 Greetings, a very hearty good morning to everybody and a very hearty welcome to today's workshop organized by the national program on technology enhanced learning at IIT Madras. This some of these courses that we are going to review today have been thrown open for certification and students can register for these courses and get a certificate from the NPTEL and the condition is that everybody is at least 13 years old so I hope that all of you are at least 13. So anyhow I would like to first thank the NPTEL the NPTEL staff here the administrative staff and the technical staff is absolutely superb and these courses would not have been possible without the expertise and commitment of the NPTEL my colleague here at NPTEL. So we're going to have two sessions the morning session will be mostly on the course on classical mechanics which I gave for the NPTEL and this is at the level of the first course I often teach at the IIT Madras so the prerequisite for this course is just high school level mathematics and physics so it is not at the level of Goldstein's classical mechanics as we teach at MSc so this is above high school physics but not quite at the level of Goldstein's classical mechanics and in particular I would like to begin with this consideration as to what one might want to do if you were to meet a situation in which there is a calamity there is a huge disaster and just coming two days after the earthquake in Nepal it is particularly ironical that we should be talking about this and if we were to leave science in just one line what is it that we would leave behind for posterity now this is an immensely arrogant proposition as to how could one pack the entire findings of science in just one line but then the person who attempted this is a very eminent physicist and he I'm going to quote him what he said is that if in some cataclysm all of scientific knowledge were to be destroyed and only one sentence pass on to the next generation of creatures what statement would contain the most information in the fewest words so this is the question which he raised and it is really challenging to give a thought to such a consideration now he not only raised this question he even went on to answer this so let's look at his answer he says it is the atomic hypothesis that all things are made of atoms so this is just one line and this is what he proposed would be the best sentence to be left for posterity if there was a cataclysm of sorts and he you perhaps recognize the scientist from this quotation let me give you a little bit more of this because what he goes on to explain is that in that one sentence you will find an enormous amount of information about the world if only a little bit of imagination and thinking are applied now this is of course a little thinking and little imagination by his standard so of course we have to take it to the pinch of salt but so it's a lot of imagination perhaps but then yes this would be the one sentence that he would recommend to be passed on to posterity now the person who said this is none other than Richard Feynman and you will find this quote in Feynman lectures now I have no idea why Feynman chose this one sentence but what I would like to share with you is my speculation about why he might choose the atomic hypothesis to be the most important thing in science so that it could be left for posterity and my question would be why would Feynman think that the atomic physics is so important okay so that's a that's a very interesting and a challenging question to consider and what he says that okay in that one sentence if you add some imagination and thinking you can get a whole lot of science now the reason one would think so is because atomic physics has played a huge role in developing quantum theory it has played a big role in the development of relativity and it has played a big role in the development of statistical mechanics and of course there is one man who contributed to all of these three which is quantum theory theory of relativity in statistical mechanics which is Einstein himself right so let me illustrate how atomic physics really has in it the seeds of these three major developments in the last century and from that it is easy to extrapolate and see why this would be such an important one line sentence that Feynman may have chosen so let me show you a picture of a spectrum which all of you have seen in your physics labs in the sodium vapor even d2 lines and obviously you have quantum mechanics in the atomic spectrum here right so that that goes without saying you have the spin orbit splitting and that would not be there without the theory of relativity okay even if the sodium atom was at rest it doesn't have to be traveling at the speed of light or even at a fraction of the speed of light even if it were completely at rest you would have the spin orbit splitting so it has got relativity built into it it has got statistical mechanics built into it because the spin orbit splitting would not be there without the fermion nature of the electron right it's the fermi spin half of the electron which is responsible for the spin orbit splitting so you can see that okay if you just look at just one simple aspect of an atomic spectrum you find in it the seeds for quantum mechanics for relativity for statistical mechanics and from that you can develop so much in science which is what makes atomic spectrum or atomic physics so immensely important not only that if you study certain detailed features of the spectrum you could actually discover even the coulomb's law and by the end of the day today you will see how okay so these are some of the very fascinating aspects of atomic physics which I have made an attempt to address in these courses and the reason you can get the coulomb's is because you have the sodium d1 d2 transitions in the sodium from the 3p to 3s because the energy of 3p is different from that of the 3s in the hydrogen atom the 3p and 3s levels are degenerate they have the same energy so in the hydrogen atom there would be no transition from 3p to 3s and then if you ask this question why is it that you have a 3p to 3s transition in the sodium atom but not in the hydrogen atom the answer this inquiry would lead you to the coulomb's law so these are some of the fascinating questions that we will address in these three courses and the connections come from a very important and underlying principle in physics which is the connection between symmetry and conservation laws and this is enunciated in the very famous theorem named after the lady whose picture you see on the screen this is Emily Neuter and this is known as the Neuter's theorem so the Neuter's theorem connects symmetry and conservation laws in an extremely intricate and subtle manner and many of these connections can be seen right in classical mechanics in Neuter's laws in Hamilton's equations and so on which often high school physics does not reveal and