Physics+ Nonconservative Lagrangian: UNIZOR.COM - Physics+ 4 All - Lagrangian

Hi, I'm Zor. Welcome to Inor Education. I have decided to put a little bit more information into the grand chapter um especially for uh non-conservative systems. Before we were talking about only conservative systems like for example um some kind of a object is moving in the gravitational field. Well, gravitational field is potential field. It has a function which is called potential and the force depends on the position. But in some cases the force depends not only on position but also for example on speed that complicates the picture and in the form which we were using it before cannot actually be used. So today is this particular example of non-conservative system and how to build lrrenjan for this system. Okay. So this lecture is part of the course um physics plus it's a chapter called lranjan and the lecture is called non-conservative lanjan. Um now this course has prerequisites. Um before physics plus I had physics for teens and also I have math courses math for teens and math plus. So I presume that all these uh prerequisites are totally familiar with you because I will use some concepts which I was basically using introducing before. So this is just a continuation. Now the website unisur.com is totally free. There are no advertisement. Um you don't have to sign in, no subscription, totally free thing, pure knowledge for your consumption. Um every lecture has written part like notes and they are written usually in a style of textbook. So you can um watch the lecture uh video, you can read the corresponding text which is on the same web page on unisord.com and that would help you to understand material much better obviously. So you have a textbook and you have a video presentation of the same material basically on the same web page. Um now there are some exams not not in this particular course but in previous courses there are exams which I um which you can take as many times as you want just to selfch check yourself. Um there are some provisions for supervised education uh on the website again totally free. All right so let's get back to business. Um now as I was saying our uh lr John we were familiar with which typically was like l is equal to t minus u uh lran john um kinetic energy and u potential energy. Now kinetic energy was more or less like 12 m² or maybe some other quadratic form. Um the potential energy was just a function of position. So if that is true then well the gjan of this type more or less fits. But now let's consider a completely different problem. Let's say you have a surface of water and you drop a stone. Now the stone goes down the water. Why? Well, there are two different forces. The first force is obviously its weight. Now let's say we have x um axis uh directed downwards. So there is a force which is its weight and another force is the force of resistance of water. Now the water is such a um media that um the resistance actually is not constant. It depends on the speed. it's actually proportional to speed more or less within certain um kind of precision things. So I assume that the force which is actually u directed against the motion is proportional to uh to its speed. Okay. So there are two forces and first let's do what u classical Newtonian mechanics is doing. In this case we have two forces in this case they're uh one-dimensional. So introducing lranjan will not actually do a lot of new things and uh will not be very useful. Granjan is good in the complex systems. This is a simple one but it's just an example just to show how lenjan would be working in non-conservative case. So simple case we have two forces along the same line x axis this is uh the positive direction this is negative direction. So the um um uh the Newton's equation the Newton's equation would be what? That the force is equal to acceleration right mass time acceleration. So mass time second derivative of time x is coordinate. So this is m and this is acceleration a. Now this is force basically uh mass times acceleration. And what is the force? The force is mg minus k first derivative. Speed is first derivative of position. So this is my second Newton's law. And uh it's actually rather simple differential equation and I'm going to basically not not to solve it. I'll just give you the result because solution is completely outside of the scope of this lecture. It goes to differential equation and it's easy. So for those people who are familiar with differential equation, they can actually try to solve it. And the first step would be uh instead of u first derivative you can put just another function y which is the first derivative. So now you have the derivative only to the first degree right. So my is equal to mg minus k y. So this is differential equation of the first order uh relatively uh easy to solve and I'm not going to do it and I'm going to write immediately the result of this just for you to know that this differential equation has an interesting result and what is the result of t is equal to mg this is G gravitational constant divided by K T minus M² G / K² if K T / MUS well it's rather complex formula but look >> [clears throat] >> It satisfies u basically this equation and it satisfies x sub0 is equal to zero and uh speed at the time is equal to zero. My position is on the surface and there is no speed. So I just put it there and let it go. So this is the equation which basically corresponds to this particular uh condition. Oh I think it's plus here not minus. Yeah. So it's plus. Now if t is equal to zero this is equal to zero and this is zero. So it's zero. And then I can put the expressions for the first and the second derivative. MG divided by K 1 - E minus KT / M. And the second derivative acceleration is equal to G * E minus KT / M. Okay. So I'm just giving it to you as pure math thing. I'm not going to spend any time on uh actually solving this or taking the derivative. It's all basically kind of elementary stuff. I'm not going to uh spend any time on it. But what we actually can do from it, we do some kind of a logical conclusion. Look, acceleration is always positive. Although with t going to infinity this thing goes to zero. So my acceleration while positive still going to zero like reverse exponential thing my acceleration graph of acceleration would be like this. This is one at t is equal to zero. Okay. Now, which means if acceleration is positive, although it just goes to zero, but it's positive, which means speed is increasing. Positive acceleration. So, speed is increasing. So, this is an increasing function. However, since acceleration goes to zero, my increase of the speed is also going down. And uh the whole thing actually as t goes to infinity so this is zero it goes to maximums basically it's a maximum speed the limit speed limit is mg divided by k this is 1 minus so this is zero so it will be only mg divided by k so this is a limit speed eventually as the stone goes down in the water it goes faster and faster and faster however Ever it's not an infinite increase of the speed. The speed will increase up to this limit not reaching this limit. So it will be faster and faster but still and eventually it will be from from the observe uh from the observer after certain amount of time when this thing is really substantially zero it will look like the speed is constant. So it will constant with a constant speed going down uh in water. So this is my basically research based on equation and its solution and uh an analysis of the speed and acceleration. Fine. This is Newtonian thing. It's not very difficult at all. So there is no need