Lecture 1: Introduction/Overview

追加: 2026-05-12

[00:00:01]Hello everybody, welcome to ordinary differential equations.

[00:00:06]Uh first of all to introduce myself I'm Hassalakarnata. Dr. Cardinal Ratna and uh due to Sri Lankan long Sri Lankan or lengthy Sri Lankan names um you can find me in two different uh I mean under two different names either hassar that is my Sri Lankan Sri Lankan recognized name and then Hassalu or Haluag that is exactly what you I mean where you can find how you can find me uh if you look it up on the web like if you Google me or uh at this point like I'm um at Roger Williams University and I'm an associate professor of mathematics there and uh uh there also like uh I'm recognized by the name Hustal Galu. So uh either way uh we can have uh that part of the introduction uh simple because we all know uh Sri Lankan names I mean usually are lengthy and uh some of us have to suffer through this lengthy name name process appeared in uh couple of different ways in uh different places. All right so let's get to our work.

[00:01:22]Uh first of all I'm very excited to be here.

[00:01:27]Uh first of all I would like to thank uh Dr. Prabhad Silva for giving me this opportunity to uh I mean to um be a part of this uh excellent work and um especially uh I would like to uh thank again like uh for the choice of the text also like because I mean this was not a course um on a different on differential equations like it is not just the topic that we that he came up with like when we had this conversation he said like uh uh let's have ordinary differential equations by using Arnold's book so let me say something about this because I mean this whole story or the the entire conversation that we're going to have will be uh depending on this this context Next. So, first of all, uh let's look at uh these two. I mean, let's let's have this housekeeping stuff like uh easy so that we can recognize like exactly what we're going to do and how uh we are going to do in terms of uh these textbooks.

[00:02:47]First of all, the primary textbook is going to be ordinary differential equations by vi Arnold, right? Um and the primary textics is edition three.

[00:02:59]And the interesting part is like uh I do have a secondary textbook for this conversation and that is none other than just the edition one of the same book.

[00:03:09]This is kind of like I mean this is the first time that I I uh I'm having this kind of experience too, right? Because I usually when we have uh a text book and a supplemental textbook it's going to be like completely something's going to supplement. There should be a significant difference between one and another but in this case like it is just the same author same title and two different editions. What's the big deal?

[00:03:41]And that is exactly the point.

[00:03:44]And to recognize this, first of all, I would like to invite you to uh have a quick Google search.

[00:03:54]Uh first of all, you can uh search you can Google VI Arnold.

[00:03:59]I'm not trying to uh just get one personality out of this context and say like, oh, we're going to craft this entire course around the person. No, it is not that, right? So we should know exactly like the the background a little bit uh that background can provide a lot of information.

[00:04:21]So if you go ahead and let let me show those things like u I may have those um Google searches search tabs with me.

[00:04:35]So let me stop sharing this for a moment and All right.

[00:05:08]So now you can see my screen where I I looked up and the first two tabs are exactly about uh the Zoom session that I'm recording this on. Um and let me tell you like uh one more thing now in this session like this is pre-recorded as an introduction and in the future like I'm hoping I'm willing to have some conversation like uh with some interaction if possible like if the timing works out if not that's I mean yet we can work it out in this way. So uh the first step right if you check who vanol is like he's uh uh he's he was a mathematician and um and um uh one of the pillars in dynamical systems and ordinary differential equations. So these two directions like dynamical systems and ordinary differential equations the the there are like two different areas. One I mean like both of them are having applicationoriented um front and at the same time highly theoretical end. So Arnold had that work mostly in theoretical land like uh so you can just uh check like one way of uh checking uh the profile is like to go to Wikipedia. I mean you can do that and uh one interesting one very interesting fact that I find is if you go ahead and click on like this is something that you should get to know if you get to know about like this this web page or like u the mathematical geonology project that will provide I mean that provides the heritage like where each academic like whoever's is having a PhD like where they are coming from like uh who's the advisor who's the advisor of the advisor for instance like if you go go to this and if you click on this like this is the advisor of V Arnold right so if you click on I mean this is a well-known character in um uh Paul Magnar is a well-known character in dynamical systems and u like many many different directions in even in analysis So here what I want to point out is this.

[00:07:41]Uh Vanold produced 46 PhD students and 244 descendants.

[00:07:51]That means like students of students and students of students of students. Uh that is what you can find here. So it is like it is not only just an isolated uh um presentation. So um Arnold is a wellestablished uh personality in that um area of study. So you can find more information. I'm not going to say much.

[00:08:20]So why do I need to emphasize that to get to this story?

[00:08:28]Because uh now if you check now I do not know whether you have this book with you already. Um if you do not have it like um I recommend you to find it and there is an easy way of doing that. I mean I do not know the legal grounds of that though. Um here we have like if you check ordinary differential equations edition one Arnold PDF right simply the PDF if you search that there is a free I mean there is an addition I mean there's a PDF available for edition one the only uh drawback in this case is if you open it up and if you go ahead you will see the readability it is not like really well readable for some reason it is not well scanned.

[00:09:24]So in anyways it is there you can uh find it and I do have edition one of this book with me actually I bought it from Sri Lanka uh when I was like there's a uh book um shop called Expographics I don't know whether it is uh there or not like many things could be different so uh I bought it from hexographics I guess like uh around 2009 or something right it was edition one now that is exactly the edition that I've seen and then in preparation to this course when I check for a newer edition I found another one of course there was an addition two and then it moved to edition number three now if you check uh if you Google the same thing ordinary differential equations edition three on PDF uh the second one. All right, you can check this uh title, right? When you click on that, it's it is giving you the third edition of uh this book.

