the basic equations are given by the Newton's laws and if you really uh write the Newton's laws in a smarter way and then you can actually see that In classical mechanics you have two dynamical variables one is the position another is the velocity I mean the position sometimes I can also write X Vector as well but I will write R for being being a little bit more general and this is the velocity vector and in general you know that r dot is equal to V then this is the definition and up to this we said this is kinematics this is describing the motion and then the second one is V Dot where basically your Dynamics comes where you are trying to explain the motion and where basically you are trying to relate the motion of a particle with something external which is a force okay and this is Newton's second law this one Newton Second Law and here m is the mass of the particle and F is the force exited and so you can actually see so this is the set of equations which we have to solve so this is a set of coupled uh first order differential equations ordinary differential equations and because there is no partial der and of course if you can write in a more compact way you know this is equivalently can also be written as rouble dot is equal to f/ M which is another equivalent second order uh or ordinary differential equation in both cases as I mentioned last time that we need two initial conditions and if you know two initial conditions okay most preferably one for x and one for V at some T is equal to t0 then you can actually accurately predict In classical mechanics the position and the velocity of a particle given given the condition that you know the force you can trace the particle that means the position and the velocity at any uh posterior moment or even actually any uh any moment which is Meaningful okay you cannot of course try to trace the particle before T is equal to Z that is not meaningful so then we started doing something some simple interesting calculations for onedimensional motion and we said for onedimensional motion Newton's law is nothing but dvdt is equal to F and this dvdt for one-dimensional motion we can write DV DX dxdt because only one space coordinate is there and then you can write simply this is nothing but V DV DX because dxdt is V is equal to F then I can now uh today I can give you an example of such uh condition where you know if as a function of X for example and I say that this is very useful this form when you have f as an explicit function of X or V so I can now take an example where your force is given by Alpha root X where Alpha is a constant let's a positive constant for instance okay and then of course you can say that for this case I have V DV DX which is equal to Alpha root X then what you can do you can take V DV on one side and Alpha root X is nothing but x to the half DX and then you have D of v² / 2 is equal to alpha x to the 3x 2 by 3x2 okay so finally you can write D of v² by 2 is equal to uh so you can actually write I just check where I have uh did the thing correctly so yes I should write like this so since I have already done the integration I should just get rid of this yeah and then you can say actually here you have an arbitrary constant of integration which should appear here okay and then once this is appearing then you can say in the Second Step you have uh V ² / 2 is equal 2 Alpha / 3 x to the 3x 2 plus some C from that in the next step you can do V is = 4 Alpha / 3 x the 3x 2 + 2 C whole root and that is now your DX now DT DX by DT and with this is a function of so 4 Alpha by 3 can be another constant beta X the 3x 2 + 2 C which is a constant and then what you have to do you have to just smartly in this thing and that I leave up to you to do and where you can have basically X I mean you have a function of X will be equal to T and if the function is invertible then finally you will have X as a function of T finally so you know that then you will do dxdt to find V okay okay so this is the strategy for this type of problem okay now in this case you know that whenever you do this type of thing there will be another constant which will appear let's say C Prime and this C these two constants can be eliminated just by using the appropriate initial conditions okay such type of problems you will uh have in your assignments I am now going to another Point Let's say I have a case where this dvdt is a constant let's say the force is non zero but it's constant then what happens then can I say something about its trajectory or something like that so okay let us start with the simplest one if C is z a constant but a zero constant okay then I know dvdt is equal to 0 and d2x dt2 is equal to0 which simply say that DX DT will be constant and X will be c t + C Prime which is nothing but a linear function of time okay so in an XT plane it will simply be represented by a straight line okay now you can say that if at T is equal to 0 my initial displacement is zero X is zero then simply you can say that this C Prime will be zero and you will have simply X is equal to CT if x is equal to CT then still this C you have to determine for that you have to know the second initial condition on the velocity okay now if you say at T is equal to Z so okay so now you know what will be the velocity velocity is x dot see okay now for this case for this particular case x dot is constant so it is uniform at every instant okay so you don't have to play with this thing so this is constant so this c will be a constant and this is a constant of motion okay so this type of thing is also a constant of motion and we will encounter a different type of constants of motion also in uh today's discussion as well a little bit later okay so you see then the whole solution will simply be x equal to CT so it will be a straight line passing through the origin with slope C okay because dxdt is C now I would like to do something for the case where C is not zero so dxdt is equal sorry d2x dt2 is equal to C is not zero then dxdt will be simply CT right and there will be some C Prime as constant I will again integrate and I will find X will be equal to CT ² by 2 + C Prime t plus C Prime so this one I know this is my known constant the two arbitrary other constants would come from here okay the C Prime and C Prime they will come as an arbit as