Lecture 22

[Music] hello and welcome to a small discussion on a very interesting topic which is the topic of lapas ralen vector so this is a vector which is conserved only in the case of an inverse Square Force law this is not a conserved quantity for a general Central Force motion but it is a specific conserved Vector for inverse Square law Force okay so in this lecture I will simply in this small discussion I will simply show the conservation of this and I will put some questions which will be very interesting for you guys to think so a is given this is the lapas rang Vector which is given by the uh this expression which is the cross product of the linear momentum vector and the angular momentum Vector minus M K * RC vector k is the constant of the force okay m is the mass of the particle now why this is uh conserved in a motion under an inverse squared Force so for that we start just by doing this uh demonstration so DDT of P cross l I can always write this as DP DT cross L plus P cross DL DT remember when we do DDT of the cross products the order should not be change of the vectors okay now this one we already know this is equal to zero for any Central force that we will use and we will simply have this is equal to DP DT cross L okay and then in the next step we will simply write again so what is dpdt we have to understand this is nothing but the force itself so I can easily say this is the force law so which is nothing but I mean I can write here is f of R * R cap okay for the time being I am simply writing this expression over here so this is uh the force so it will be F of R * R cap cross L now L instead of L I can also write this F of r r cap cross R cross MV so this is L this one is L that I can write I can write F of r r cap cross R cross M Dr DT okay and then you can write of course this is equal to m f of R now I take an R over here and I make this one as a r Vector okay so then I can simply say that I will have R Vector cross R cross Dr DT okay if I can do I can then use for this part this is a vector triple product I can use back cab rule as you all know MFR by R then I can First Take R do Dr DT time R minus r² Dr R Vector DT okay now the first one you know whenever I talk about this R do drdt this is a very important conclusion is equal to also R do Dr DT because both of them are actually equal to DDT of r² by 2 now R square actually comes in two ways r² is equal to mod R whole s and it is also R do R okay so that exactly we use here to get all these things okay I can then write this is equal to m f of R by r r Dr DT r r Vector sorry R Vector minus r² D DT okay I have this thing and then I can write that okay I have MFR I take this R inside I will have Dr DT R Vector minus r Dr R Vector DT okay I then do something very interesting I will multiply and divide the whole thing by r² and then what I will get by this interesting multiplication and division is the following then I can take MFR time r² then this 1 by R square I will include that inside the bracket and I will simply have this 1 / r² Dr R DT R Vector minus 1 / R Dr Vector DT okay and I take a global minus sign now common MF of R * r² and I write the expression as 1 R Dr Vector DT okay I just take a minus sign throughout common okay and I will have plus minus of 1 R by r² Dr R scal DT R vector and if you just check this expression carefully you will see this expression is nothing but DDT of R vector by R okay and so DDT of P cross L comes to be so I can just write so DDT of so DDT of P cross L is equal to Min - m f r * r² DDT of R vector by R now what is f f FR is if you remember Min - K by r² so f * r² will be minus K so there is a minus it will be M K DDT of R vector by R okay so you see DDT of P cross L minus these both are constant so I can write M K what is r vector by R this is nothing but R unit Vector okay that will be zero and you can finally say that is my a vector so which is the laas ralin vector vector okay now lapas ralin Vector is conserved for for example when something is conserved something is zero something is conserved we have some uh intrinsic meaning for example we say that okay whenever there is no torque the angular momentum is conserved okay and uh whenever we say that there is no force that linear momentum is conserved so actually uh in some later lecture I will also discuss some Symmetry and the conservation some relation between Symmetry and conservation and you will see that always with some this type of uh conservation principle there is a continuous symmetry Associated huh and uh so my question is can you also find some other meaning so is this just an accident the conservation of uh laas rang Leng Vector all this is just a mathematical uh output which doesn't have any meaning I mean at least or can you at least make some geometrical meaning geometric uh geometrical significance huh from that related to the nature of orbits and everything like that and uh tell me that what is the meaning of this uh conservation of this laas rangling Vector there are two very important points for that I am giving you a hint okay what you have to do you have to find three things one is a do l so what is the and that that means actually I am interested to know the angle between the angular momentum vector and the laplus rang L Vector then a do r or R cap okay so the radius vector and the laplus rang LED vector and finally modulus of a okay or its square if this is a constant Vector of course you have to know that this is also a constant the magnitude is also constant now my point is that using these three elements can you find some interesting conclusion about the nature of the trajectory or something under an inverse Square law and the significance of the conservation of laas ralin vector okay thank you very much [Music] [Music] [Music]