the undergraduate courses also do not reveal so what we are going to discuss today and there is a four page brochure handout which has been distributed to each one of you and in that I have pointed out that these courses are not really designed to meet the requirements of any particular university or institute's curriculum but it is possible to have a curriculum based on these courses what they do is that they bring out to the surface many features which we miss out in undergraduate physics and which will make learning physics very enjoyable and also very robust so these are the things that we are going to address in today's workshop so symmetry and conservation laws are extremely important just a few years ago there was a Nobel Prize related to some studies in related areas so these are sophisticated topics in which I will not get into but I will only highlight the importance of these subjects which I think we can teach right from the point when we discuss Newton's laws and similar undergraduate physics so we are going to go down to the very foundations of classical mechanics and the foundations lie in these principles as I write on the screen here which is the principle of causality and determinism the linear response theory the principle of variation and symmetry and conservation laws so it is really not necessary for you to take notes because this whole procedure proceeding of the workshop is being recorded okay it will be uploaded and a PDF file of this will be available to you so that is one of the major advantages of the courses this is really technology enhanced learning so I would like you to focus on the discussion itself rather than be distracted and be occupied by taking notes so you're quite welcome to write whatever you want or draw cartoons if you like but so far as the material itself is concerned it is going to be available to you and a PDF file of all the PowerPoint slides that I'm using will be uploaded on the NPTEL website so anyhow let's go down to the central problem in mechanics and very few undergraduate courses really deal with these issues in a at a very fundamental level so the principle question here is how is the mechanical state of a system described is it described means here I've got an object so do I describe the mechanical object by its mass by volume yes all of these are mechanical properties in a certain sense okay but that is not what we mean by the mechanical state of a system so how exactly is the mechanical state of a system described and then it turns out that we need to require for a system having one degree one one coordinate we need to specify both the position and the velocity to describe the mechanical state of the system and there are many undergraduate students who often wonder why is velocity independent of position because after all it is the time derivative of the position so they don't see that it has to be specified independently to describe the mechanical state of a system and you can see it if you just have a look at this picture and if you think of an aeroplane which is flying from Chennai to Bangalore at 300 kilometers per hour roughly speaking which is the distance between Bangalore and Chennai then in an hour if it started in Chennai it would reach Bangalore but if at the same speed it started out from Bangalore it would rather reach Bangalore if it were flying toward the west so where it will be found at a later time can be predicted if only you knew both the position as well as the velocity one of these parameters would not be sufficient to describe the mechanical state of a system so which is why you require both of these parameters the position and the velocity and these being independent you can map them on a plane and on orthogonal coordinates the position on one axis and the velocity on the other obviously the dimension of this plane will not be l square as is often the case if you were plotting x versus y but what is on the y-axis is not a length parameter but it is length divided by time so the dimensions of this two dimensional space will be not l square but it is l square t to the minus one since velocity can is also contained in the parameter momentum you can also specify the mechanical state of a system equivalently by position and momentum so now we have our answer that a mechanical state of the system is described by position and velocity of an object or equivalently by position and momentum and this is the central question in mechanics that tells us how the system is described and how it evolves with time will be given by the rate equation the rate at which the position changes is the velocity the rate at which the velocity changes is the acceleration so if there is a relationship between position velocity and acceleration then we have the equation of motion which tells us how the system evolves with time and you can also do it by giving the rate equations for position and momentum because that is an alternative way alternative and equivalently of describing a mechanical state of the system so this description of how the system evolves with time is what we call as a law of nature because there is something fundamental about it if we discover how the system evolves with time and that would be applicable to every system no matter what it is it will describe how this object falls how it moves if I were to throw it it will describe how the moon revolves around the sun it will revolve everything in the universe and that is why it would be called as a law of nature so the law of nature would be the description of the relationship between position velocity and acceleration and these are given by what we call as the equations of motion and these equations of motion are the Newton's equation or equivalently the Lagrange or the Hamilton's equations and this is the technical description of what an equation of motion is that it is the mathematical and rigorous relationship between position velocity and acceleration so if you look for fundamental questions in physics these questions are what are the laws of nature that is what we endeavor to see we want to know what exactly are the laws of nature and how do they influence the physical world around us in physics we rarely ask why are the laws of nature the way they are okay that that that question typically goes beyond the realm of physics it perhaps gets into metaphysics and I will not address that question you can sometimes raise these questions but most of those questions if you analyze them they will reduce to what the laws are rather than why the laws are what they are now what are the laws of nature what are the laws of physics so that we can think of the Newton's law the coulomb's law the ampere's law principle of uncertainty theory of relativity all of these are laws for which no exception is found these laws are applicable at different levels it's depending on whether you analyze things within the domain of classical mechanics