Domain, image (Poincaré) (contd.)

0:00 Thank you. Thank you. We're going to take a 10-minute break. Can we... I'm going to talk to Marco and Marie. Can someone take care of the coffee and tea? Who wants coffee first? Who wants coffee? Anyone? Who wants coffee? Coffee? A coffee? Yes, please. Okay. Well, then... Can I trust you? Thank you very much for your time. I made a name for myself, but you have to send it to me right now. Well, come on. She's got more to say. Come on. You have to send it to me right now. Yes, but you have to send it to me right now. I'll... I don't know why I think this one is clearly for the next lecture, it's a bit too long, I think it's too long. Wait. You have to tell me before. On the other hand, in a much more projective way, the question is the same. I have officially named, but I think there is a nomination that is not there. So, I took the nomination. Renaud Chorlet, responsible for the Chinese restaurant, today, in place of Anne, who is the most... Listen, Renaud, it's an extremely heavy responsibility. And because of that, the salary of Converse Revis will be increased by 20%. Exactly. Now I can sign everything if you want. Do you confirm the nomination? I confirm the nomination. Thank you very much. With an exhibition that will wake us up from a nightmare of Rio.

2:30 She doesn't know that Harry has just woken up herself. She thinks she comes from the summer of Rio to... It's difficult to be the last one. ...to come and talk to us in the open air. We're in full strength. No, it won't sleep at all. No, I... Raffaelle, I'm with you. You'll have to come with me. Well, I had talked about the concept of stability at Poincaré and I would also like to go a little further than Poincaré. And I will quickly talk about Diagnostics and Discourse, which will highlight the definition of stability proposed at the beginning by Paul Carré. And in fact, I will try to show you that this is a concept of physico-geometrics, that is to say, or physico-topological theory. And in fact, I will talk more about the geometric thought part in the title of this day. Because, well, the domain that interests me and on which I work for my thesis is the theory of the dynamic system and which is normally conceived as a theory that starts from the consideration of the entire solution of differential equations from a geometric point of view, that is, instead of trying to solve a particular differential equation, we are going to try to define the entire solution of differential equations from a geometric point of view. The theory of Poincare is associated with Poincare's work from 1881. It is a study of memory on the curves defined by a differential equation. Then, the study of linear differential equations. This study focuses on non-linear differential equations, that is, differential equations in general. And we also associate the genesis of qualitative points of view on differential equations. So what I will try to show is exactly what these qualitative points of view are about, based on an example that is the example of stability. We have already noticed in some works of history For example, Jelani Gray. Poincaré used topological tools that were available in his time in his original works, such as the Parcels and Cantors ensembles, the Manus of the chronicles, the genus of the surface, and the characteristics of the universe, etc.

5:00 But in fact, what I'm going to try to see is not how Poincaré used geometric tools, but to see how, and this is about the introduction of a geometric vision, if you want, of a geometric style. In fact, I don't know very well how to recognize that from a physiological point of view, because it's not a point of view in the sense that Bruno spoke about. But it is a geometric vision, that is to say that the definitions themselves, the methods, are motivated by a topological thought that comes precisely from the problem of instability. And in fact, this topological thought is not only the use of topological tools. There is this quote by Hadamard when he analyzes the work of Poincaré. It is even surprising that we had to wait until Poincaré to see that the conduits invented by Riemann were subject to all the problems of local to global transition, which is the case in particular of exponential equations. So, Adama observed that... We had to wait just one square minute to use topological points in the work on differential equations and that's where it can be surprising that we didn't notice before that it would be useful for differential equations problems. So, do you want to write? And it is especially here that, when this occasion is badly divided into a lecture in an incredible way, that the success of the lecture is not limited to a particular subject, but rather to a lecture. But, after the second definition of Poincaré, this concept is clearly established, that is to say, the geometry of situations, the concept of the use of the geometry of situations. It is so closely linked to these passages of Locale, in general, that it is the greatest confirmation of the primordial nature of mathematics. In every passage of this interior, we can only see, when I write the situations, the symbols. Just to make the ad for the 3rd April general lecture, we will ask you to comment on this sentence, the passage of Locale, in general. Very good.

7:30 Excellent, we can't hear anything. And there, in fact, a woman in particular is the classification of the singularities. We will see a little later and quickly what it is about. But I come back to this introduction of topological points by Poincaré because, in fact, as far as the notion of stability is concerned, we will see, in fact, what I will try to demonstrate, which is Matel's, let's say, is that Poincaré is a bridge between two different traditions, that is to say, between two different styles, one which is the theory of quantum mechanics, that is to say the traditional theory with which Matel dealt with the problems of celestial mechanics at the time. and the introduction of new topological effects. There is also another quote by Adama Valsu, a mathematician who makes square-shaped tables. It's always like that. Yes, because this is the final part. So, he says, the Thesis of Poincaré already contains on differential equations a remarkable result, and he decided to be prepared to live with it in the early stages of his research on metaconstellation. From his first work, he was, on the other hand, led to perfect the main tool that has been used so far to develop differential equations. Tools that would have been used better than anyone else, while at the same time as the previous ones, they would have taught the theory of quantum mechanics in the past. That is to say, it is precisely the two points of view that are at the same time, it is the work of Poincaré, the one that is in the traditional field of quantum mechanics theory, and the other that the new topologies have taught in the past. Well, in fact, we will see this tension between the two points of view in a specific problem, which is that of stability, which has, in my opinion, a constitutive role in the development of these technologies. So, what I mean is that the motivation that comes from physics...