[00:10:40]Now, this is the interesting thing.

[00:10:45]These two editions are very different from each other. I mean the first edition of this book it was just the raw presentation where you can find like exactly uh the theoretical framework and then starting from that it's going to it's not going to talk about much about like anything related to solving differential equations nothing right so this kind of a book now if you try to find this book and if you try to buy uh the first edition and if you do not find it I'm not surprised because not many people would buy that book except for whoever's uh whoever is willing to have that uh deep con conceptual understanding and who's going to have that sentimental value of the book. Uh so that is one issue uh not an issue we know how the business would work like uh when it comes to textbooks right so that is why if you do not find it frequently that may be the reason and that is one of the things that that's being addressed that had been addressed in edition 3 they introduce something some solution type at least like the the way that it's going to work. But first of all, when I saw like since I saw the first edition at the beginning and then I saw the second edition, the third edition, I felt like, oh no, because the first edition has that abstract value in it. And the third edition, the nice thing about it is it has the same value, but it has something more. So when we read the third edition now nice thing about this PDF that's available right in front of us is this is very readable and I highly feel that this these available copies I mean these are well recognized uh and the legal ground should be fine. Uh the reason is like most of these books like if you try to find them I mean even when I try to find it to purchase even it is I mean they're not frequently available some of them like uh I need to buy a used copy I mean I I do not see like u ongoing printing of some of these books.

[00:13:21]So uh this is what I'm going to say. I'm saying too many things about the books but let me tell you the reason the reason is the addition one has the framework that framework is simple that framework is simple but it could have been explained a little further and that little further explanation happened in addition three. I did not find uh I did not read edition two in between. So if you find it you can work you can take a look at that too but I'm not going to do that.

[00:14:00]In addition three, I find more elaborative presentation of that theoretical framework. At the same time, it provides like little bit of that concrete layer of solving differential equations. But I'm not a huge fan of that part of the presentation because I would like to maintain that abstract framework in this conversation because that is the beauty of this approach of the approach that Anol provided in his first edition.

[00:14:35]So that is what we are going to do. So this is why I mention I gave one comment right here in the lecture.

[00:14:46]Let me go back to my writing interface.

[00:14:55]I wish I had a faster way of moving back and forth with with this sharing.

[00:15:04]Unfortunately, it is just the same old way that uh works. All right. So now look at this here. I mentioned like these two books and then I gave a particular comment uh do not read either of these texts until we have a conversation. This is this was the conversation that I mean the conversation is exactly what we had what we mentioned a couple of minutes ago because if you try to read addition one you may feel it is abstract and you may feel at a certain point like if it was explained a little bit and if you read uh addition three uh sorry if you read edition three actually you will find more elaborative um version of it. However, it has like other like concrete considerations. But when you consider addition one on the other hand, it has the abstract form, right? So I would like to look at the framework from edition one and try to map that exactly to edition three to work with that abstract version of this conversation.

[00:16:29]So that is the point. So when you read edition three, let's try to do it in this way. I will point out exactly where I would be working at, which parts I'm looking at, which parts we are going to have conversations on, which parts we going to just hand wave and say like, oh, you can in fact read and understand it, right? So, we can have that and that's the plan. So we will have a very close look at it and at the same time when we work with this um presentation I mean when we are going to uh when you're going to have this conversation I I'm planning to do this uh by directly following some of the sections I will follow even sometimes I will accept some of the content some of uh the parts from the book and then I will elaborate and I will work with some of the problems some uh expansions in certain cases and like that right so a con concise course plan let's try to make it like simple of course this is very dense we cannot have like in four four or five different points like what we're going to do in this uh entire conversation it is like too short So we will expand each component and even like when we consider like let's say oh we have an introduction we start it here right this lecture or this conversation is exactly the starting point but it is not going to be uh over just after this lecture we are going to continue that conversation because I mean there are certain components that we need to We need to um let me see.

[00:18:41]Oops. Right. So, we need to uh maintain in terms of each uh component. Right.

[00:18:48]So, um let me give you a quick idea about like these sections also just an overview.

[00:18:55]First of all, we will have an overview an introduction where we are going to see what what we going to do pretty much what we going to do and how we are going to do and then in the second part we will learn phase spaces phase flows vector fields and direction fields like we're going to have that have the layout for this and uh in a moment I'm going to talk to you more about this approach. So you will see I'm not talking about differential equations like in this part one nor part two and then when it comes to part three I'm going to talk about uh basic theorems where we have like when when we say when we talk about differential equations we have a bunch of um theorems where we we have where we establish exist distance, uniqueness, continuous dependence on initial conditions and then extensions of the solutions which means like when we find a solution how can you find the maximal interval where that solution can be defined on.

[00:20:14]In certain cases the solutions may not go all the way up to infinity in terms of the time variable because there could be a finite scale. the solution has a singularity just explodes.

[00:20:30]So in either case in each case we need to uh have this um conditions established.

[00:20:42]Now in a conventional differential equations course there's a frequently asked question like why we are doing this existence uniqueness so I mean when we have a differential equation if we can find a solution what's the big deal like would this even matter now I will show you exactly the importance of this we will talk about like first of all we will talk about this results and then we will talk about the proofs of these results. Right? So when it comes to the proofs of these results, what we are doing is basically analysis.

[00:21:25]So I will show you the the entire conversation of ordinary differential equations that we are going to have in this is an extension or a branch of real analysis.