or quantum theory so i'm already referring to phenomena which demand the theory of relativity in quantum mechanics but nevertheless this is what we call as the law of nature then at the high school level students also learn conservation laws like the conservation of linear momentum conservation of angular momentum conservation of energy and then you also learn about the conservation of mass energy which are interconvertible then there is a conservation of charge and so on so so these are number of conservation principles that one learns now the question is can we get one from the other and this is an interesting question because in physics you always try to learn the least and get the maximum information out of it so if you can get the conservation principles from the laws of physics there is no need to learn them separately vice versa if you can get the laws of physics from the conservation principles you don't have to learn them separately and in any case it is very important to get the interrelationship because knowing one you now have the a mechanism or a technique to get the other and both are obviously extremely important in physics so these connections between the laws of physics and the conservation principles come because of an intrinsic and a very powerful connection between symmetry and conservation law so symmetry plays a very important role which is why I highlighted Neuter's theorem a little while ago and let us look at Newton's laws we always say that okay Newton's laws cannot be derived from anything you can verify them you can carry out experiments and so on but now let us ask a question can we actually deduce it from symmetry can we deduce the Newton's law from symmetry and what I am going to illustrate as to how we can deduce Newton's third law from symmetry consideration alone so this is a new approach which started out with Einstein's work in the last century and it was encapsulated very nicely in Neuter's theorem very beautifully elicited by Wigner using group theory and other advanced theories and quantum mechanics and mathematical physics so these it is these are extremely important ideas which you can communicate to students at the very early stage of learning physics when they're just fresh out of high schools and undergoing their training at the undergraduate level in physics so we will talk about how you can first get conservation principles from the laws of physics so let's take a simple example of two objects m1 and m2 two masses which are interacting with each other through gravitational interaction so you've got two bodies m1 and m2 and they have this g m1 m2 by r square interaction between them you can write the equation of motion in the center of mass okay so all of these things I will not go into details you're very familiar with this so you can set up the equation of motion force by one on particle two is given by g m1 m2 by r square and this being a force you have to write the direction of the vector very clearly and the force by the mass two on the particle one would be given by an equal and opposite force which is over here right so the direction of this force is opposite to that of the previous interaction so you can rewrite this in the center of mass coordinates and essentially you introduce the proportionality which is g m1 plus m2 as kappa which has got the dimensions of l cube g to the minus two and you can write this equation in a very simple manner for the acceleration and essentially you see the one over r square law which has been used over here so this equation at the bottom is nothing but the one over r square gravity law so we start out with a law of nature that this is a law that we know we know it from Newton and we will use it in our analysis as as we develop it so given this law of nature which is the equation of motion which is packed into this equation of motion you take the dot product of this equation with the velocity so there's a very simple vector algebra nothing nothing very fancy no advanced mathematical techniques all you do is to take the dot product of the velocity okay now the velocity is a vector so you must take the time derivative of both the distance and also the direction so when you do this you get two terms put these two terms in the expression for the velocity which is here and you have the velocity appearing here as well so put it in both of these equations and essentially what you're going to find is a result which is v v dot plus kappa over r cube r r dot equal to zero so this is a very simple relation that we get simply from the equation of motion by doing some simple scalar product with the velocity so this is the relationship now and we can integrate this okay because it has got a derivative term so we can do some very simple mathematics so we have got this relation and if you integrate this with respect to time you get v square by 2 minus kappa over r equal to e and if you multiply this by the mass you see the kinetic energy plus the potential energy is e what is e this is emerged as the constant of integration okay so it is really very interesting that we started out with an equation of motion and all we have done is to take the scalar product so no major physics principle or mathematical algebra inserted in our analysis so beginning with the equation of motion which we call as the law of nature we actually get a conservation principle because the last equation here is nothing but the statement of conservation of energy that the sum total of the kinetic energy this is the kinetic energy per unit mass because I've removed mass from our analysis and likewise this is the potential energy per unit mass right so this is the net energy of the system which is a constant and we get a conservation principle directly out of the equation of motion so we don't really have to learn it as a separate principle or something we can get it straight out of the law of nature now it is also interesting that the integration is with respect to time okay because when you integrate with respect to time the conservation principle that you have deduced is the conservation of energy and this happens because energy and time are canonically conjugate in the in the sense of classical mechanics so energy and time being conjugate variables so these are related through this particular relationship that the differentiation of this relationship will give you the previous equation so the relationship between these two equations is a very simple mathematical inverse relationship and there is no new mathematics or details physics principle which is incorporated in this so this is what I call as the mechanical energy because it is the energy per unit mass so I've just taken out the mass to simplify our discussion so this is because the energy and time are canonically conjugate variables and before we proceed I would like to pause here for a little moment because we are already talking about scalars we