10:00 The problem of stability must have a constitutive role in the development of mathematical notions that are more adapted to define this topic. Poincaré, to talk about these two traditions, Poincaré himself, I'm sure, uses a method of two different ways. He says that a comprehensive study of a function, here he talks about functions defined by a differential equation. This is the analysis of the scientific work of the human brain. And this is the part where he talks about functions defined by a differential equation. So he says that a comprehensive study of a function, which is the solution of a differential equation, composes two parts. The first is qualitative. For geometric studies of the curve defined by the function, there is a second quantitative or numerical calculation of the values of the function. It is naturally in the qualitative part that we must address the theory of all functions. And that is why the problems that occur in the first place are the following. Constructing the curves defined by differential equations. When they are constructed, they are constructed in a geometric way. This qualitative study, when it is completed completely, will be of the greatest use for the numerical calculations of the function. And so it will be easier to say that we already know the convergent equations that represent the search function of a certain region of the planet. Well, we think that this qualitative study marks the second of two types of justifications that will be noted, and that this qualitative study will have by itself an opportunity to take orders. Various questions of great importance in the field of mechanics are in effect ceramic. But let's take as an example the problem of three bodies. And so they ask questions such as whether a body will always remain in a certain region, whether the distance between two bodies will increase or decrease, whether it will always remain within a certain limit in a certain region, and many other questions of this kind. In fact, we see here that Poincaré himself proposes two types of arguments to legitimize the new points of view. One, which is for those that the new points of view will help to find quantitatively a function according to the old points of view.

12:30 And another, which is really new, that is to say that qualitative tools have an interest in themselves because they allow the use of these functions, which are useful in celestial mechanics. This text is from 1901, a catalogue of works by Poincaré himself, but it is a very particular passage where he quotes the introduction, which was already that of 1881. It is not something late in Poincaré's thinking, and that is how, at the very beginning, when he has some problems, he presents them. So, Paul Carré, he speaks, he starts his thesis, he declassifies the similarities of differential equations according to the four types, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, But in fact, in this aid, he used, to quantify the values of these singularities, a notion of equivalence that took into account the differential structure of solutions. That is to say that now, precisely, what is going to happen... In his new work on the differential equation, i.e. in the 21st century, three years after his thesis in 1877, he will no longer demand that the differential structure of the curves remain unchangeable. This property of all the differential structures of trajectories in these neighborhoods will no longer be so important if we talk about the problems of celestial mechanics. And it is only then that we will see why topological views become important, why it is possible to use topological descriptions. and its satisfactions. I don't get it. Is it always in an analytical framework?

15:00 No, but we'll see. Is it defined by a differential equation? No, in the courses defined by a differential equation, we'll see. Because I'm not talking about a point of view. I'm not talking about mathematical results. I'm talking about a point of view, we'll see. Well, so... So we're talking about two physical problems that are, for stability problems in general, we're talking about two physical problems that are, one, the problems of the stability of the solar system, and the other, the problems of the stability of the figures of equilibrium in a fluid of rotation, and in fact we're going to see that these two problems have different roles, genetic roles, let's say, different motivation rates for the definition of stability that will follow these two problems. So, for the first problems, that of the solar system, the traditional points of view, which were the series of the transverse, were developed in series. We tried to show that the terms should remain between certain limits, and thus we would guarantee that the great axes would not align much with its initial position. But we only took into account the first terms of approximation, that is, linear terms. After, with Poisson, we took into account non-linear terms, terms of the second world, and we came to terms like that. And so these terms could oscillate, and the oscillations could be bigger each time. So we would return to the neighborhood of the starting point, but after, we would be further away. This is where this study of Poisson demands that we redefine the notion of stability. Concaret will therefore propose the third part of his memoir. Instead of defining stability by the strict periodicity of the axes, variations, observations, we define it by a quasi-periodicity. A more specific and less specific composition, in fact.

17:30 We say that the trajectory of the mobile pole is stable when, at the starting point, a circle enters a sphere with an R-ray. The mobile pole, after being a very small circle in this sphere, enters an entire trajectory, and it is there that the particle is R. Now we no longer demonstrate that they are exactly periodic, but quasi-periodic, that is to say that the points can diverge a lot and then come together in the formation of the point of departure. So this is the definition of stability that Poincaré presents in the third part of his memoirs, on the But in fact, he adds that the importance of this definition is purely theoretical, since in practice, it is necessary to look for regions or trajectories that are continuously contained. So from these points of view that I am talking about, it is not in the definitions themselves, but from all points of view, there are, in his memoirs, these two points of view mixed. So, at the same time, he demands that the properties of these series, these types of cases, be notified, but at the same time, he says that this is not important, it is only a theoretical element, because in practice, what needs to be done is to find regions where the trajectory remains infinitely continuous. So they use data analytics, but the results are not... No, I mean that they are not only interested in the results that are used in data analytics, because precisely it is these types of properties, the fact that we only require that the trajectories remain confined to the needs of the people... And we study the limit behaviors of trajectories, but not the whole history of trajectories, which allows the use of topology and the development of what I call this topological vision. Because topologically, precisely, because to analyze... The whole of the unit, and to know these two trajectories, if a trajectory, for example, enters or does not exit a region, or if these two trajectories have the same behavior in the unit, we do not need to know the structure of the differential of this trajectory, we only need to know its topology.

