 Nu, we hadden allemaal Newton's formulatie van mechanieken. Het is een tijd-invogelheid van een mechanieke systeem. Er is een alternatief, en ik heb dit genoemd, er is een alternatief formulatie die we nie moet kiezen op determinisme en kaas. Nu, wat had Newton ontwikkeld? Hij ontwikkeld, er is ekceleratie. En, hij sête dat proportionaal om die forse. Nu, wat is ek volk? Ik kan het nie een volk zien. Ik kan het effectie van een volk zien. Ik kan dit ekceleratie omhoog. Er is ekceleratie onder ekceleratie. Maar wat is dit ekceleratie? En hoe had hij het gekozen dat dit ekceleratie, wat ik dit ekceleratie kan kiezen, het ekceleratie onder ekceleratie, het is hetzelfde als het ekceleratie van het son of het oud. En u weet dat dit ekceleratie, en die is hetzelfde als het forse. De constantie van de proportionaaliteit was inercia, de inerciaal maat. Ook de inerciaal maat en de inerciaal maat eindden in het hetzelfde. Dus, er is dit constantie van de proportionaaliteit. Dus, van dat, kan u dan het ekceleratie zien, nou, als ek de maat dubbel, want u kan 2 identieke objecten hebben, en ek dezelfde forse oefening, u kan zien wat m'n ekceleratie werkt, want nu is m'n ekceleratie het anders gaan gaan, en u kan ekceleratie kiezen van het ekseleratie, het is hetzelfde als mx dubbel, het is hetzelfde als ma. Maar het is het determinisme, en het is hetlele, het is hetlele in effect, in Newton's 1 en 2, hetlele, hetlele 3, maar het is dit alternatie, want het is hetlele in effect, hetlelele is hetlele, hetlele is hetlele, oning woens het meitaal, wat is hetlele, en het een proton, we kunnen nes jeorijn,agaat, we kan Newtoninde Wing Tip Zhong, Newtoninde Fey误 subsetув, Kenien en Selung, dit is hetlele你可以d BigQuery, daaribl geen ouding van, die van die me quartzanimhag, of die van die laws ek flirtatie gaan, meten. En het is mooi, het is metemetisch, maar er is ook wat in juwisje aan het doen. Als u op het waterkastig, nie, het is een groot waterkastig, dit is het waterkastig, op een klein hoogtje, het is het verhaal, waarom is het dat verhaal? Wel, u kan het zeggen, er is verhaal op het, maar er is iets anders gebeur hier. Waarom is het dat verhaal? als ek nou de ronde oor de koin en die ronde oor in de cirkel en wabbel en voeldoen, waarom het dat doen, dat specifiek path, waarom het dat geschoen, dat specifiek path, en het dutie van het denken dat het in Hamilton's principle uitkomt. Als u dit met de Mathematica van Riga wilt uitleggen, moet u het het calculus van variatie, dat het calculus van variatie is, is het nie het topic die dat is gelde in oudere kaas in die universiteit, maar het verhaal om iets te kiezen over de kaalteleurste van die variatie, so wat gaan we doen, ek ga ekasies en eksplonations van die kaalteleurste van die variatie, maar het verhaal in een oudere field van het oudere. Dus nu gaan we dit Hamilton's principle, ook die principle van oudere actie, principle van oudere actie, wat we van universiteit, van van m'n eerlijke dag, is dat, als ek een function heb, in X en Y, en ek neem het function eerlijke van daar tot daar, hier is het function die, dat is m'n, het eerlijke domain dat ek neem, hier is het function, ek kan gelukkig zien, die is een globale maximale, het seem dit is een globale minimale daar, maar in het between, every way you remember this where the, where the derivative, first derivative or the slope is 0, there is a local, a local extremum, local extremum and this local extremum is where the first derivative of our function, first derivative equal to 0, the slope is 0 and it can be either a local maximum or local minimum, we need to make use of this extremum, it's usually a local minimum and that's where it comes, there's something is a minimum when the water decides to take that path, something is a minimum when the coin rolls in a certain pattern, something, some nature tends to choose the path of some minimum and that's what we're going to deal with. I want to remind you of this equation, remember we said that if I have a function and I take its derivative, if it's a single variable, it's just going to be the limit, this delta x approaches 0 of this change in function divided by delta x, that was the limit and that was df dx and we have this delta f equals df dx delta x, we can write like this, as I say I'm not going to go into regular mathematical explanations, take this foreground if you don't know the change in our function f that means it's y axis change is going to be partial derivative of f with respect to x, now this is a function in x and y, it's going to be partial derivative of f with respect to x times delta x, the change in x, plus the partial derivative of f with respect to y times delta y, that gives us our change in f, just remember that equation, we're going to use it in this derivation. Lastly we're going to make use of, we've got to think about the phase space and the world line and I've tried to do it here, now we can remember cannon ball with an initial velocity, initial force on it taking this path, but that would have been in x and y, now this is phase space you'll remember from earlier video lectures in this phase space we have position on the x axis and we have velocity on the y axis and any point say for instance I go from this point towards this point, my initial point and my final point nature is going to decide on a certain path, nature will decide on a certain path in this phase space and this is time evolution, both are functions