are already talking about vectors and we should know exactly what they are and I'm very grateful to my colleagues Dr. Arvind and Dr. Vijayan over here and they will be helping us with some lectures during the course of the day and Dr. Vijayan will have a more detailed discussion on vector calculus in his lecture but I would like to caution you over here that we often define a scalar as a quantity which is defined by magnitude alone and a vector which is defined by magnitude and direction now this is all this is very nice but if these definitions fail then with all the phd and the training and everything that our students may get you know they may feel very frustrated if these definitions actually fail and these definitions actually do indeed fail they are not appropriate definitions because these definitions would fail if you are able to find a physical quantity which has got magnitude and direction but it is not a vector okay then this definition won't work or if you find a quantity which you know for sure is a scalar but you can't describe it without attributing to it a direction okay if it doesn't have a directional attribute it won't work and let me give you an example over here and which is why the definitions of scalars and vectors will have to be sharpened so here here let's have a picture of the mount Everest and you look at this point over here and you ask at what rate does the height of this mountain changes okay what is the rate of change of the height as as you move away from this point okay so it is delta H which is the change in height divided by delta F which is the step size you take from that particular point and you take the limit delta H by delta S in the limit delta S going to zero and obviously if you take steps in different directions you're going to get a different answer right so the numerator is delta H which is height the denominator is delta S which is distance on the side right and this is a ratio of two scalars it is obviously a scalar and it will not have a meaning unless you specify the direction in which you take this ratio so this is a directional derivative which is actually a scalar and it requires it has a directional attribute so you cannot properly define a scalar just by magnitude alone or a vector which requires just a magnitude and a direction there are some additional features which one must specify but professor vision will elaborate on this in his lecture so I just want to caution here that we are already using these ideas which we should use with an enormous degree of caution so this is our fundamental question in mechanics how does this how is the mechanical state of assistant described and how does it evolve with time and Newton's mechanics tells us that the system it was with time according to an equation of motion which we call as the Newton's law and it tells us how what is the relationship between position velocity and acceleration which is f equal to mass times acceleration or rate of change of momentum dp by dt which is a better way of stating the Newton's law but then any other equation of motion which will describe the rate at which q changes with time and p changes with time because q and p the position and momentum do specify the change of the system so you can also describe the mechanical state of the system by a function of position and velocity or by a function of position and momentum okay and these functions could be your Lagrangian and the Hamiltonian so they do exactly what Newton's laws do they tell you how is the system described and how does the system change with time but the two are based on completely different principles so the Lagrangian and Hamiltonian formulation is based on the principle of variation it tells you that the mechanical system evolves in time in such a way that the action integral is an extremum so this becomes an alternative but equivalent formulation of mechanics and the condition that the action integral is an extremum is stated either in terms of Lagrangian's equations or equivalently in terms of Hamilton's equations so both of these are conditions that action would be an integral would be an extremum so you have got two alternative formulations of mechanics one is the Newtonian or the Galileo Newtonian formulation of mechanics which is based on causality and determinism and the second is the principle of variation and there is no reason why students who have just the background at high school physics in mathematics and physics cannot grasp these ideas so these principles can be introduced at a very early stage of learning because they are so powerful and they introduce the students to the language of the principle of variation and variational calculus so this is now our answer to how a system evolves with time but now we are confronted with the question that if position and momentum cannot be determined accurately under physical actual laboratory conditions okay not because of certain limitations of the apparatus but because of certain intrinsic manner in which nature reveals itself to an experimental probe then you have to find an alternative scheme and that is where you require quantum mechanics and this is again now an answer to the same question how is a mechanical system described if it cannot be described by position and momentum because they cannot be determined simultaneously then the system has to be described by a completely new quantity it cannot be q and p and what is this new quantity it is a state vector or the wave function and how does it evolve with time by its rate equation which is the equation of motion for del psi by del t and that's what the Schrodinger equation gives us so we can introduce the students to all of these ideas at a very early stage so now we will ask how the laws of physics were discovered or how they were how they are to be understood and can we deduce them from anything that is more fundamental so here is an interesting question which many of our students fail to answer correctly and I think it's it is a good exercise if you look at Newton's laws and you write Newton's second law as f equal to ma which is mass times the second derivative of the position vector and if you put f equal to zero then you get the velocity to be a constant and then one can say that okay what it is telling us that a body address remains the address whereas if it is going at a constant velocity it will continue to do so and we may be led to suspect that the Newton's first law is only a special case of the second unfortunately some books also say that it is but then if one were to ask that if it is a special case of the second why do you learn it as a separate law it's not even an oral problem okay so there is a good reason why it is a fundamental law it is a separate law it is it cannot be regarded as a special case of the second and the reason is you cannot even touch the second law unless you recognize what an inertial frame of references