20:00 So this is precisely what happens in... A topological point of view that will later, we will see from this table in the Diaphanos-Tircotte check, which will require a redefinition of the notion of stability. So, we already see there a tension between two types of... There are two types of definition of stability. One is formal, which is the quasi-periodicity of the series, which comes from the properties of the development of a series of solutions, and a second one, which is purely qualitative, which is to be derived from a direct analysis of the set of solutions and the qualitative properties of the set of solutions. So, what you're talking about right now, you're not talking about the qualitative side? In which the trajectories may differ, yes, but if analyzed from a series of developments, no. And normally, at Poincaré, it is there to analyze everything from a series of developments. But it is said that this is not very important. You classify according to the tool rather than according to the type of formulation of the property. Exactly. That is why I say that it is a point of view. I am not talking about topological tools. I am talking about a topological vision, that is, that the topological results would be sufficient. At this point of the development of the square point, when we talk about development in the circle, does that mean a convergent development or only a development in the circle? We'll see. We'll see that. A convergent development? No, no, it's convergent, but we'll see. We'll see soon because I'll give you an idea. So, in fact, the square point is two-dimensional. What he calls a topographic system is a partition or domain of trajectories, a cycle that he calls the limit cycle or the contactless cycle. The limit cycles are those towards which the trajectories reach infinity. And the contactless cycles are the cycles that are not solutions to the differential equation. So if we have a contactless cycle, it means that...

22:30 If the trajectory, if a real trajectory, because there are no solutions, so if a trajectory that is a real solution crosses these cycles in one direction, it cannot cross in another. So, the partition of the domain into cycles without contact is important because... It allows us to analyze whether a trajectory remains confined to another region or not. That is, if this trajectory crosses a symphonic cycle, it must remain inside. So it allows us to know what is the stability associated with this trajectory. This goes in two dimensions, so we use strongly the properties of the algebraic curves in two dimensions. In addition to the two-dimensions, precisely, these are the problems. This type of analysis that he had done, that he had succeeded in two dimensions, and in two dimensions, this is the famous theorem of Poincaré-Bendixson, the fact that it was proposed by Poincaré and worked by Bendixson shortly after. And it is a theorem that is not only a co-dimension, because in dimensions we have the theorem of Jordan, we have the properties of the co-dimension 1 for a curve, which is a very particular and very strong property of dimensions that allows us to demonstrate all this. And the problem is that there are more two-dimensions, there are more theorems, and we can no longer use these results that come from the theory of coulombs and bricks, which are results that Poincaré calls geometric, and so he says that to push this analysis, to extend this analysis to more two-dimensions, we have to create a new tool, which is the analysis of the situs. And this is the introduction of his article on the analysis of the situs. And it is precisely in relation to these problems, that is to say, in order to do the same kind of qualitative studies in more than two dimensions, it is necessary to develop the tools, it is necessary to do with the tools of institutional analysis. Well, so, in two dimensions, the problems of...

25:00 To know the curves defined by a differential equation, we have, and it is practically solved precisely by the terms of Pocahontas and Dickson, which say that All characteristics, which is the way, as they call the solution at these moments, which does not have a knot, is a cycle or a cycle. So, like that, we know, globally, the behavior of all trajectories, because trajectories are either cycles or spirals, which spiral towards a cycle. So, from there, we can know the overall dynamics of trajectories. All of this allows us to know the solutions of a non-linear state, because we have the overall structure of the solutions described from this. When we want to increase the precision of the approximation, when we want to push the approximation beyond the approximation of nerves, there will be a problem because for the case of singularities, of foyers, of collars, of knots, this is maintained by perturbation, but in the case of a center, this is not maintained by perturbation. So we will have to do a direct study of the case of a center that is in a state where we have all the closed corners, we have a singularity that all the neighboring corners are closed corners. And so for these cases, we will have to do a direct analysis of what happens in nonlinear cases. And so Poincaré will propose a process that is quite traditional in fact. Thank you for your attention. We'll just read it out, so it's just all good.

27:30 These are the ones that we're going to use. So, we have here, we want to find a function that is constant on these curves, on these So we start with the terms of order 1 and order 2 because we want to know the linear approximation. So the first terms we find that they are of this kind. And then, as in the process of celestial mechanics, we try to find the terms of order i. I found the terms of order i plus 1 by writing this term in the form of the polar. And f of i has only the terms of the trigonometry. So we want to use R as a trigonometric term so that we can guarantee stability. And we find a problem for F4 because the integral of this, because we are going to obtain this, and the integral of this, we are going to find the omega here, that is, we are going to find the secular term, which is not in the trigonometric terms. And so on and so forth. So, if we have these zeros different from zeros, not zero, we have a proportional term to omega in our table. And at this point, it is precisely the problem that, under methane, we fall into celestial mechanics and which is related to the circular term. It is only that Poincaré will produce a new one because... In the positional processes of celestial mechanics, if we find a zero, it is a zero, we cannot conclude the instability because these terms are only due to the development of a series of sinuses. So, we can't conclude for instability when we find a T0 different from 0, while with the time-square approach, yes, we can conclude for instability when we find a T0 different from 0, since it shows that a T0 different from 0 is equivalent to the existence of a 50th cycle. So, if we have a 50th cycle... What we have here is that if a trajectory enters this contact cycle, it does not want to exit, so it cannot arrive in its starting point, in the starting point of its starting point. So, by the presence of a contact cycle, we show them the instability of the trajectory, which is that the presence of different zeroes and zeroes and zeroes before was not the conclusion.