[00:21:42]I will show you that approach because one thing that you can do is like uh uh if you have some computational background I mean this is the whole point of like hiding this computational component right at the beginning if you have a computation I'm some of you may already have that background so you know how to compute a differential equation I mean a solution of a given differential equation for a simple is and you may feel like oh what's the big deal this is like existence I mean the solution right there look at what I've written down that's the solution right you can say that but on the other hand one can I mean if one thinks about mathematics as a um let's say um in a process of reasoning So just like um a foundation of analysis like you're constructing the foundation.

[00:22:48]If that is the case, one can say like, "Oh, I do have existence and uniqueness." And then you're going to look at a very simple differential equation like let's say that you might have seen in many cases like let's say exponential growth dy dx equals dydt equals just y right so we have seen that kind of a scenario and then uh even if you have seen it in application somewhere else right so if you consider that kind of a differential equation and now if you have existence and you weakness I mean existence of a solution. Now you can name let's pretend that we never seen exponential function before.

[00:23:32]Now you're going to have a point where you can introduce these functions into analysis in a formal way through this uh approach. So once you guarantee existence for instance like let's say let me give you a brief idea now you use s and cosine functions frequently if you go back and check where you have seen s and cosine functions right at the beginning uh you have seen them in like let's say in advanced level somewhere at a certain point like or even before that right so in seventh eighth grade somewhere you have seen in like let's say a triangle and then with a triangle you you know the opposite divided by the hypotenuse and it's going to be uh sign you wrote it down and then in advanced level time you extended that to the unit circle and say like oh that is like the same triangle it's just extended now let me ask you when you learned analysis you even treated like continuity limits very carefully with epsilon delta definitions So you did not assume much right? You did not even assume uh uh the construction of I mean you did not assume real numbers even there's a construction there's a piano I mean there's a approach called piano uh um construction to the real numbers and then there's a koshi approach like where are you consider like in terms of completions there are different versions of like even constructing the real number line where you stand on Right?

[00:25:19]You start from there and then you try to uh formalize exactly the real number line. Now think about this. If that is the case, how can we have that? How can we have that kind of like a sloppy uh definition for a trigonometric function like triangle and like opposite divided by hypotenuse? Is it going to I mean who's who has the definition for opposite or hypotenuse? That whole thing is depending on a triangle.

[00:25:47]So now uh without doing that here right. So if we prove or if we establish this existence uniqueness uh to a very simple system a second order system uh we will be able to come up with two functions which would mimic exactly the properties of s and cosine. And the nice thing is they do have that origin already. The existence is guaranteed by the existence uniqueness. In other words, what I'm saying is we are going to give these theorems existence, uniqueness, continuous dependence, uh the extensions and rectification all of these like all of these theorems their corresponding place. Those are not like some background noise that you're going to have in a differential equations course. It is it should be exactly the focal focal point of this entire conversation.

[00:26:57]The whole thing is revolving around this like you are going to have that analytical approach to define or define your entire pathway to talk about solutions of differential equations but also at the same time existence of some of the functions you can do that. So there are different entryways into these functions. I mean I'm not saying like this is the unique way of doing that because one can define exponentials or like uh s cosine even by using power series approach.

[00:27:33]That is one way from analysis. But there are certain components where you might need to struggle to establish like simple stuff right sin square + cosine square equals 1.

[00:27:47]Think about writing s and cosine in terms of like two power series and squaring them, adding them up. How it's going to work with like all the cross terms. There could be some some stuff and there could be different approaches where you can see it in a better way. So that is the beauty that we going to see in this conversation. So sometimes I mean not having that not having the example right at the beginning is the way to have the value to have that example later.

[00:28:27]So that is exactly what we're trying to do.

[00:28:31]So why am I elaborating all of these pieces like all of these uh components on day zero?

[00:28:41]The reason is this. Now when we talk about this abstract conversation I mean when we are going to have this abstract conversation there there is a certain point one might feel tired a little bit like because theorems proofs theorem proves is it going to going to be the way that we're going to do this uh we need to find a purpose of doing that right we need to find why we are doing that I mean that is exactly the beauty that you can find in this approach approach that combines both that theoretical background and at the same time the geometric intuition coming from uh Arnold's approach and then when it comes to the next part like where you can see like linear systems right so linear systems uh are going to be like exactly uh the higher order uh higher degree um differential equations com combined I mean like um diluted down to first order systems and uh then we can talk about those systems and we can do many things by considering those systems and sometimes you can talk about like different parameters or different variables interacting with each other uh to the uh evolution rates right so how can you describe that the only way of course when we have a system we are going to have linear algebra right so we are going to do we are going to have that approach and um so that's where we are going to have that combination of linear algebra together with um story of ordinary differential equations and then at the same time don't forget like even when it comes to existence and uniqueness we are going to have that entire conversation not only to a single differential equation we are going to establish that for uh for a system. So I'm not going to say that all of these uh parts are going in a linear order because at a certain point we might need to go back and forth a little bit.

[00:30:55]The point is uh when I prepare these lectures um especially since I I mean this this is a course that I'm going to uh I mean I'm offering by using this book for the first time which is a very very novel thing I mean which is a very cool thing but at the same time uh we are going to work it together. So sometimes if I feel like oh I should have mentioned something even like with further elaboration I'm going to revisit that topic and we will have it like the point is to have this entire conversation as a complete as much as possible a complete entryway to differential equations and at the same time uh later when you use it somewhere with uh some differential equations in terms of like application orientation, you're going to still have the the theoretical part running inside you. Or on the other hand, if you consider this as an entryway to have like let's say a combination or supplement to the the analysis component, that's exactly going to be uh the most valuable way of looking at it. And okay I forgot there is a very important third entry point.