and these are some of the things that we have highlighted in our course on classical mechanics so I will not go into those details because the course is already there on the web and you can go through those lectures we can continue to have discussions based on that but you need Galileo's first law the first law which we call as a first law of Newton is actually due to Galileo because he discovered that where an object would fall if you would drop it when you are on land or if you were going on a ship which is moving at a constant velocity that depends just on the state of the motion which is described in an inertial frame of reference so he deduced what are the conditions of an inertial frame of reference it is one in which motion is self-explanatory it does not require any external cause so you must first identify what is an inertial frame of reference and then go on to explain why there are departures from equilibrium and those answers are then provided by Newton's second law so the first law is a separate law and we need to highlight this in our undergraduate courses and then we need to recognize how to find how to identify an inertial frame of reference and unless there is sufficient familiarity with an inertial frame of reference as opposed to what a non-inertial frame of reference one really does not get a good grasp of what an inertial frame of reference is because it is very unsatisfactory to define an inertial frame of reference as one which moves at a constant velocity with respect to another frame which is an inertial frame of reference so you end up defining it in terms of itself which really doesn't take you anywhere so it is important to elucidate these ideas and that is something that we attempt to do in these courses over here that you have another frame of reference like f prime which is moving at a constant velocity you see with respect to this red frame of reference and if you look at an object in this red frame of reference and you look at the same object from this blue frame of reference as well then you can relate the position vectors in the red frame and the two and the blue frame by this very simple triangle law of additional vectors and you can take the first derivative and find the connection between the velocities which is the Galilean relativity that you find in this statement over here and then if you take the second derivative you find that the second derivatives in both the frames are exactly the same which means that if you multiply both sides of this equation by mass you get the mass times acceleration and the cause effect relationship in one frame is exactly the same as it is in the other so f equal to ma will hold good in one frame and the same f equal to ma will explain the departures from equilibrium in the blue frame as well and this is essentially the principle of Galilean relativity that laws of mechanics are exactly the same in all frames of references which move with respect to each other at a constant velocity so to some extent the first law of inertia is counter intuitive nevertheless I think it needed the genius of somebody like Galileo to discover that and here you find the departure from equilibrium which is which reveals itself as acceleration and here you have got a linear relation the force which is the cause of this departure and the acceleration which is the effect of this force and you find that the effect acceleration is linearly proportional to the cost so here is the principle of linear response okay so the linear response is at the very heart of Newtonian mechanics and this is how Newtonian mechanics is formulated it is better stated as the rate of change of momentum rather than mass times acceleration the two are completely equivalent in most situations you will also notice that the equations of motion are completely symmetric under time reversal if you let t go to minus t you get exactly the time the same equations whether it is Newton's equation or Hamilton's equation Hamilton's equations of course are first order equations but then there is a minus sign here which takes care of the symmetry and both the Hamilton's equations as well as the Newton's equations and of course the Lagrange's equations are symmetric under time reversal one has to be careful because time reversal has got a different meaning in quantum mechanics but I will comment on that perhaps in the afternoon so we introduce the laws of nature as fundamental principles which we need somebody like Galileo or Newton to use their insight and their genius to enunciate them or we do not derive them from anything more fundamental and likewise Newton's third law would also be introduced in a similar way in the Newtonian scheme as such so because as long as we did we did not enter the 20th century and the world of quantum theory these there were no exceptions which were found to these or even if the exceptions were found they were not understood in terms of failures or limitations of classical mechanics then these were called as the laws of nature because they were universally applicable so let us look at the third law that the force on object one by two is exactly equal and opposite to the other force in the backward direction so now this is introduced as a law of nature in Newtonian scheme however we will illustrate today that it can actually be obtained from a different principle which is based on symmetry and conservation laws so let us introduce that and this brings us to an important question are the conservation principles consequences of law of nature or are the laws of nature consequences of symmetry principles that govern them and these are very fundamental questions which had one answer until Einstein but then after Einstein's work it was understood that the symmetry plays a fundamentally important role and that is the fun that is the important role that I will like this example to illustrate so let us take the example of the third law and if you begin with the third law which is f1 2 equal to minus f2 1 you immediately find that the time derivative of the total momentum goes to 0 which means that the linear total linear momentum is conserved so you get a conservation principle from the law of nature as we did earlier now let us look at it in the reverse sense what we will do is to begin with a symmetry conservation we will consider a certain n number of particles in homogeneous space and we consider the space to be homogeneous in the sense that okay all the property that's the space if you move things translate them one way or another backward or forward upward or downward right so if you if you have translational invariance in this space so that is the kind of homogeneous space that we work with and in the system we consider a system of n particles okay n particles and all of these n particles we consider all of them to undergo a displacement through an infinitesimal extent like delta s a small tiny displacement in homogeneous