30:00 And, well, that's it. Well, we see that in the case of a center, Poincaré will use a fairly traditional method, which is the characteristic method of celestial mechanics. And then he will have to increase the dimension of these problems, because this is for two dimensions, but then for three dimensions Poincaré will study... Problems in the neighborhood of a periodic solution, so this is part of the novelty of the New-Banker approach, is to study the neighborhood of a periodic solution for what it can bring to the study of solutions in its neighborhood. So, what makes these periodic solutions so precious is that they are the only breach that can help us penetrate from one plane to another in an unavoidable way. In dimension 2, we have a global description of trajectories. In dimension 3, things are much more complicated, and we have to start with game-solving algorithms. We have to try to write down what happens in game-solving algorithms. And in fact, this method has been evolutionary in these areas. And who founded, in a certain way, the idea of the systemic system, but that's what the engineer does. It's the method of sections, so we have a periodic solution, and he's going to cut this solution into a section. And then he's going to study what happens in the design of this section, which we call today the Pocarette section. And so, we have this. In other words, in a periodic solution webinar, all the solutions must cover the section and in fact we are going to study the dynamics of intersection points on the section.

32:30 And this dynamic will describe all this space in the neighborhood, all this interesting space, and this will be the most important point, in the neighborhood of the anterior formation. That is to say that all this space here does not interest me, but I am interested in what happens in the neighborhood of the starting point, that is to say on the intersection point. And that is, once again, a topological division, that is to say that the behavior of the transients and the geometric structure of this trajectory... These points are not interesting. I am only interested in the way these points come back on the function surface. And this is, once again, motivated by the problems of stability, because it is the problems of stability that suggest to me that the transient behaviors are not the same, and that I have to look at the mimetic behaviors, the asymptotic behaviors, the behaviors that are good, that is to say, all these are interesting tricks because I am interested in solving the problems of stability. He proposed the same type of analysis for the boisonnage of a center because there we can have on the surface of the section the same four types of descriptions of the boisonnage of a singularity, col, foyer, center, that is to say we can have again this case of a center, only that now the closed surfaces are a question. I would like to return to this question to see how these points are distributed on the sections, that is, according to the nature of these distributions, we have very little information on the... The points on the section, this is the result that was very surprising at the time, because on the section, the points are exactly the same type of points that we had in the storage of a singularity, a two-dimensional, that is to say, that. That's exactly what we have on the section this time, that is to say, we were analyzing the curves, that is to say, the solutions in themselves in the

35:00 This is an example of a two-dimensional state, an example of a two-dimensional state in the state of state in the state of state in the state of state in the state of state in the state of state in the state of state in the state of state These are the points defined by the transformation of the points, which is a transformation that brings a point of the section over a point of the section, in the arrangement of an invariant point, which is the point that represents the solution of the problem. This is very surprising. All the mathematicians who have worked on it, Poincaré, Lattès, Birkhoff, etc., will repeat that it is impossible, says Poincaré, not to be struck by the analogy that this analysis presents with dimensional states. Yes, but in fact, the solution that has been found in the Wilcox's thesis is because we are doing functional equations, so we need a functional space to see that it is banal. Well, we will come back to this point, but it is not the most important point because... I am going to analyze only the central cases, so we have, for the central cases, a closed loop in the We have no way of concluding, a priori, that all these constants, these zeros, are null, but in three dimensions, Poincaré will also serve as a novelty, it is not he who invented the so-called original optimization, which is the notion of integral invariant. So he says that if we have an integral invariant, we can guarantee that all the constants of these terms, which we talked about before, are null.

37:30 Why? Because if we have an integral invariant, all the curves that are solutions of the differential equation will be on tors in the evolution of this periodic solution, because we have functions that are constant on these tors. We can't have contact signs between the energy solution and this gold, so we conclude that the C0s are null. Before, we had said that C0s different from zeros indicate the presence of a contact sign, so the strength of a contact sign here indicates that the C0s are null. So that allows... I wanted to integrate this series, in fact, the one we had before, without having to worry about the secular terms. However, there is another problem. This does not allow us to conclude all the following for stability, because the series, in addition, may not be complete. And this is the cause of the problem of the small denominators, which is equivalent to the problem of the commensurability of the great axes of the three planets in the problem of the three poles. So you understood, you eliminated the term secular? Yes, if the cedars are null, we don't have the term secular. Because in the series, the cedars were the only terms that appeared by themselves in the language. In trigonometric terms, we had these zeros alone, plus, for the others, it was the constant terms that multiplied the cosinuses or the sinuses, etc. So if the zeros are null, we have an infinity. Well, then... And it is precisely these problems... In fact, I am very happy that I had the opportunity to speak to you earlier, because it is precisely these problems, the fact that stability depends on the quasi-convertibility between the creases of the valence, that Reichstadt means that it cannot end up in a physical problem, that is to say, there will be an error in our way of analyzing problems, because the resolution of the physical problem of stability will not depend on the questions of convertibility.

40:00 These are mathematical questions. So we have a result that allows us to conclude that for complete stability, in the case of integrability, that is to say, if we have an integral, if the integral converges, that is to say, we are in the integrable case, we conclude that for stability we have a certain property. Otherwise, and this is also part of my thesis, is that another notion of stability will come into play, that is to say, we will need new mathematical definitions of stability as soon as we no longer have an integrable problem. That is to say, in the world of integrability, everything works very well, we have a definition of stability which is the quantitative definition, We can conclude on this property from qualitative and quantitative methods, but the two qualitative and quantitative notions are not very different, they are both mixed up. But from the moment when we no longer have integrability, these solutions will have to be well defined in different ways. So, this will become clearer. This is an example of the study of the stability of periodic solutions in these works on celestial mechanics. So there is an interesting thing in these works on celestial mechanics, and in the introduction it says that the result of the triple-format of the three bodies, which may be established with absolute rigour, requires mathematics. Theorems on periodic solutions and on asymptotic solutions and asymptotic demands, etc. So, all the qualitative methods he uses to analyze these types of solutions, he considers that they are part of the absorbed river that requires mathematics.