[00:32:19]The third one is there is a different area in mathematics called dynamical systems or dynamical systems or dynamical systems and uh actually my work I I I do some research on dynamical systems and u control theory. control theory is a sub subcategory subp part of um dynamical systems. So we are pretty much like one can just simply say like oh it's just the math behind the control system just robotic systems where you have like let's say adaptive cruise control in a car how do you have that kind of a control system defined uh I mean you can just simply have it it's an engineering story but behind that engineering story there's like heavy mathematical component going around so that's exactly the part that I'm working on so if If you try to think about like at a certain point an entryway, I mean control theory is very specific.

[00:33:26]But when you consider like the the dynamical systems uh the uh realm of dynamical systems that has a wider scope and what Arnold does in this approach is bringing the elements of dynamical systems to start with the abstract version of differential equations. So in other words it is just like this.

[00:33:54]All right, one last thing to go from this uh lengthy, boring uh introduction.

[00:34:01]I uh apologize because I mean it seems like I've been talking too much about like what it's going to be, but I guess it's worth.

[00:34:14]So um what Arnol does in uh this scenario is to bring the dynamical system approach into uh this all conversation in a way that you're trying to prepare the dance flow.

[00:34:36]You're not trying to define dance moves.

[00:34:40]Right? If you learn differential equations like just an application orientation like to solve this differential equation to solve this solve that differential equation it is just like you one step at a time you try to learn a dancing move right but forget forget about that for a moment. Where are you going to dance if you do not have a flow? So this is about preparing that dance dance dance dance floor and if you do that with a very good intuition eventually you might not end up being a dancer. Instead you will be able to become um a dance who's it called like a manager who can who knows exactly how the dance floor works and he knows he or she knows like exactly how um it's going to work right so you can you can be just behind I mean not not behind like you can be in front like far ahead of just like uh defining these dance moves.

[00:35:47]So that is kind of like uh the approach that we are going to have. So that is what we are having at the end like OD is in on manifolds like that's like what are manifolds like on differentiable manifolds we we will talk about that uh with less amount as much as possible like not to have like two overwhelming unknowns into the picture. we will try to just uh make our path the way uh from point A to point B right as much as possible to get the best we around. So that is a plan right and of course I cannot do it without uh your interaction I mean because you will provide me the feedback and we are going to do that together and then in terms of like uh the the housekeeping stuff I might have covered some of these already the approach uh and of course we know like we're going to talk about this in terms of like we're going to have these conversations and then uh along the way I will provide like some assignments ments interesting problems some some of them could be even concrete but I will not provide some concrete examples with like computational aspect like it's not going to give us u that taste if I ask you to compute something I'm going to I'm going to expect some some intuition behind it otherwise like I mean the point is you will get this computational background in many other places and even if not like if you just read a book you will get you will be able to get that um I'm pretty sure some of you may have this background with differential equations maybe a little bit like in terms of like solving differential equations so what I'm going to do is like uh not to have that right as I mentioned before and then um uh what else right so What else? Uh exactly this is a good question because this is exactly the parts that we have not seen uh like the the issues we may figure out like what to do uh how to present certain components. I mean like let let's say like the unknown it's the unknown until we figure it out. So let's have that and the main goal is to retain this uh taste of abstract approach to this entire conversation.

[00:38:29]All right. So uh that that is the that is the important part. Let's keep the main goal in our mind. So at a certain point if we get a little tired of uh what we're doing that's exactly when we should consider to revisit and see what we're doing and that's going to give us again an energy boost so that we can start with a fresh mindset. So with this I am going to invite you to this uh conversation. Um and I'm very excited to have this conversation for next uh couple of weeks or months like in that way like we we are going to have this conversation as much as possible because I'm going to enjoy it in my end because before each and every lecture I'm going to do some work uh I'm going to uh think about like how should I present I mean even though I'm working on dynamical systems this approach to differential equations. I mean this is a course that we don't usually have in this approach. So that is why it's going to be a fresh experience for me too. Now let's talk about a quick motivation. All right. So uh take a quick I mean like let's say a deep breath that old story of the introduction is now over.

[00:40:01]Now let's talk about uh all right what we are going to do I mean I briefly mentioned this also already but let me let me tell you again let me give you the motivation in a totally relevant way which means like let's say usually in a differential equation course we are going to find solutions right uh for a given differential equation we are trying to produce uh analytical solution or analytic solution or uh a numerical solution something right so you have seen this already in different uh levels so without talking about differential equations let's talk about this I mean you if I ask you to solve this is kind of like a joke.

[00:41:01]We can all laugh, right? Of course, this is correct, right? Everybody knows what's happening here, right? So uh yeah I can ask this question or I can ask like oh let's do something further a cubic equation or I can extend this to uh the the fourth order.

[00:41:33]I just make it up. I mean some of them could be solved easily, some of them could not. I mean like I didn't even check. I didn't even bother like thinking about solutions of these equations.

[00:41:47]But look at this the story of like these are not differential equations. These are algebraic equations right? So the story behind algebra if you think about that for a moment was to work with these equations. If you if you go ahead and check like the history of algebra the way that it worked was like to I mean challenging uh the renowned mathematicians one another like saying oh can you solve this kind of like a cubic equation can you do you have a formula for that? I'm going to challenge you with this equation. Can you provide me the solution?

[00:42:30]That was the story. But then people kept thinking, okay, can we do that for any order? Right? The fifth order, fifth degree, I mean, this is quadratic, cubic, and um um right. Uh so all of these orders, right? For each one you can find like there are formulas.