of time, so it's time evolution, nature will go from a certain position, now it's only y on the x axis but it's position anyway in space, I can jot that down and jot that down and that's usually what we're going to end up with this function not f we call it l and it's going to be a function in q and q dot and sometimes even t and we'll speak about that t a little later, so a function is going to be like that so I can still write the function of this equation but anyway along the line there's going to be a point where I'm going to have the q value in a q dot value, a q value in a q dot value and this is the equation that we are after, so we want to develop this equation instead of f I just call it l which stands for the Lagrangian so your wife there which will be a function initially we just going to make it a function of q and q dot which will be the same as a function in x and y I don't have two independent variables so it's written with 3 q q dot and t and that function will be a function of this graph called the world line in phase space so anytime t my particle is going to be on this line and I need to find this equation and the way that we do go about it is just to consider again this principle of extremum action least action that major decides that this is the path that the particle is going to tank I don't know what this is, now I could choose any other path for the particle to go from the initial let's say for instance the particle goes like this that's a path I chose that is another possible world line now it can cross itself, it can go to another galaxy and come back there are a lot of paths that can be taken what I want to do, what I want to notice is that from there to there should be the same time irrespective of what path the particle takes it starts at the same time and ends at the same time and in my experiment it starts at the point and it ends at the point so irrespective of the world lines that is taken the world lines are actually the actual one that we are interested in but we don't know what it is at that time and at that time beginning and end it's at the same position it's in the same position in space and time now if it moves in this direction at a certain time it might be there and at that same time it's got a long way to travel so say for instance at the same time it would be there so if I just look at that and I look at that difference here that will be delta q that will be the difference at the same time if it went on the path that it will take or this other path and what we want to know and why this is called the principle of extremum action is I want that to shing to nothing if all of these and remember there will also be a delta q dot big delta q dot here so there's this delta q and delta q dot difference between these two paths the real one and any alternate one and what I want to know is what I want to do if I shrink all of those differences too you can well imagine that this equation will shrink right up and go lie exactly on that now in order to get that we have this function called s that s is an action we call that an action and we say that equal to a definite integral and going from time initial to time final let's call it time 1 to time 2 of our function lq and q dot I'm just going to leave it there now with respect to time usually we'll have a t in there as well I'll say a few words on that so that is called the action and what you can see is on this equation for this world line that will just be the area under the curve I call that my action now that is this lq and lq in q dot I'm just going to leave it like that that is the actual equation for this graph but I can now have my alternate one so that will be q plus delta q and q dot plus delta q dot that will be the area under the curve for this alternate now this is a calculus of variation so if it doesn't make too much sense just take it on face value that will be the equation for this function of mine now even if I don't put t here I just want to say a few words about that I don't put t there as a third variable q is a function of t and q dot is a function of t those will both be equations because for q I can have an x y plot and that will be a graph on its own and I can parameterize that as a function of time so it becomes a line with the direction and same with velocity so those are both functions of time so implicitly my Lagrangian function my l function is a function of t and if I put comma t and that will explicitly be now when will we have this l, q, q dot and t that means there is some change in time now we are supposed to be talking about time invariance that there should be no difference in the laws of physics from one time period to another if there is change though something happened and we refer to that as a hidden degree of freedom if I push this board eraser if I push it like that there is some frictional force at the bottom so something is happening I'm just going to make it simple without going into deep explanation there is only two forms of energy potential energy and kinetic energy any other form and the friction that is in there now that is not a frictional force per se that doesn't really exist there are hidden degrees of freedom there there is interaction between the surfaces that touch each other so there is something happens and I have to take consideration of that and I would have had to build that into my very long equation my Lagrangian equation otherwise if I