42:30 So, in this book, he will mainly study the strange problems of the three agreements. Because it starts with the movement of two bodies, we have a Keplerian problem, and then it introduces a third mass that depends on the mu parameter, which is an infinitive parameter. It will tell us what happens when we introduce this third mass. And so we need to know what happens when we disturb a periodic solution, precisely, because before we had the Keplerian of a periodic solution. And we are going to build and coordinate, adapt, in order to study the resolution of a periodic solution in the same way, or in an analogous way, to what we did with the sectional equation. So we will have this problem formulation. So we have the starting equation, we have a periodic solution i. We are disturbed by a sexist term this time. Yes, I told you. We forget the non-linear terms. We will write a new equation, anxi, this time. And we will have a linear system with periodic coefficients. And it is actually a linear system that writes the voids of a periodic solution on the section surface. The alpha and i are the exponent characters here, such that any solution is a binary combination of terms of this type, where the s, i, k are periodic, so it is a matter of knowing if the... If these characteristics are pure imaginary, because even if they are pure imaginary, we can conclude that the x, y remains finite, so we can conclude the probability of periodic solutions, that is to say, a solution of the equation of periodic solutions, but not so far from it. So Poincaré defines at this moment the stability of a periodic solution for this particular property, that is, a stable periodic solution if this characteristic explosion is imaginary.

45:00 But this definition is only valid for linear cases, because the equations of variation, which are these linear equations, are only valid when we have a linear approximation of the problem. For non-linear perturbations, such as the torus example that we saw, it is possible that a neighboring solution escapes. So, we have to make sure that there are always invariant closed curves around the fixed point that prevent the projectors from escaping, just like in the case of a twistor. Remember, if there are invariant closed curves in the coordinates of these invariant points, In other words, we have solutions on a torus, and the intersection of this torus with the two-square section is a closed group in variance. If we have a design of this type on the section, we verify that the solutions do not shift, because the solutions cannot cross each other. So a solution here cannot exceed the variance curve. So we verify that the solutions are confined in a certain region. This is precisely where the property of nature is essentially topological. These were not verified directly by topological methods. Why? Because the Poincare definition is still an analytical definition, which refers to the possibility of writing down these trigonometric lines. In cases where we have an integral theorem, we can guarantee that the solutions are in this order around the periodic trajectory, because there is a constant function on the trajectories. But in non-integral cases, which are called non-linear perturbations, even if it is a small perturbation of the integral case, we cannot conclude anything.

47:30 So these problems will be solved by... And we are going to see that it is precisely from these terms that we are going to see what is the definition of stability used by Pocahontas. It was not yet an explicit definition, even if implicitly he always said that it was necessary to find a region where the trajectories were enclosed, etc. Explicitly, his definition was not yet fully valid. But before we move on to the Utilitas, I would like to quickly talk about the Lyapunovs, because the definition that the Utilitas will employ in their community was inspired by the work of the Lyapunovs. He is an English mathematician who wrote his first thesis at the end of the 19th century, but his thesis was not published in French until 1907, so we will not know his thesis until 1907, but it is known that before that he exchanged letters with Poincaré, I will not comment on these letters because I do not have the time. But he exchanged letters with Poincaré precisely on the definition of stability, which is interesting, it is the force. And there is an interesting problem because, in fact, Poincaré worked on the stability of the solar system. He also worked on the stability of the figure of the ellipse in rotation. And the definition of stability that he used in both cases was completely different. In the case of the stability of the key terms, it is not a definition that was that of Thomson and Type, i.e. the classical definition, which is derived from the existence of a potential function, the theory of the maximum value of a potential function, the theory of the range, while Lyapunov proposed a new definition.

50:00 For this case, for the stability of this case, which will investigate directly what is happening in the surroundings of an equilibrium point. So, by using the work of Poincaré on the stability of the solar system, he cites in his thesis, will propose a new definition of stability that will be more useful. In the work on solar system solidity and in the work on the eclipsed solidity of the solar system. That is to say, the apnoea, one of the different methods of Pocahé, proposes a definition of the eclipsed solidity of the eclipsed solidity of the eclipsed solidity of the eclipsed solidity of the eclipsed solidity of the eclipsed solidity of the eclipsed solidity of the eclipsed solidity of the eclipsed solidity of the eclipsed solidity of the eclipsed solidity of the eclipsed solidity. So, the definition of S-N is just for today, to find a solution. If I'm not mistaken, it was wrong, because I thought it was a French translation. When I read it, I thought it was a French translation. It's not very difficult. A solution x0 of t that has the initial condition x0, c0 is stable. Here, all the solutions that go through the initial functions, x, y, zero, are close to x zero and y zero, and are close to x zero t, for all t greater than p zero. So this is precisely a definition of stability that meets the set of trajectories in the vicinity of the trajectory that we have analyzed. That is to say, a trajectory is stable if the solution next to it is not stable. So stability is no longer the property of a trajectory, but the property of a set of trajectories that is defined from the set of trajectories in the vicinity of the trajectory of the departure trajectory. So, it is this definition, which will be used by Levitica, which will explicitly say, in the first part of the lecture, which allows us to pursue the next part of the lecture, which says, so...