[00:42:54]One can provide a formula for the quadratic. Everybody knows cubic there is a formula for the fourth degree.

[00:43:01]There's a formula which is extremely complicated. Nobody is going to even have them in their mind. Right? So the third degree and the fourth degree we we are not going to do that. We're not going to memorize them. We do not have that kind of a capacity in our brains.

[00:43:18]Uh some may I definitely most definitely I do not. So if you think about the fifth degree right one can say like oh I'm going to try creating this formula but if you know the story what I'm going to talk about a little bit like as a an irrelevant entry point starting from the fifth degree onwards there is no formula gen no general formula that you can provide. So this is uh this method uh the solution by formulas is basically like you're trying to take the roots right you're trying to do the radical right solution by radicals is not a possibility from degree five onwards if you know what I'm talking about this is exactly what you have seen or what you may you you're going to see in your uh abstract algebra sequence when it comes to galwa theory right so this this genius way of thinking about this story right with the contributions of ael and uh um de galwa right so they did not try to answer this one equation at a time no they try to see the abstract playground behind this which opened up uh many other subdirections like group theory, ring theory and field theory and then the whole section of like galwa theory like the whole entire story is going in that abstract layer abstract playground right so that abstract playground is exactly where They're going to say like starting from the fifth degree there is no hope of having a formula. You can individually work with some possible equations polomial equations for higher degrees but it's not going to work for all cases. Now why did I talk about this completely irrelevant irrelevant uh approach to get to this story? Because what we do in differential equations in most cases when we introduce them by like let's say like this solve a differential equation. Let me write down a differential equation.

[00:46:10]So prime we know we all know what prime means right in the previous case when we considered those equations we know like we are trying to find a solution for that x right a specific value for x in this case all right what is your x what are you going to find one can say oh it's a solution oh what do you mean by a solution So if you give it a little bit of a thought one can say oh it's a function it's going to be a function that would balance this equation. All right if it is a function where does that leave right? What is the domain? Can you say something about it? Right? So there are many formalities that you need to have behind this equation and yet we can just hand wave and say like oh we are going to solve this equation. So we can extend this idea to like let's say you can again say uh write down a different different equation right increase the order uh bring any fancy terms into this like this or that right so solving these equations it's just the same story if we handwave and if we say like uh if we just mute this technicality It is what we going to do is like to blindly find solutions some functions that would balance this equation.

[00:47:43]At a certain point that might work at a certain point it might not.

[00:47:48]So when it doesn't work right so the the point is this this is just like dancing right so dancing without having a flow.

[00:47:58]Now if you prepare the abstract layout in this case, what is the abstraction?

[00:48:05]What is behind? Where are these all entities living in? If we can, if we do have that conversation at the beginning, then it is going to provide us more not just the solution. Even if we do not have a solution, it might provide us a better intuition why it does not what is happening here. Right?

[00:48:28]That is exactly what we are going to do in this right. So this abstract layout is exactly this is the dance flow that is your dance flow where this is just the dancing without having like uh any target right so you can just you can dance right I mean if one gets high with some alcohol the I mean they they naturally define dance moves right so Just like that. But it could be uh it can be done by many. But preparing that dance flow should be done carefully. You cannot do it uh while you're high in alcohol, right? So you need to do it with some care. So that's exactly what we're trying to build here.

[00:49:26]And of course that's that's not going to be just abstract.

[00:49:29]later on that abstraction can be converted to and application oriented I mean applications as well because it's heading towards dynamical systems where you can have like a whole bunch of applications in front of you so the f focus is going to be abstract however it's going to it's it's going to be fun that's what I wanted to say in simple words all Right.

[00:50:00]So, um that is exactly the introduction, the beginning of the story.

[00:50:07]Um I'm just thinking should we just um think about like uh let's say the approach the starting step I guess like we can um in fact now let's let's have a little bit of yes little introduction or like how we are going to do this. Now let me tell you how I prepared this part of the lesson.

[00:50:39]To prepare this part of the lesson, I just uh I went ahead with uh the first edition of Arnold's book where it's providing I mean like where it's provided this matter is provided just like the in the abstract form with no much information around. So I started in that and then of course we may need more information. So for this basic setup we are going to have like two different levels of the conversation.

[00:51:14]First of all let's try to do it. Let's try to identify what we're trying to do and let's try to do it in any ways like no perfection guaranteed. Let's try to do and then once you go through that process later uh when we revisit the same topic we are trying to have the formalities we trying to add it like the precise structure. So to begin with this entire story right the basic setup these are what are the essentials uh essentials that we need to have we need to have uh a set right uh now according to even according to Arnold at the very beginning the way that this is mentioned is like let m be an arbitrary My handwriting is horrible arbitrary arbitrary set that is what it is mentioned right so uh it says like uh at least Arnold mentions like it's an arbitrary set now this is an alarming point everybody so when we say arbitrary there's something sloppy be going behind that right so there is I mean you cannot say just arbitrary because especially when it's a set it could be a set of cats and dogs right it it cannot work in that way so there should be some structure so eventually we need to introduce the structure but at least like uh with little contrast I'm going to have little more addition uh saying let's call it a non-MPT at least so if it is an empty set there are no elements. So whatever the dynamics that you're going to define on this set there there is no dynamics because it's a it's void.

[00:53:17]Now what we going to do is this. All right.

[00:53:21]So let's pretend at least at this point like uh arbitrary un empty set we have something it could be anything and when we have it right and then we are going to have um also let t be a subset of r.