space through a translational vector okay so that is the situation we are considering I would like to highlight some important characteristics of this displacement that we are considering we consider these particles to be such that the only interactions which govern their dynamics is the inter particle interaction so in other words I'm stating that there are no external fields which influence them we also consider the displacement to be instantaneous the entire n particle system undergoes the displacement instantaneously okay we consider the entire medium to be homogeneous so the properties of space are the same here as over here okay there is absolutely no difference and we consider the entire displacement to take place at a specific instant of time as and not as a displacement from now until then if I were to take an object and move it from a certain point to another point even if it is infinitely close I would pick it at even and leave it at t2 so there is a certain time interval which is involved in such a displacement but the displacement that I am considering is is one that takes place at a specific instant of time so obviously it is only a mental process it is a virtual displacement no real physical displacement can be affected at a particular instant of time in the manner in which I have described right so this is what is called as a virtual displacement and this is a virtual displacement which takes place at a specific instant of time and because the displacement is virtual it becomes redundant to ask what is the agency which caused it so there is no agency which are okay so this is just a virtual displacement that we consider and because there is no external force which is involved in this okay internal forces do not do any work in the virtual displacements or work done is zero or as I like to call it a work not done is zero right so let us write this in this mathematical form that the work done is zero the network done is the total force times the displacement so you take the scalar product of the force with the displacement and the net force on the kth particle is the force due to the ith particle your sum over i going from one through n to leave out i equal to k and then sum over all the particles k going from one through n so this is a very simple relationship that we get for this virtual work which is zero right so now if you look at this result and these are very nice mathematical techniques developed by um uh d alumbert or i don't know how to pronounce this french name good thing that french know how to do that but i certainly don't and we simply ask a question that under what condition do we have the scalar product of two vectors go to zero okay now the scalar product of the two vectors would be zero of course if those two vectors are orthogonal to each other but that is not something that we need to consider because the displacement was in some random direction so it is not obligatory that it would be in a direction which is orthogonal to the sum of these two so now we are left with only one possibility that the two vectors are not necessarily orthogonal to each other but the displacement being there is not zero so delta s is not zero and therefore this sum total of the vectors in this beautiful bracket must be zero so that is a conclusion we cannot escape and now let us remind ourselves that in deriving this result we have certainly used used Newton's first law because we worked in an inertial frame we have also used Newton's second law but we have not used Newton's third law and if you look at this result that dp by dt is equal to zero if you specialize this for just two particles you'll find that dp2 by dt is equal to minus of d by dt of the momentum of the particle one as essentially we have discovered Newton's third law in other words if Newton's third law was not initiated as a separate law of nature we could have deduced it from the consideration of homogeneity of space and translational invariance so symmetry would lead us to the law of nature and it provides us with a mechanism to discover laws of nature here is another example that if you consider the two body motion between earth and the sun we know that the solution is an ellipse and we could actually use symmetry and conservation laws to discover the gravitational law the law of gravity so let me illustrate how so if you now consider the equation of motion so this is the law of nature that we begin with this is the law of nature and we have the angular momentum over here which is r cross p I take away the mass so take the momentum per unit mass so instead of p I've got the velocity right so this is nothing but the angular momentum per unit mass so h is the angular momentum per unit mass which is called as the specific angular momentum what I do is take the equation of motion and construct the cross product with the angular momentum so again no major involved mathematics being done over here so we just take the cross product of the angular momentum with the equation of motion what do we find so let's do it step by step so you've got one term here a second term over here let's bring it to the top of the next slide over here here you have got h which is a cross product of the position vector and the momentum or the velocity momentum per unit mass now you've got a vector shipple product so you use what is famously called as the backup rule and expand this bracket so you've got what do you call it outer dot remote adjacent minus outer dot adjacent remote or something like that right so you use the backup rule and you get this relation and this is a very simple algebra vector algebra that we are doing no involved mathematics and all you have to do is to find what this r dot v term over here is so fine we find that r dot v is nothing but the product of the magnitude of the position and the time derivative of the position r r dot and we put it back in this equation so now if you look at this relation once again you have this r h cross r double dot in the first term which is nothing but the time derivative of the cross product of the angular momentum with the velocity right so you can put this back over here okay so this being the same term as here you replace the first term by the time derivative of h cross v so here we have got the time derivative of h cross v coming in the first term and now you write the second term as it is but you recognize that the second term itself is the time derivative of this ratio r over r okay because this will give you these two terms the second derivative this is the derivative of a product of two functions this vector divided by the scalar and this will give you these terms now you combine these two and essentially you'll find that the time derivative of the vector which appears in this rectangular square bracket which must go to zero and what happens when the time derivative of a quantity which goes to zero that quantity must be a