arbitrary constants of integration we have to remove them okay and who will help us the initial conditions so once again if you say that at T is equal to Z my X is zer then your C Prime will go to zero at T is equal to 0 x is equal to 0 then CP Prime is zero then you are left with x equal to C t² by 2 + C Prime T and then you say okay this is okay but then what is my dxdt or velocity now velocity is CT plus C Prime then you say at T T is equal to 0 velocity is also zero then C Prime is also zero okay if you say if at T is equal to 0 DX DT is also zero then finally your final Rel equation will be x equal to C t² by 2 now you see what will be then the trajectory is of the particle in XT plane so in both cases they will be Parabola this is a parabola where the origin is shifted and this is the parabola which exactly starts from the origin now with all the things the final thing which I will now try to address is the case where this force d2x dt2 or DV D DT is a function of velocity so the force is now a explicit function of velocity the simplest case is the force is proportional to the velocity itself I mean I would say the magnitude of the Velocity itself so the speed okay so what happens see if my dvdt is equal to KV then I actually know that D of if I just do a I mean multiplication with v so then it will be V dvdt and that will be K v² and then I say this is nothing but d v² by 2 of DT I mean sorry DDT of v² by 2 and that is equal to k v square if in this point at this point if in this position I make my K to be less than zero you see that such a force is actually responsible uh for d v² by 2 by DT to be less than zero and that means just by choose and on the other hand if you choose your K to be positive then your situation is different then you actually find your uh DDT of v² by 2 is greater than zero so such type of force directly gives you I mean directly plays a role like a energy Saker or an energy sink or an energy pump okay so this is called a syn of energy I mean because your system is actually losing energy huh so this not this is but I mean I I'm just saying that this this type of force basically uh says that there is a sync of energy where your energy is getting dissipated or leaving the system that's why your system is losing energy in time and this is M mainly saying that your your system is gaining energy with time so there is a source of energy okay so you see most of the cases when you do several experiments for example one raindrop is moving through air one ball is moving through water like this type of thing most of the cases you see in general properties of matter even in mechanics we model the frictional force to be proportional to V and that is the case where we actually say that we have a dissipative force so this this is that is why KV type of force is a force which can guarantee dissipation okay here you see I am just uh talking both these cases in terms of the accelerations that I can do because in the last lecture as I said I am considering only a particle with constant mass m okay if the mass itself is a function of T then this case would not be that I mean convenient then you have to take the whole thing huh but here I can simply say that mass is a constant so even if I have a mass over here a mass over here I can just have to redefine the constant to be keep Prime let's say that in this motion I just said that for for example when in your in the motion you have a force is constant so that means this constant is true I mean this constant uh remains constant throughout the motion so this is a constant of motion okay then also when I say that okay the force is zero then velocity is a constant okay this is a constant of motion because this is also remains constant throughout the motion now in general there are some other constants of motion which are a little bit of different type I will not exactly penetrate into that now but I mean but somehow they will also come out so you see that if we have a constant of motion where you can when you write the constant of motion so for example here you can write this now that this V is equal to I mean DX DT is equal to CT plus C Prime and C Prime is equal to zero huh so you have a constant of motion of type of uh like this huh so this type of constant of motion they just uh I mean ordinary constants of motion but if we have some constants of motion which are coming as a result of some integration then we also call that as a integral of motion okay and the basic difference between a constant of motion and an integral of motion is that in an integral of motion the time does not appear explicitly okay so this is something which we'll talk a little bit later I am now somehow trying to uh give you another uh I mean a very basic example of an integral of motion so let's say I just said for an uh one diim for a onedimensional motion you have M dvdt which is equal to f i now say what happens if if is so this is a scalar because this is 1D everything is scalar over here if if is the DDX the space derivative of another scalar function function I mean it is not impossible we will just ask this question what happens if we have this do we have any interesting thing so the first thing is that what you can say that yes you have to just look this thing a little bit more carefully so M if I say that this is V DV DX that I just said uh some while ago that that dvdt is nothing but DV DX DX DT so this is V dvdx and I simply now said okay let us do something a little bit more attractive so that people can relate I would say if is not dvdx but it is minus dvdx actually okay no problem I mean this all depends on how you define v okay so I can write that so this is M Small v d small V DX is equal to minus d Capital vdx what happens I would say this is nothing but DDX of half M v² and this one is minus V so it will also come to this side that will be V and that will be zero okay so you see that this DDX of this thing is zero now if I simply say DDT of half in m v² + V then that will be actually so uh I will have some DT DX is equal to Zer and what is DT DX this is 1 by V now if I consider a motion which is where I mean it's more most reasonable to consider the V not to be equal to zero huh otherwise uh there is no I mean no meaning of studying the motion okay okay so 1 by V is not equal to