[00:53:46]uh now when we say t all right so when we introduce these symbols of course the these did do not need to give any intuition but t maybe at a certain point we may see some intuition but I'm not going to say anything right so it's any subset of uh r any subset again I'm saying that means that is also little bit tricky and then consider A collection collection of maps right I'm going to introduce these maps like this G supererscript T this superscript is going to identify what function I'm what mapping I'm talking about and this mapping The meaning of a mapping is like basically like you map from that set to itself.

[00:54:51]So in other words, these mappings are going to be um in fact right. So we do have this right. So um arbitrary mappings in this scenario right. So if we consider collection of maps um uh let's say and we will indicate that mapping the set of maps and the collection of maps.

[00:55:52]Let me indicate it by like a fancy F symbol. Right?

[00:56:00]So in other words, what I'm saying is this F is none other than a family of these family or like the collection of these maps.

[00:56:12]So I do have g sub uh g supererscript t from m to m and such that I mean for all t in r or uh in in particular in this case like okay let's consider like t in t because that is the uh set that we considered right here there are many things that we need to uh introduced into this structure yet. I'm going to say here it is to say this this is the mapping or the collection of functions that we are considering and uh right maps such that uh let's give little bit of like some speciality I mean this could be any mapping any collection other than saying it like arbitrary I'm going to say this collection of maps is going to have this uh special property g of t + s.

[00:57:26]So if you consider the superscript or the index of that function I mean t + s that should be g of t * g of s. Now when you consider this one, you have to be very careful because now this is going to demand a little bit of more out of this set set T. One cannot say like oh it is just a we can we can always do this right because one can say like oh t and s both are coming from this collection and then when we consider t + s t plus s is going to be again like oh if it is living in this t then there will be a function like this otherwise there is no function like that there is no mapping like that.

[00:58:19]So at this point one can say oh this arbitrary subset is not enough. We need to say a little more. We need to give a little more structure. So that's exactly where one can say like oh let's try to make it like algebraically close. You can make it like a group subgroup or even without considering the subgroup structure there are different ways of introducing this uh dynamical system. So what we are trying to do here is this right I'm trying to give you this sense where we have like all of these possibilities.

[00:59:01]If one is going to say like oh this is going to be um this t is going to be the collection of all integers from negative to positive infinity like even including zero in the middle. So that is going to be an additive group right if if if you're familiar with the concept of uh groups from abstract algebra already. If you do not have that background, let's say simply it's going to be closed under two things. First of all, it's going to be closed under addition. Closed means like whenever you get two elements and when you add them, it's going to get into that same set.

[00:59:44]So, it's not going to go out of that system uh that set. And at the same time, if you'd get a number and if you make it negative, it's going to stay in that. Right? So those are the two things two characteristics that I would like to have out of this set T.

[01:00:03]Now one of these characteristics is evident right here when we demand this property it's going to give you just this scenario. However, at the same time we need to understand like we we need to have like two different ways of looking at it. one part is uh right. So in terms of that whatever that closeness of uh closeness under addition we are going to guarantee uh at least the platform for this equation and then what is really happening behind this equation? What are we trying to do with this? What you're trying to do is to make some kind of like a playground where we have consistency. So what you're trying to do is this. We're going to see this uh hopefully in the next couple of minutes we're going to see this. What you're trying to do is to start with a point somewhere on that set M.

[01:01:04]Right? So let me draw a picture for this and let's let's be hopeful that we're going to see that picture uh in mathematical form in a moment. Right? So if you start from one point like let's say you start from one point x and you apply this mapping and you're going to jump to like let's say with g sub uh g super t you're going to get get to a certain point um let me do this let's start with s you're going to jump to a certain point and then at that point if you apply again once again like if you're going to measure t and you're going to apply apply gt to that.

[01:01:47]So you're going to get g sx here and then you're going to apply again the second mapping that's gt then you're going to get to a point. Now the action of this whole point is it is like even if you ignore this midpoint or the the point that you found on the way that that uh uh stop that you had in the middle even if you ignore that you're going to have this directly from here to there. If you just directly measure that and apply G S plus T, what's happening is basically we we make it consistent. If it is kind of like a trajectory, it doesn't really matter if you just uh stop at a certain point and if you start moving again, you're going to have the same same journey even if you stopped or not. So that is kind of like a trajectory that we are trying to plan here. That is a geometry that we are going to have out of this wonderful property.

[01:02:58]So in anyways that that property was pretty abstract right. So it is just like yeah one can say all right second observation one can say like oh this seems like some kind of an exponential form right one can say like that but it is not in fact just the exponential property it is much more general I mean it is not an exponent it is not an exponent do not mix it up because the same thing would work if you consider it in the subscript notation I mean it is just a symbol it is it's a continuation So it is uh the consistency of like the trajectories.

[01:03:38]So um by the way hold on I didn't say what I mean by this dot right that dot is none other than the composition right I should have mentioned that very carefully do you see like when you try to prepare uh the abstract background we need we need to work through that couple of times right so this is exactly the composition you you compose right this product structure that you're going to see is just that composition. So this is going to give us a very nice u nice setup. Right? So now without making it too wide let's narrow it down. Now the setup let's revisit the setup we talked about like uh the general setup that you're going to or general or the setup that we are working on in this case other than considering any arbitrary time set or time set. Oh, I gave you the word, right? Don't say say it loud. It is not just I mean it is playing as a time set.

[01:04:53]But what we are trying to do is to get this structure since we heading towards ordinary differential equations other than considering set t I'm going to say like our structure is like this.

[01:05:09]Consider this.