constant so what have you discovered by taking the cross product of the angular momentum with the equation of motion you have discovered another constant you have discovered a conservation principle the constancy of this new vector is a conservation principle now this vector which is conserved is called as a Laplace Rooney lens vector and what it does is it plays a major role in two-body problems because it keeps the orbit fixed and let me show you how so if you take this vector which is the Laplace vector and if you take the dot product of this vector with the position vector okay all you are doing no major involved mathematics just the dot product with the position vector and you organize these terms reverse the signs and so on the algebra is very simple all over these slides will be on the NPTEL website so you can look at these mathematical equations there's no involved mathematics that we are doing so just focus on the basic ideas that all we are doing is taking the cross product of the angular momentum with the equation of motion the next thing we do is to find discover a constant of motion which is the Laplace vector and now we take the dot product of this vector with the position vector and what we find by doing a little bit of simple algebra is to discover nothing but the equation to an ellipse in polar coordinates so we actually get the orbit by doing this very simple algebra and if we were to do this using conventional techniques how do we get an orbit or how do we get the trajectory of a particle which is governed by some dynamics what we would do is to set up the equation of motion right put the initial conditions integrate it okay do differential calculus and and and by taking all this we will find the solution which is the equation to the orbit so here we have got it directly from the properties of the Laplace vector and this equation to the ellipse that you get is in polar coordinates and it involves epsilon which turns out to be the ellipse eccentricity of the orbit so you can illustrate it using polar coordinates and professor vision will help you get acquainted with different coordinate systems and so on so I won't spend any time over yeah but essentially I will point out to you that this vector which is a constant of motion we already found that it is a constant of motion but let us discuss for a moment what is the reason why it is a constant of motion okay that its constancy is geared to one very important property that its time derivative must vanish and if you analyze its time derivative take the d by dt of this vector angular momentum is already a constant so this term goes away and if you look at the remaining terms take the time derivative of the unit vector which is not a constant so on put in all these terms we find that its constancy requires us to know what is the force law because unless you know what this dv by dt is which is dp by dt actually right so it is the rate of change of momentum per unit mass which is dv by dt acceleration is nothing but the rate of change of momentum per unit mass and the rate of change of momentum per unit mass or dp by dt is nothing but the force right so we must know what the force and if you put the force to be the 1 over r square so this is the rate of change of momentum per unit mass which is the 1 over r square law right if you put this 1 over r square law you immediately find that this 1 over r square must be inserted here so that you find that the Laplace vector is a constant of motion in other words the Laplace vector would not be a constant of motion if gravity was not determined by the 1 over r square law okay now if you work with this argument bakwa okay and argue that okay you do know that the Laplace vector that there is a vector which is constant and how do you know that because the ellipse does not possess okay you know that the ellipse does not possess you can have an ellipse which possesses and that would look from a distance like a rose like the petals of a rose which is why it is sometimes called as a rosette motion and the reason why the orbit does not possess is not because of the constancy of the angular momentum but because the major axis is fixed and the major axis is specified by the direction of this Laplace vector which is a constant in other words might you have discovered the law of gravity from this property it's very fascinating to think about that possibility okay and these are very interesting questions because now we all know that the law of gravity is governed by the one over r square Newton was the first one to discover that and he did it as the story goes Haley asked him this question what would be the force between the two masses and Newton said that it has to be one over r square and Haley asked him how do you know that and Newton said he had invented calculus he had he had discovered gravity he put everything together and did the calculus and then discovered that it has to be an ellipse and that is how one would solve a problem but here is another way of discovering that the orbits have to be an ellipse because otherwise the ellipse would not be a fixed quantity so the ellipse is fixed only because of the inverse square and for no other reason so these are the connections between symmetry and conservation laws which I wanted to highlight and as you can see you can acquaint an undergraduate student who is just fresh out of high school to you can introduce him to these very fascinating principles in physics the principle of causality and determinism in Newtonian mechanics or the principle of variation in Lagrangian and Hamiltonian sense and symmetries are of very many different kinds when you get into quantum mechanics and so on you deal with not the continuous and the discrete symmetries you also have the dynamical symmetry that you have discrete symmetries like parity charge conjugation time reversal and so on so I will be happy to take some questions here essentially I will like to point out to you that the Feynman messenger lecture as it is well known from the project tour is a very nice lecture to listen to it is available on the internet I'm sure that all of you will absolutely enjoy it we have a few articles which discuss the connections between symmetry and conservation laws these articles are available on my personal webpage and I will stop here I will be happy to take some questions but essentially the idea over here is not to meet the requirements of any particular syllabus of a university or any institution or how to help your students take their exams so that's not what these courses are about these courses are about enjoying physics learning them from first principles and keeping focus on very rigorous mathematics which at the same time bring out the principles in physics in a very direct and very natural way so that you really enjoy teaching and learning physics we have some time for questions and then we are going to break for tea I