zero and that's why this should be zero if that has to be zero the whole thing the product then this should be zero and that means this half MV ² plus v is constant in time and also in space as you have seen over here okay in 1D both are same no problem in there okay and what is this this is we know the kinetic energy of a particle although we have not defined in this class still now but this is a part which is kinetic energy and this is a part which we call potential energy okay and this scalar function whose space derivative with a minus sign is the force we call that as a potential function okay so you see if this is the if f is equal to Min and and the sum of these two sorry that's what I was I was trying to say the sum of the two they are always keeping constant although individually they are not constant in time or in space okay that is a very good information now what exactly is this and what exactly this is so what I just said that I have a mathematical conclusion huh so that I will discuss later so the first thing is that you know that the sum of two things they are constant and you also see this constant does not contain uh time the time explicitly of course time is there through velocity uh through uh V huh through v v is a function of X and X is a function of T so always all of them are intrinsically or implicitly depending on T but T is not explicitly appearing and that's why this is known as an integral of motion since I have this relation I can actually back calculate to find V so if I say what is v v is nothing but minus integration F DX okay you will always see there will be a play of an arbitrary constant like an initial potential or a boundary potential that you can choose at your uh convenience and it is the difference between this uh I mean difference of this V values between two points which are actually of importance in physics because if you understand that finally the DV DX which I mean Associates DV is actually something which is giving us the Dynamics okay so uh you see that um we have two things till now we have two con I mean conservation one is when the force is DVD I mean force is equal to m d vdt is zero then MV is constant that is very uh that is very evident to see and another constant is now if we have the force has this type of nature then I have another constant this is very general this is not general for that you have you have to have a force to be of this nature okay finally a smaller one another conserved Quant or another constant of motion we have to search for and uh before doing that maybe it is good to just tell you so what is this this is called the linear momentum conservation linear momentum conservation and this is the conservation of energy actually it is called mechanical energy okay now uh at this point I will just try to say one thing that if you now increase your uh dimension of motion huh the uh dimension of uh the whole thing to be more than one then of course you don't have this dvdx huh then you have to have something a more General uh expression for that and and for that what I was trying to say that I have to need uh am dvdt which is equal to then F which can be coming from all the special differentiations or all the I mean space I mean uh due to the space variation of all the spatial coordinates okay and so for that I have to somehow generalize this expression so one can easily show that what is this then you have to so in 1D if this is minus DV DX this is a capital V DV DX uh then X capap okay in more than 1D let's say in 3D where you have 3D cartisian system then you simply have minus d v DX X cap minus d v y y cap minus d v z z cap okay if you have this then actually you can say okay this is nothing but minus I have this del d x x cap plus del d y y cap plus d d z z cap which is acting on V now what is this this so what is this first of all this is the change in V change in the scalar potential function when you are changing infinite asmal your x coordinate this thing is the combination in more than 1 D let's say in 3D this is the combination of three terms this one just saying that you are only changing infinite aim x coordinate and you are keeping Y and Z coordinates to be constant and you check how V is changing only with X this is the same for y this is the same for Zed if you combine all the effects then finally you will have this in general we denote this thing by a nabla sign so it is a minus nabla which is itself a vector operator Vector differential operator working on V okay now if you see I if I just take this thing dotted with this thing dotted with Dr Vector where R is nothing but the position Vector then I have a very interesting result huh then you have DV DX X cap plus d v d y y cap plus d v d z z cap dot DX X cap plus Dy YC plus DZ z cap if you do the calculation what it gives since this is a scalar product between I mean two vectors what do they give they give simply DV DX DX plus DV Dy Dy plus DV d z DZ and if you know the rule of chain rule of partial differentiation this is nothing but the definition of a total differentiation D of V okay D of V that is exactly what we will use for def I mean what we will use for defining the something like another conservation of I mean something like a conservation of this type conservation of mechanical energy in more than one dimension okay how to do that how to do that so we will start by saying that okay I have I have M dvdt and that is equal to now what I was just saying is that minus gradient of V that is my assumption my force is in this form okay I will write both sides V Dot so I can say like this no problem and this one I would write this dot Dr DT because V and Dr DT they are the same thing vectorially right this is M DDT of I would say this is DDT of M v² by 2 and this is you just have seen this part is nothing but dvdt so you have DDT of M v² by 2 plus V is equal to Zer and you have M v² by 2 + V to be constant even in in more than one dimension okay this is known as once again kinetic energy this is known as potential energy okay here our assumption is V is a function of the space coordinates only okay with this I just uh wrap up this lecture so in the next one we will discuss some more properties and some more conservation principles okay regarding this type of motion thank you [Music] [Music] [Music]