[01:05:13]consider this combination. I mean one can say it like uh as a set but uh other than making a set I will make it an ordered um combination. I do have t set of t but in this case I will consider the entire collection of r. This is our choice of t from this point onwards t.

[01:05:39]So t is uh being played by the real number the entire real number line. So simply our time set is going to be r and then where these points are jumping here and there or moving smoothly would be a certain set called m which is an unemp. We just generically defined that right at the beginning and we said it's an arbitrary set which is kind of like too generic.

[01:06:12]We still need to add more structure to that. We're going to do that one step at a time. And then the third one is exactly the collection of those functions or this collection we know that is exactly the functions G from M to M or mappings from M to M.

[01:06:40]Um let's let's say more about this right. So this collection now here let's try to make it uh more precise here for every t in R such that such that inside we are going to say this let's consider like T1 T2 Right. But the same thing that that was played by TNS early when you have that it's going to be T1 and then composed or like you can simply indicate it by a dot. Uh but what do we mean by the product of two functions?

[01:07:29]It's none other than the composition.

[01:07:33]So this is the family and this this family is called.

[01:07:41]Um the I mean uh these functions or these mappings are called transformations.

[01:07:55]Transformations.

[01:07:57]That is our starting step. It's not a question mark. It is just an exclamation. transformations.

[01:08:04]And even if you consider like this in general setting there is something called the group of transformation if you consider any arbitrary and unemp.

[01:08:24]So here this is what uh we are going to have as the first proposition. Actually you you can try to prove this proposition or if you feel like this abstraction is going a little too fast like at this point try to understand this story because we will definitely revisit this this entire story with like a different angle again right but what we need to understand is like the first step proposition one Let's have something out of this. What we can see is like okay so we do have like a certain product structure in this function right so among these functions. So what if we consider this this family f together with uh this structure.

[01:09:29]Oh uh for this case let let me do something uh without having this um property included in it. Let's let's try to have this this set separately.

[01:09:43]Let's have the set separately and let's define this structure separately. It's going to make things much more clear.

[01:09:50]define this uh I mean define all like uh it satisfies let's say consider this set and for all t1 t2 we do have as an additional condition we do have should not be m it should be r the time right we should have this this set and then the transformations of course The transformations are none other than these functions that we consider. Now let's consider this set the transformations and uh uh together with this product structure. The product is none other than the composition. If you consider this this is going to be this is a commutative group.

[01:10:46]This is this is going to form a commutative group commutative group and it has a special name right. So it is called hold a one parameter group of transformations.

[01:11:31]Now this is the starting step. we need to think about now by any chance if you're not familiar with like let's say groups I recommend you to uh take a look at like look at the definition because you are already familiar with the abstraction right so if you have not seen this the there are like only four points that you need to show uh the closure associivity and the existence of an identity and existence of the inverse for any given element. Right? So this story, this whole thing is exactly giving us some uh structure called a group. But this is exactly where I'm going to say this um as a result. But then we will revisit we we we should think about like let's let's try to work with this right uh in terms of like a particular proof and we need to have that we should have that conversation and then uh maybe like as a less um demanding structure one is going to say something like this proposition number two. So some of these propositions I mean the way that you're going to prepare these propositions is like you're trying to make this abstraction right. So when you try to do this what you're doing is basically when you feel something is going to like a certain property that seems like that that's appealing you're going to just write it down. you're going to craft it carefully and then if you can see exactly that's true right then then you're trying to make it a proposition you're trying to make it a result you're going to write down a proof so this is one of the properties that I'm going to see in this case every mapping every mapping in f is one to one. Uh uh this is not the best way of writing it, right? One to one. We know what one means. We know that that uh in in basic um algebra and set theory, right? So and at the same time you have the background from real analysis. So you know what one mapping is.

[01:14:20]Now uh how why this works?

[01:14:27]I mean other than doing too much here like I'm I'm trying to introduce this this this the basic layer of this collection right so the basic background here we introduce this uh formal setup where we have a time set and an unemp set and a family of mappings from that Z M to itself together with that uh specific structure that would maintain like the the trajectories and that is what we had right here. So this is exactly this property and with that this is what you're going to get.

[01:15:07]We're going to get two two components.

[01:15:10]One party is like the structure is going to be a group. That means you're allowed to do algebra on this. There's an algebraic structure on this among these mappings. You can play with that. And then the second one is uh every mapping if you consider any mapping it's going to be one one to one and it is like I guess like we should have mentioned that earlier as a condition but I mean we did not have it as a condition and we do have it right here.

[01:15:42]Now uh what I'm saying is this um let's see let's try to see why this proposition two works if we can get some kind of an idea for that maybe like we can work around like proposition number one too I mean of course you you can work with that as an example let's see let's try to have the outline line outline of the proof.

[01:16:18]Let's try to prepare. Let's try to see like okay of course what is a one to1 mapping. Uh one to one simply means like if you have two different origins at the beginning they should uh produce two different outcomes.

[01:16:36]they cannot produce the same outcome. If you start with x1 and x2, they cannot provide the same value. Right? When you map it, it should be like different f of x1, different I mean different g of x1, different g of x2, something like that.

[01:16:54]But the way that we're going to establish this factor is like uh exactly the contraositive, right?

[01:17:01]other than saying like g1 is not equal to uh right let's say the general definition for a 1:1 is write this right x1 not equal to x2 should imply uh whatever the function I'm not going to use f here to confuse it with uh the family I'm going to say g right just say gx1 should not equal with uh gx2 but look at this in mathematics like when you work with like things. When when we work with uh mathematical entities, working with equations or inequalities, it's easy.