don't build that in there something different between the first and the second something was lost and we amalgamated in an easy physics if I can call it that I think there is some average friction of force it is actually just hidden degrees of freedom as far as time variance with respect specifically to T is concerned but for practical purposes here we are only going to consider the Lagrangian with respect to position and velocity so we have that now I want back to my phase space diagram here I want least extreme action usually a minimum and how can I do that I need for these delta q's and delta q dots to go to a minimum so what I really want is I want this delta s to equal zero so if I subtract the integral area under the curve of this one minus that one I want that to go to zero that is the extreme action or least action I want that to be a zero so what is that going to be well it is this integral going from t1 to t2 of the l of q plus delta q comma q dot plus delta q dot dt minus the integral going from t1 to t2 of the l of q and q dot with respect to time so that is what I am looking for I want this difference to be zero now it's not simple calculus or this calculus of variations which as I said I won't explain here but I just want these two at any time for these two to shrink and be on top of each other and that will be a calculus of variations and it will be an extreme if this goes to zero so what I am just writing here is zero equals delta s equals the integral going from t1 to t2 of delta l q dot dt so it is this one minus this one all the difference between these two is delta l it's the change in l because of a different pathway and I write here delta l and I want to refer you back to this delta if it was a function if an x and y and I have a function in q and q dot same thing and I can write it in that form so zero equals delta s equals the integral going from t1 to t2 of that so that is going to be delta l delta q plus delta l delta q dot delta q dot ok now we remember easy problems in physics in in the elementary calculus so I am just going to break this up t1 t2 and I am going to have delta l delta q dot delta l delta q dt plus the integral going from t1 to t2 of delta l delta q dot delta q dt I am going to clean the board and we will carry on with this the derivation so we continue with our derivation we start there now with these two integrals now on that side I think we can see that we can it's a product of two functions and we can use the product rule for this integral and what I can do is I make use of this stating which one is u which one is v prime let's make v prime this delta q dot delta q dot which means v equals delta q because q dot is the first derivative of q so that means this must be the l by l or del l del q dot which means u prime is the d dt d dt of del l del q dot so I can use the product rule I am going to have u v minus the integral of u prime v as far as that I am not going to go into the details of doing the product rule for integration so 0 equals delta s and that is going to be this integral going from t1 to t2 l del q delta q dt plus now I am going to have u times v u times v del l del q dot l l del q dot times delta q and I evaluate this at t1 and t2 minus the integral going from t1 to t2 of u prime v so that is these two so that is the dt of del l del q dot delta q dt now just think about this I am going to evaluate this at t1 and at t2 what I said in the beginning though is at the initial and at the final points there is no delta q there the two values are exactly the same here the two values are exactly the same there is no delta q there there is no delta q dot there so you can clearly see that this term is going to go to 0 that term is going to fall away evaluated it because there is going to be a 0 in the first one and a 0 in the second one so negative 0 is 0 so I am left with these two terms and I can just very quickly simplify these two and I can combine them together en I see I have a delta q is a common factor so I can idea there is going to be this minimum it is going to be the integral I am going from t1 to t2 of I am going to have a del l a del q a minus the dt of del l a del q dot in name both of these has a delta q in them delta q dt delta q dt now I have got to do this integral of this function this expression I should say and that has got to equal 0 now what is the only way that this integral can be 0 if I look at this term and I look at this term I don't want this term to be 0 because I've got my world line in any other possible world lines I want there to be delta qs otherwise we won't have this concept of the two possibilities en then going towards the real one so the only way that I can really have this integral being 0 is this term is that I want this to be an arbitrary any possibility as I say to another galaxy in that as far as the alternative one is concerned only one I have is this equaling 0 so del l del q minus the dt of del l del q dot has got to equal a 0 and that is where we have that is where we have the Lagrangian of a system setup in the next video I'll explain to you what we can do with this now I left you off with Lagrangian's equation del l del q minus the dt of del l del q dot should be 0 I could also just take this to the other side they take this to the other side those two has got to be equal to each other now of course people sat with this and just said what can this l be what can this function be so