think and then yes I think teachers as teachers we should impress upon the students why they have to learn the other coordinate systems other than rectangular coordinate systems because that is the way nature is yes rectangular coordinate system is a very artificial thing yes maybe only you find it because it's drawing and yes yes yes yeah professor vision brings up a very important consideration that if you look at objects around you very few things have the Cartesian symmetry if you look at objects around you and the world would not be beautiful if everything had Cartesian symmetry okay it would look ugly it would look terribly ugly can you think of a rose which is rectangular so it's very important that we look at nature the way it is and that makes it necessary to acquaint ourselves with other coordinate systems professor vision will certainly be commenting more on this and it is important not only to be aware of these coordinate systems but it is important to do physics in these coordinate systems you have to be able to write the law of nature the equation of motion which are whether it is Newton's equation or Schrodinger's equation you have to be able to write it in cylindrical polar coordinates spherical polar coordinates and even a simple problem like the 1 over r potential the hydrogen atom to get the complete and correct solution for the 1 over r potential even the spherical polar coordinate system is not enough you have to use the parabolic coordinates and then unless one uses to solve equations of motion in different coordinate systems it is not possible to do physics you had a question yes you can be seated please so I thought we could deal with some systems of space time euclidean space or mid-coast space for example if a fish living inside a vessel curved vessel if it looks at the universe and has its own experience and absolutely these are very good ideas and it is often interesting to look at space in either restricted dimensions or with different curvature properties as you point out so all of these can be done in undergraduate physics so you are absolutely right so as a matter of fact in this course we introduced vector coordinate systems in a two-dimensional space in what is called as a flat land right so we work in a flat land as if the whole world had only two dimensions and how would physics appear in this flat land and then you can add additional dimensions but you can also add a curvature to the space yes we will also explain finally students will ask you are considering a work the force acting between the particles and you say it is a virtual work and finally you say 0 why do you want to consider it so consider himself we can go ahead and do it so why do you take it up the students generally we how much we put the formulas into them they last yeah without considering why don't you do that yeah I think the importance of the shunya has has been known especially to us Indians okay I think it was one of the most brilliant ideas okay and and the shunya is an extremely important quantity at the fact that there is a zero in a mathematical equation which has got two elements which you are adding which tells you that the two quantities are equal and opposite and there is a lot of information about the other side that you can get from the zero and you can't get it unless you put those two sides on the other sides so I think absolutely absolutely because there is no way you can extract properties of the other sides of the equation unless you use the enormous weight that is there in the idea of the zero yes yes I will be doing this in the next class actually after the T what I will be doing is give examples of non inertial frames so that everybody recognizes an inertial frames in contrast to that but essentially an inertial frame of reference is not to be defined as one in which Newton's laws hold that is a property which is absolutely correct but that is hardly a definition which reveals what is the nature of an inertial frame of reference so an inertial frame of reference is best defined as one in which motion is self sustaining but no other agency needs to be invoked to explain equilibrium as a matter of fact you require you seek a cause only when you find that there is a departure from equilibrium so if there is a departure from equilibrium then you have to ask what would cause it that is what you call as a physical interaction that is Newton's idea of a force what now we call as a physical interaction whether it is gravity or electromagnetic or whatever so that is a cause which explains departure from equilibrium and inertial frame of reference is one in which motion is completely self sustaining it requires no cause at all yes then some philosophy and then start explaining like do some experiments to establish to prove it then we confuse again to think that whether this character not for example this scalar and vector scalar with direction and vector without direction so every day we confuse with our definition well what what physics is attempting to do is to describe nature find what are the laws of nature and it turns out that physical quantities are tensors basically tensors special tensors have got special names tensor of rank 0 is what we call as a scalar tensor of rank 1 is what we call as a vector but essentially physical quantities are tensors and that is the way nature is and that is what I spelled out at the very outset that most questions in physics boil down to what are the laws of nature how do you describe physical systems and how do the laws of nature govern the evolution of the state of the system so that is what physics is about and because these are tensors scalars and vectors being only special kinds of tensors they have to be defined absolutely accurately and rigorously like I pointed out that if you define a scalar just by a quantity which is what magnitude alone then you run into a directional derivative for which you have to provide a directional attribute so that you can get the complete meaning of that scalar likewise a vector if you define just by magnitude and direction it is possible to find a quantity which has got magnitude and direction which is not a vector so which is why I think as a physicist we have to define these quantities very carefully so you have to go down to the basic definitions that these definitions involve how these coordinates transform when you view them from a different coordinate system which is rotated because if you rotate a coordinate system how do the components of a vector transform and if you write those equations you will automatically run into the cosine law right so the cosine law is what defines what a vector is and then you will find that the law is different for a vector like the momentum but it will be again different for a vector like the angular momentum because it transforms in a different way so we have to define these things in a very precise and careful manner