[01:17:41]But when you work with nonequalities, right, when something is not equal to the other, right, it could be the whole lot of spectrum. So if you start from this end, it's it's pretty hard to do something, right? So this is exactly why if you have seen the other way the contraositive uh behind this statement I mean the contraositive is none other than a way to uh work around this right so this right so uh for the entire domain right for every x1 and x2 x1 not equal to x2 uh would imply gx1 not equal to GX2 is uh going to be if and only if the contraositive g(x1) equals g(x2) by any chance if they started with I mean if they end up with the same value they must start with the same value in the domain.

[01:18:51]So this is the contraositive part. Look at this. Here we are playing with equalities.

[01:18:56]When you have an equality there are there are things that you can write down about it but the inequalities like not inequalities nonequalities like it's not the equality when you just contrast the equality then there there is a confusion right a confusion or less things that you can write less things that you can present. So this indirect approach is really helpful in that way. So let's look at this. This is what we need to prove. But look at this. Let's pick a function in this.

[01:19:31]So, uh, since we have this background, let me just get rid of this and let's say uh, let x1 x2 be on this um collection and even before even before doing that what are we trying to establish we are trying to establish a certain or like let's say selected or any given mapping in the collection of f is going to be one to one so let's start with that let g T GT right we know the form of this right it's going to be a G together with the it's called by that structure um or the index so let GT be one of the elements of not M the collection of transformations yeah and then I'm going to consider like X1 equals X1 and X2 from that's collect ction and let's assume suppose that g x1 gx2 what we need to establish is this what we need is uh we need to show that x1= x2 now starting from this statement what we need to establish is x1 = x2 How can we do that?

[01:21:27]Now, this is where we need to get a little bit of an idea about like uh the the structure here, right? If you compose minus t, right? Don't forget this is valid for all values, right?

[01:21:59]Therefore, you can just do this.

[01:22:04]If you do that, what can you say in the next step?

[01:22:16]And then by the assumption that we made related to this product structure right we know this is none other than g of -2 + t + t and x1 and then this will end up getting g x1 on what is this G0? This is going to be exactly like the structure.

[01:22:56]Now if you try to think about this this uh the action of this mapping what we indicate by G0 is none other than the identity mapping in this. What is the identity map? It maps to itself.

[01:23:12]Now let's try to think about like let's say if this t is playing the role of time what it does is this you start from a certain value let's call it x1 and if you apply gt into that right in this scenario what you're seeing is like let's say after time t where it should be you're trying to get that value right so what it says is like you start from two different points and you you end up in the same time you you will end up somewhere the same point.

[01:23:51]If that happens what we are saying is it's it should have started with the same origin.

[01:23:59]So if you start from one single if you start if you at if you're at a certain point the simple story is you can look at what happened in the past along the trajectory and you can recognize where you started it from and you can identify it uniquely and what this gives us is x1 = x2. The simple fact is g0 is none other than the identity function. ID simply means the identity. Identity of what? If you go back and check now proposition number one, I told you it is already a group structure. For a group structure, there should be like there should be one mapping in this collection which acts as a group identity. Who's that group identity? That is none other than the identity function. And that is exactly what's playing this role. So this tells us this function is going to be I mean this mapping is going to be one one. So now look at this.

[01:25:05]We simply now what we did was this. We didn't do much except for introducing this first of all we introduce like exactly the cause and then we we had like right away without having any computational example. we just immediately jumped into this abstract form of the story and we started proving stuff and this is exactly what I wanted to point out now in this in the next uh session this is what we going to do so consider this whole thing I mean any ambiguities that you might have had like you are having right now with this this end point wait a minute how did we get here like with the subtraction how many things like when we say like the set m I have to reset what do we mean by that so all of these I mean if we feel that there are missing points behind this and we need to make it more formal and smooth as a conversation from that level all the way down to like what we're trying to describe like do we see any differential equations here not yet right do we see any background for differential equations? Not yet. So preparing that background is exactly what we are going to do in this conversation. So my plan for the next lesson, I mean for this lesson right after the introduction of the course, I wanted to give you like a direct uh structure what we the way that we are going to do it. This is just a test and then in the next lesson what we're going to do is uh in the next lecture we are going to have like let's say according to the imperfect approach uh or where we need to have little more details added into provided by Arnold's first edition.

[01:27:08]We are going to have that conversation.

[01:27:10]We're going to make it as much as possible. we will try to find where it's lacking more information where it needs to be supplemented and then that's exactly what we're trying to do with uh addition three or like the way that we are going to do it like it need not to be just the textbook it it should be our intuition the way that we are going to prepare uh our understanding on our notes and our um in our work in our presentation. So that's exactly what we're trying to work on the second uh lecture onwards.

[01:27:48]All right. So uh in that way we are going to have this introduction and overview for two three sessions that would be natural in this scenario and then we will get to uh the relevant layer of uh conversation where we we are going to have I mean gradually we will progress to next levels. All right. So with that I'm going to conclude this session and I will see you in the next uh lesson. So take a moment uh to skim through Arnold's book.

[01:28:23]uh but my recommendation when it comes to uh certain types of solutions like uh especially in addition number three if you feel like certain components I mean solution oriented uh conversations do not worry too much about it because we are not planning to do that uh in our work and then uh on the other hand if you're looking at addition number one if you get confused by the whole uh abstract conversation. Do not overthink because that's exactly what we're going to elaborate in upcoming lessons. All right, I'll see you next time.

[01:29:03]Have a wonderful uh week and I'll let's meet the next session.