that if I take that function and I take its partial derivative with respect to position that that has got to equal the derivative with respect to time of the partial derivative with respect to velocity what can that be now there's a bit of explanation I can run through but I'll just give you the answer Lagrangian equals half in q dot squared minus the potential energy so what am I trying to say there it is kinetic energy minus potential energy as simple as that and if you plug that in you will see that this works so let's just have a look at that now q dot is velocity q dot is velocity and the potential energy does not contain velocity remember when we just looked at I have something here it's a meter down so for all the day it has potential energy and that potential energy was just mgh mass times reputational acceleration times the height that was its potential energy or I could have put it in negative there depending on how I set things up but see there velocity doesn't play doesn't play a part there it's only q height it's just the position it is so high up so that will be mg q so this is a function just of q and not q dot okay so let's do that let's take del l del l del q so just with respect to there's a q dot and not a q so all I'm going to be left with is the derivative of this so it was basically going to be minus v at q and that was going to be the d dt of that so what is the first derivative of energy you might remember this from calculus well that's just the force that is just the force the first derivative of velocity gives you a force what is del l del q dot what do we need let's del l del q del l del q dot sorry so what is that going to be well there's a q dot only in this so that's going to give me m q dot m q dot and what is the d dt what is the d dt I still have to get that of m q dot well that's m q double dot that's m a that's m a and that's equal to the force as well so force equals force on both sides and they better work out like that because if it didn't come to to be this then this initial this initial equation that I chose here kinetic energy minus potential energy wouldn't have worked out for me but beautifully there I have never used in any of these derivations but to get to this in this derivation have I ever used causality or determinism there's no force there's no effect I will use this thinking beautifully about nature chooses some kind of minimum and the minimum it chooses is that the actual world line that it takes I can construct any other possibility but nature will choose the line that's at some form of minimum so if I choose any alternative path and I subtract those two and delta q dots an extremum in this case a minimum I'm going to end up with this with this as long as I have the same initial position in the same final position in the same time and I get this which is Lagrange's equations I've set up all I need to now do if I have some problem that I have to solve I set up the Lagrange and off the system and I can do these equations and it'll apple pop for me an answer in other words I can still see what the acceleration will be I can still derive what force would have had to be applied for that but without using force and acceleration and all initially to start off with I can set up I'm only looking at q and q dot now once again I just want to reiterate I don't have t I don't have t in these functions I only have them in here with velocity I only have them implicitly not explicitly but I have q q dot and if I had t if the Lagrange changed over time and I didn't put that in in my original IMP and if I set up my Lagrange only with q and q dot and there was something that I did not take into consideration for instance these electromagnetic interactions between these atoms of the surfaces I'm going to get the incorrect answer there's not going to be the force and acceleration that I predicted through Newton's laws because there is this hidden degrees of freedom I did not take into consideration if I didn't write my equation this equation properly more in a more expensive form to take into consideration all these things obviously then my equations and my outcomes of experiment are going to be different because they are hidden degrees of freedom that I did not and that is where we don't say energy changes to frictional energy or it escapes in heat this is hidden degrees of freedom that I do not take into consideration in my Lagrange's equation in setting up my Lagrange I should say so the beautifully the Lagrange's equation with a little bit of mathematics skipped over a few parts as far as the calculus of variations was concerned but not too difficult so for the first time now you know how to what Lagrange's equation is and I have to set up the Lagrange and it's basic for very basic physics it is this potential kinetic energy minus potential energy and I have Lagrange's equation here so in the next lot of videos we'll go to the Hamiltonian of the system this is the Lagrange and we're going to set up the Hamiltonian give you a sneak preview just to wait the appetite that the Hamiltonian is going to be nothing other than the total energy it's going to be t plus v so it's the only type of energy we're considering two forms of energy that you cannot you cannot make or destroy this type of energy so the Hamiltonian is going to be t plus v and we'll see the two equations here we just have one with setting up the Hamiltonian we're going to get two equations