like inventing John etc. But I'm using this as an example of um using lranjian mechanics to get some formula for lranjan and oiler lrange equation in this particular case which will be definitely different from conservative system. Why? Because because of this force which is which depends on u on the speed. All right. So let's go to the current job. So this is my Newtonian second law. Now if lranjan exists for this system then the oiler lrange equation which is this now first we go by time right of partial derivative x x prime t by d x prime = to d l x prime d x. So this is my oiler lrange equation. So if Ljan to this system exists then solution of this equation and this equation I mean L is some kind of a formula which depends on X X prime and T then solution would be X of T some kind of a function should be the same as solution to this thing. So these two equations must be equivalent. Now how can I come up with this particular function L expression which depends on X X prime and T. How can I come up with this in such a way that this equation is equivalent to this equation? Well from the first glance it doesn't seem to be easy. But um some smart people actually um came up with a very interesting approach to this problem. Let's take this equation and try to modify it in such a way that it will look like this equation where else some kind of a function. So that's what I'm going to do. I'm following um those smart people who have already done it for me. So first of all we will um modify this equation. We will divide it by m that would be easy. And instead of K I will put K / M and I will even do it better. I will put this to the left. So it will be plus and G would be to the right. So it will be plus this equals G. So this is the same Newtonian second law. I'm just rearranging a little bit. So I will use this as my starting point and I will try to change it in such a way that it will look like this one. Okay. So what actually was uh kind of a ingenious um guess was let's multiply both sides of this equation by e to the kt / m. So what I will have is this. I will have this um I will not use uh t it's assumed basically plus this equals this right just multiply by this multiplier which is never equals to zero by the way. So I did not really introduce anything new. It's equivalent um it's equivalent equation equivalent to this one. Whatever is solution to this is solution to this and vice vice versa. Now look at this expression. I'm just saying that I can recognize this as the derivative by time of this function. Why? That's the um product of two functions of t. This is a function of t. This is a function of t. So it's this one times derivative of this which is this plus this time derivative of this which is e to the power of kt / m times the exponential coefficient at expon at at exponent it would be k / m. So this is the left part. Okay. Okay, [clears throat] I need this to be equal to this, right? Well, that's easy because this is function of t and this is partial derivative only by x prime by the speed, right? So, I can actually have dt d by dx prime of what? Now this is function of t I can put it as is and for having okay so partial derivative by x1 x prime now this is not related to x prime so it's like a multiplier where partial derivative so it's only depends on x prime and partial and derivative of 12 of x prime squared is obviously two uh cancel with two and x prime. So this is what it is. So this is my left part and it looks very much like this. Now what's on the right part? Well, on the right part I have to have a partial derivative by x. But this is doesn't depend on x. So basically if I will I I will have it as a partial derivative by x of e ^ um kt / m g * x partial derivative of x would be this right now I can say that these are equal this is equal right so I'm just converting converting this thing into slightly different format but now it looks much better much closer the only problem is this is the same function L and L here I have two different functions however what's very important this is partial derivative by X prime and this thing has no x prime so I can always add this to this without changing anything. So if instead of this I will put this plus this derivative by x prime will not change because this thing is not dependent on x prime. So I can always say that this is d by dt d by dx prime e to the power kt / m x prime² um uh plus e to the kt t / m gx that's on the left side. Now on the right side I have partial derivative by x. This thing doesn't really depend on x. So I can add this to this and that would be d by dx of again of the same thing which means this same thing this one e to the power kt / m is the same on both sides. So I can have this as 12 x prime squar plus mgx. It can be my l of t of x x prime and t. So with this L this particular oiler lrange equation is exactly equivalent to this one because I was just converting with uh with absolutely equivalent transformations. Uh no no this is not m this is just gx I'm sorry m is out. All right. So this is my function but I would like actually to be a little bit more physical so to speak. Now if this is function if I will multiply by constant both sides it will be the same thing. So instead of this I will put this 12 m and m here. This is my lanian. Why did they do this? Because this looks like a t. This looks like minus u minus because x is going down. So as x is increasing, potential energy is decreasing. Right? So what is my final formula? used to be remember L is equal to T minus U for conservative system and for this particular system it's E to the power KT / M * T minus U looks like it but there is one multiplier which depends on on on time. So that's what's very important. So we have come up with a lanjan which is kind of resembling in some way the non-conservative system but be uh the conservative system but because this is a non-conservative and time dependent this time dependency this particular case this particular problem is reflected in this multiplier which is explicitly depends on on time and since it's explicitly depends on time there is no such thing as um conserv conservation of energy and there are some nice you see many very nice features of lrrenjan and lrrenjan mechanics are actually for conservative systems non-conservative systems present problems now in this particular case we have come up with certain solution which results in a very interesting kind of a function which I I I I like that it resembles the conservative system with just one multiplier In the more sophisticated cases, it's probably much more complex. We are not going to go. However, this is an example of using plunjan mechanics for a non-conservative system. So, it is possible although it loses its um many of its nice features like uh conservation laws and and and some others. So lanjan mechanic is nice for conservative systems and in many cases it gives you a lot of advantages like for instance generalized coordinates. This is not generalized coordinates and again in some more sophisticated case uh it will not be possible to do like a relatively simple conversion from a Newtonian torren mechanics. So that's that's what it is. I just wanted to give you this particular example. I do suggest you reading the uh the notes for this lecture. Uh they are a little bit more detailed but not by much. But still what's very important is approach approach is take the Newtonian second law and convert it into format which is looks like lranjan. That's what actually was done kind of artificial. Uh I agree but well whatever I mean it works. All right. So thanks very much and uh good luck.