52:30 Thank you for your attention. In summary, there are sufficient ordinary criteria for stability. From there, it has clearly resulted that, already in the case of the S-systems, which contain T-manifestations, as in discrete, the stability of the characteristics are very stable. I think that the conditions of inequality are sufficient to reassure that stability is at the same time a character that requires, that is to say, conditions that require the nature of the functions of x and y. So here we can see that it is useful. The utilitarian says that instability is a qualitative characteristic, and stability is a quantitative characteristic, because to ensure stability, conditions on the functions x and y are required, so the integral is what we call the integral. So, you need analytical conditions on functions x and y to conclude for stability, while to conclude for instability, you just need to have a number without contact. So, for that, instability is qualitative and stability is quantitative. So, we associate quantitative terms with analytical conditions, i.e. analytical conditions on the nature of functions x and y. Is the question of qualitative quality conditioned by inequalities? Yes, but it's not. He's talking about synchronicity. Because in fact, that's what he calls the thought of generalities. That is to say, you just have to know that if the projectors are smaller or larger than a point, then we already know what's going to happen.

55:00 It's in that sense. That is to say, the presence of a synchronicity is enough to reassure the instability, because we have an analysis intelligence. I'm sorry, I'm not sure if you can hear me. Yes, but we're going even further because it's really qualitative in the sense that we have a geometry, a topology criterion to ensure and verify stability. That is, for stability, we need the equivalent of an integral, that is, a condition over a condition. So, I'm jumping a little. This is an article from 1901 by Ricci, by Ricci d'Itta, and in fact this article will be cited many times by Kirchhoff, who will say that the property of fish, the stability of fish, which has been defined as being the property of stability by Poincaré, In the paradigm, it is quite correct that a system of this type possesses instability in the sense of possible. This use of the word instability is, however, unfortunate. This use we wish to reserve for this word another meaning in school. We shall always use recurrence to specify systems stable in the sense of possible. We come now to the question of stability. The fundamental fact to observe here is that this concept is not in itself a definite one, but is interpreted according to the question of deceleration. We may, however, approach it in the following manner. Certain parts of the whole space are regarded as regions of stability, the other regions as regions of instability. So, we are going to talk about the definitions of stability and how they are multiplied within the discourse.

57:30 Birkhoff will clearly say that there is a more quantitative definition of the PSL associated with requirements on series properties and a more qualitative definition of the PSL which is part of a direct inspection of the entire solution. Searching for these spaces in the building of a periodic point which is square with a stable value, you remember that the periodic points on the surface of the section of the stable was this one, and for these, the arithmetic exponents are pure imaginary. This is because this condition was associated with the possibility of developing a more magnetic series. Birkhoff will say that this property should not be called stability but ellipticity. He will call a point of this kind an elliptical point. And so it is a question of verifying what happens in the formation of an elliptical point. That is, is an elliptical point really stable or not? And for the integrable case... An elliptic point is really stable, so when Poincaré defined elliptic points as being stable, he meant the condition of integrability, that is, the quantitative condition. Whereas now, even if we disturb a little bit an integrable system, and if we obtain a non-integrable system in the evolution of an integrable system, we will have... In the neighborhood of an elliptic point, this, that is to say that we have, this is on the square section, on the section surface, these are the elliptic points. In its neighborhood, we have invariant closed curves, so curves that guarantee stability. But we also have homogenous points, that is to say points that are absolutely unstable. These are points that, if we take a neighborhood of a point in Hawking, it will not depart from these points, so they are unstable.

1:00:00 So what happens in this figure is that we have stability in one direction and instability in another direction. This shows us that there are two definitions of stability, otherwise we could not have both at the same time. So, in the case, Birkhoff will say that in the case of integrable, the two definitions of stability are equivalent, because there is a family of variables on which the integral is constant, but in the case of general, there are two types of stability, a stability that we can say formal or astronomical, and a qualitative or mathematical stability. The formal is linked to the possibility of writing trigonometric series. Who is the real signification, the requirement of the characteristic exponent that is pure imagination. And qualitative is the direct analysis, it starts from the direct analysis of the composition of a variant form. To know if all the movements of the initial conditions are close, the mothers are close indefinitely. Stability. Does formal stability guarantee qualitative stability? And does qualitative stability guarantee formal stability? That's the question, to know if we have two definitions or not. Or if the two definitions are equivalent. Well, for an elliptic fixed point, as we have this figure, Birkhoff shows that there are areas of instability and areas of stability. So we have an infinite number of homogenous points where we have areas of instability, but also an infinite number of stable periodic points where we have areas of stability. And why are they in instability? Because we have seen that if we have a confirmed variant, the trajectories remain confined, that is to say, they are stable in the qualitative sense, in the sense that the trajectories remain adjacent, not in the sense of periodicity. So a stable solution in the sense of Poincaré, elliptic, can have infinitely close solutions that escape, so there is instability in that sense.

1:02:30 But these solutions remain enclosed, so there is stability in another sense, which is the sense of the diagonal. So we have seen that it is stable, not in the sense of Poincaré, but in the sense of the diagonal, which has been written by Leipzig. And this conclusion on stability, in fact, does not come from a series argument, because the small denominator would diverge the changes in the coordinates by itself, so it is not formally stable by Poincaré's method, but... Quantitatively, because it is through an inspection of the topological properties of the transformations in the storage of these points that Birkhoff demonstrates this theory. So, the conclusion, I don't know if... I also managed to demonstrate and understand what I had proposed at the beginning, which is that there is already this square... A mix of two traditional methods derived from analytical functions and topological methods, and also, secondly, that the definition of mathematical stability is motivated by the characteristic, by the very nature of the physical problem that it proposes to solve, that is to say, the mathematical definition of stability as a reality. However, neglecting transient behaviors is intimately linked to the very nature of the problem of stability, rather than physical problems. Maybe first, to make a connection with the theory of physics, to think without risks. I would like to remind you that Joan Maudrin had presented her work on George Descartes and I told you that at some point, Descartes had completely reformed the ways of the Syriac.

1:05:00 By reproducing abstract species and structures, we end up at the root of this and precisely the section of Pocahontas, because this is the effect of what Jérémie Christel had defined. Either by a dynamic associated with differential equations, or by a dynamic associated with the transformation of points, which will lead Birkhoff to say that a certain dynamic can be defined by the action of a discrete or continuous group in an abstract space. And that's what he's talking about. I think it's quite interesting as a passage in relation to my day, At the same time, there is the introduction of science, of mathematics and physics, which is very different, but at the same time, it leads to the transformation into a whole, so I think that there is an interesting passage in relation to the question that we have to ask ourselves as well. This book is one of the methods in which Birkhoff's motivation can become a general characteristic in the definition of any dynamic system. Since from the moment a dynamic system can be defined by a dynamic system, continuous or discrete, and precisely from the Poincare section, we need to have a general definition. So he makes an effort to give a definition that can be valid in both cases. But in fact this definition was not widely used, even today. And the definition of group, action in a group, for example, which I think is a very interesting definition, I don't know if you can see it. Yes, but if you want, it's even more than that. It defines a space with physical properties.

1:07:30 All the definitions are completely abstract. Yes, but I agree that it is an influence, a general definition that Victor proposed, but I don't see it in the methods themselves. In the fields of mathematics and in the fields, this does not work at all. Every time we take a differential equation or a point transformation and we can't see it. I don't see... I don't see a very important role of this cosmopolitan thought in the theory of physics. At least, from the examples given in the other seminar, yes, there we begin to have, by using classifications, when we want to classify all the courses defined by all the differential equations, how they can be studied in general cases, there, yes, we have an intervention of the differential equations that we used before. I don't know, it's ideal, it's true, it's true, it's a matter of time, but I don't think there's a more important question in the methods of... Can you give me the word? I also see a link between your presentation and mine on the question of the functioning of the ensembles. An example of the functioning of ensembles is an example of the functioning in the paradigm of variable magnitude. A case I had not addressed, which we find in your presentation, is the case of the infinitesimal magnitude. I need you to pass the panel to explain. You explain, you come for a moment. We look at the approximation in order 1. So we look at the behavior of neighbors in one point. And so, of course, the term neighbor is so ambiguous here, since there is the term, there is the term really assembly of neighbors. Yes, no, here it is too limited. And we can study the restriction of the function to a neighborhood, and therefore say that behavior is of this type, that is to say that by a kind of local idiomorphism, we can refer to this, for example. So this is neighborly or non-neighborly. There is a real neighbor, a real piece, not infinitely small, such as Pabla, and there is the non-neighborly version, which is the one that has existed throughout the 19th century, and all the same at Poincaré, including in his local study, not local in the other sense of the word, but for that matter, it's... I'm looking at... Wait, can you see or take another one? Yes. I'm going to throw this one. Give me the video.

1:10:00 The other meaning is, of course, something else, that is, this function, defined in this point, I only look at it in one direction, so I use an approximation. What I look at is not a conventional behavior, it is a behavior on an infinite neighborhood of images. So it's quite something else. And this figure is more than metaphorical. It's the infinitesimal spread of the situation, not just the local spread. Finally, to make the link, I'm not going to go into it, but I'll go into it later. To make the link between the two, there is a generation after, for example, the link between the normalization of F' and the interstitiality is the period of localization, so there is really a period of passage from infinitesimal to local, so from the infinitesimal neighbor to the ensemble, from the neighbor to the ensemble, to the interstitial between the two, it is the Horses' theorem, on the characterization of the term degree 2, and that really, it spreads like that by a few years, and so on. And so, here, we find in this idea that people look at order, time, and have a local behavior, in quotation marks, an example that unfortunately is not yet in its place, but I don't care, I don't care, I don't care, I don't care, I don't care. Yes, but at Poincaré, there is no absolute difference. Sometimes it is really local or non-local. There is a real local regression, but sometimes it is local or non-local. We look at the number 1, the number 2, the number 3, and that's it. In fact, with this example, I would like to show an example that is not at all non-local. To show that here, I think that it is really a geometric concept that is at work. I don't think there is, even up to Birkhoff, what I say about Birkhoff, a concept of a geometric and an analytic thought.

1:12:30 So how did Birkhoff start? Still, there is the idea that all the factors are interlinked, right? Yes, that's right, but the idea that all the factors are interlinked is completely wrong. That is, the idea that all the factors are interlinked is like this. When we analyze, we can analyze the trajectories on a torus, the torus that I gave, there are trajectories on the torus. So we can analyze... How does this trajectory behave? And each time, this is a method that comes back at Pontaré, we make a curve and we will see what are the intersections of trajectories with this curve. And so, to know if the curves come back in the arrangement of starting points or not, we need to know whether the distribution of these points on the surface is a perfect set, in the sense of Cantor, or not. So it can be said like that, that is to say, for the geometric consequences of the fact that the whole is perfect. Yes, but it's still a tool that comes from somewhere and that is interested in something else. Some kind of topology treatment? Some kind of topology treatment. It's more or less topology. Yes, no, no, when you talk about topology, it's topology in the sense of what you look at, the object you look at. I think I can understand the point, and we have two things. On the one hand, it was important to promote set theory because you see that this kind of theory is studied by many points that have important applications. And on the other hand, I understand the point that there is no use of satiric thinking or satiric definitions there, just one thing or the other. This is where quantum and topology come in. So, do you think that, in the end, when you have a dimension of a thumb, you can loosen the constraints?

1:15:00 In the case of mathematics, it is done in dimension 2. When we are in dimension 3, we are in continuity, we do not need anything more than mathematics. So we gain something when we go to the dimension of work. Of course, there is work, but does that mean that the need to go to the dimension of work, at the same time, gives us the ease of doing constraints in relation to the project or the precepts? I think that what makes it easier for us in 2030 is the physical nature of the problem. I would say that. Because both topology is required, but it is also allowed. That is to say, topological answers are satisfactory because we are only interested in the behavior of the university, the behavior of the students, the recurrence, the asymptotic, etc. So topology is denied by the physical nature of the problem, and it is required in a certain fashion by the... It's not enough to have topology, it's enough to have topological evolution. Yes, I know, but what I mean is that topological evolution, which is weaker than the other, suffices anyway, and it affects people anyway. But on the other hand, the introduction of topology is required because we can detect the co-dimensions, which is the main thing. That is to say, we do not demonstrate a theorem like Pocahontas-Bendixson because we do not have the co-dimensions. That is to say, we cannot say that this gives us space. So these are really algebraic properties that we discover in two dimensions that make it the same as when we arrived in 1802, 1806, 1802, and that we don't have the same intervals in two dimensions as we do now. And so we have to come up with new methods to realize, to try to describe the shape of the curve in all spaces, but in fact we can't describe the shape of the curve in all spaces, so we're going to try to describe the shape of the curve in the order of these three groups. They all have a specific type of reference, which is easier to study and is still interesting because we can solve these types of problems. I would like you to specify that what you call the physical problem is the problem of astrology.

1:17:30 Or is it more general? No, it's the problems of the stability of the solar system and the problems of the stability of the figures of equilibrium in our universe. These are the problems. Well, the stability of the solar system and all these sub-problems, the problem of the three gates, the problem of the question of the three gates, the problem of... All right. All right, so it's because of the physical problems of a different system of... No, no, these are the problems. No, no, I'm talking about these two problems, which are the problems that are at the origin, and it's very good. The law of Poincaré and Lyapunov-Virkov are part of these two problems. They are very useful. I see that there is a new quality character in the solution of these problems, here. Is it possible to look at certain characters without the physical concept of stability? Only the mathematical solutions, such as the problems of the debate, for example, exist? No, in fact, the physical problems here have a special character because the stability of the solar system is necessary. This, since the beginning of space, has been reaffirmed by Poincaré to demonstrate Newton's law of validity. That is to say, it is a theoretical problem par excellence. It is necessary to demonstrate the solar system to solve the epistemological question on the validity of Newton's laws. So, it is necessary to know that the solar system, it is not necessary to predict the composition of the solar system, but it is necessary to know that the solar system does not go infinitely, but for an infinite analysis... It is to distance a lot, to distance a lot from these conditions of departure.

1:20:00 So, the physical characteristics that I am talking about are cellular. That is to say, there are people with very theoretical physical characteristics. Well, that's it. That's it? That's it. For people who are in mathematics, if there is no secret for others, we will discuss one of the axes of our work, the history of mathematics and physics. So those who feel concerned about ASEIS are welcome, those who are not concerned may disappear. I called on Marie-Josée to discuss mathematics, physicists and physicists in the second half of the 19th century, and I said, let's take the opportunity to discuss the colloquium that I was going to do next year on this subject. So we have a slightly intra-team discussion. So all those who feel concerned are welcome, and those who do not feel concerned, well, we'll tell them next time. But did you want to say that there are five questions left? Because I have four messages for the candidates on the stage, so the minutes are... At the end of the hour. Oh no, no, later. What? So I have to print these things and I have to bring them to dinner and we'll talk about it later, in a minute. No, we'll talk about it later, in a minute. No, we'll talk about it later, in a minute. No, we'll talk about it later, in a minute. No, we'll talk about it later. But I have to start with what? Confidentiality. Ah, okay, perfect. Okay, so I'll reply to four messages that are used in all the seminars and we'll come back to that. Thank you for watching.

1:22:30 Oh, excusez-moi. C'est le 1er février de l'HP, ce séminaire sur Descartes, sur la physique, la rapport entre la géométrie et la physique. Notices of the... Ah, yes. The premier failure. The seminar which is organized by... Yes, exactly. Yes, exactly. It's this coming... The first of two weeks. The first of two weeks. So you may be interested. No, it's interrupted. I'm sorry. It's a long day. I'm afraid I wasn't listening at all to what Marie just said. She mentioned another seminar. Well, we wanted to have a meeting about research. Dealing with laboratory resistors. Purely internal. Yes, so it's purely internal in the sense that you can stay if you want. Oh, no, no, no. Very interesting as I'm sure it will be. It's just a business meeting in Minnesota. I think I've had, it's very much important. I'm really glad I came to Paris for this. This is an extremely interesting meeting. I enjoyed all the talks very much, all of them it would be invidious for some level for now, but especially, well all of them, but I have to say that one of them was very particularly, I don't know if you can take a great deal, but especially... They're all extremely impressive. Okay, well, it's very nice to meet you. Nice to meet you. And I'll see you certainly again, I hope, here in Recife, Simon. I think I'm going to come again on the... I've just been looking through the diary. I think the next time I'll be able to come will be on the... The 20th of February. Yeah. Oh, both of them.

1:25:00 Mathematical foundations of physics of the 18th and 19th centuries. It's Euler and d'Alembert. Yes, two talks about Euler, one on d'Alembert and one on Hilbert, which is slightly a shift of chronological, quite a major. So they all look very interesting, so I feel satisfied getting to that. Thank you very much. Excellent. Nice to meet you. That's the one.