 So, it's useful in this sense to look a little bit to Poincaré's career, and so I will be talking about here, about 1898, when you write a paper on the stability of the solar system. And so, Poincaré before that, he started to work, he was very much interested on one problem, which is the equilibrium of fluid mass, and that was after reading the work of Georges Darwin in 1880. And he started to work on Linsted equations, and this started really on a small mathematical equation, and he looked to the divergence of the series that were used in this resolution of this solution. And it's through that, that after, he was interested really in celestial mechanics problem, and which culminate with the paper given by the price of Oscar of Sweden. And so it's in which was then published in 1890 on the problem of the three-body and the equation of dynamics. And after that, he was really involved in celestial mechanics, and in which main work is the three volumes of the new, the new method of the mechanics. And at the same time, if you look to his professional activity, he was the first at the Sorbonne Professor of Mathematical Physics and Probability. But then in 1896, after the death of Tisran, he was on the chair of Professor of Mathematical Astronomy and Celestial Mechanics. This chair was in fact made for Le Verrier, and was then occupied by Cauchy and Puiso. And this is also for his academic career in 1887. He was elected member of the Academy of Science. And then through all his involvement in celestial mechanics and astronomy, he was also elected in 1993 as a member of the Bureau de Longitude, where he occupied himself with lots of practical problems of astronomy, but also of geodesy, and really linked to practical astronomy, I would say. So the paper I'm starting is in fact published in the Annale, the Annuaire of the Bureau de Longitude. It was when he was there. And at the end of this Annuaire, there are some scientific papers for general audience. And this one was published in 1887, and it's on the stability of the solar system. And this paper was in fact, you know, it's a popular paper, but it was reprinted immediately in three languages. It was reprinted in a scientific journal in French, but it was also reprinted in nature, and it was reprinted in a German equivalent journal in science. So I take the translation in nature here of the same date. All persons who are interested themselves in the progress of celestial mechanics, but can only follow it in a general way, must feel surprised at the number of time demonstrations of the stability of the solar system have been made. Yes, usually one thing that a single demonstration is sufficient in mathematics. So that, he says Lagrange was the first to establish it. Poisson then gave a new proof. Afterwards, other demonstrations came, and other will still come. Were the old demonstrations insufficient, or are the new ones unnecessary? The astonishment of this person would dopless be increased if they were told that perhaps someday a mathematician would show by rigorous reasoning that the planetary system is unstable. So I think it's, what is very impressive is how in a condensed way and in a very simple way everything is written. In fact, the problem of the stability was a very big problem in the 19th century. But it was also a very big problem in science in the 18th century. And it started with Newton, and with Newton when he wrote in the, in the optic books. And this is the version, this is the second edition in English, because there was a first edition in 1706. But this was in Latin, and I don't read Latin, but the, the paper I quote would be in the same as in the Latin edition. It's in the volume of optics, but in the volume of optics, at the end of the volume he writes a very thought about gravitation. It just takes the opportunity that he's making this publication to give his, his thought about gravitation. And in particular, this is what he writes. He, he, in fact, he will say that the fact that all the planet orbits are so well-ordered is for him the evidence of the existence of a creator of God. Because it could not be just by your blind faith. You see, for a while comets move in very eccentric orbs in all manner of position. Blind faith could never make all the planet move one and the same in way in. So that's the proof of existence of God for him. But then he continued. He helps concentrate some inconsiderably regularity accepted which may have risen from the mutual action of comets and planets upon one another and which will be up to increase till this system wants a reformation. So that's, that's the other part. He thinks, in fact, he, since you have Newton's law, he knows that the motion is not Keplerian, that the planet, if you have a single planet around the star, okay, you have a Keplerian motion. But then when you have two planets, the motion, the orbits are disturbed by the presence of the other one. He cannot compute precisely this perturbation but he wonders whether this perturbation will disrupt the stability of the system. And in fact, at the time of Newton, there were evidence that it was the case. Because they were all the observations that were made by the Greek and the observation made in the 16th century. And they have been analyzed by Kepler in 1725. And Kepler found that, in fact, Jupiter and Saturn were not behaving properly. Jupiter was going towards the sun and Saturn was going away from the sun. So there was some evidence of instability. And I think that when he speaks about this inequality, this alternation, it's referred to that. And this was a big question. In fact, this was the cause of a dispute with Leibniz. Leibniz thought that this was really thinking very little of the power of God. And he warned, he sent a letter to the Caroline, the Princess of Wales, to warn her not to listen to Newton's word and to the follower. Newton and his follower have a very odd opinion regarding God's workmanship. According to them, God's watch, the universe would stop working if he didn't rewind it from time to time. He didn't have enough foresight to give it perpetual motion. So you see, that's the point. He cannot, Leibniz kind of thinks that in Newton's word, God is a very poor watchmaker. So and this cannot be true. The power of God should be infinite. And he should have made a solar system that is stable, infinitely stable. So you see, you have this big problem because stating the stability of the solar system is basically stating the existence of God. So this is why this problem was so important, or one of the reasons why it was so important at the time. There's also another question, which was whether Newton's law could actually explain completely the motion of the celestial bodies. And this problem was solved by Laplace. In fact, in his comment, Poincaré forgot Laplace. And this is not right because the first one to do it right was Laplace for the stability of this size of the orbits. But then Lagrange, Poisson, you see, this is the continuity of this work. And what was the outcome of the work of these people is to say that the orbital semi-major axis are not strictly invariable, but the variations are limited to small amplitude oscillation around their mean value. This is the word of Poincaré. You see, this is what Laplace showed, in fact, is that the size of the orbit is strictly invariant except some oscillation, which one of them may have a large amplitude because of a closed resonance between the motion of Jupiter and Saturn. And if you had observation about 200 before Christ, the Greek observation, and new observation here, you had the impression of a trend that which was just because of this big oscillation due to closed commonsurability. And in fact, that was a very important result from Laplace because once he did this and once he explained this big term here due to the closed resonance with Jupiter and Saturn. He was able for the first time, just with Newton's law, to recover all the observation of the past. And he could recover the Greek observation within one minute of arc, which corresponds to the observation with naked eye. And this led to a problem that, in fact, with a single differential equation, you can if you modelize everything, you modelize everything, but then the only thing which counts is the uncertainty with which you get the initial condition. And this is Laplace-Demon. Maybe we can lower this a little bit. Which is an intellect which, at a given moment, would know all forces that set nature in motion and all position for all it and that composite would embrace in a single formula the movement of the greatest body of the universe and those of the tiniest atom. Nothing would be uncertain and the future, like the past, would be present before its eye. So can we lower a little bit the screen? No? Okay. If it's not possible, it's not possible. I asked it before, too, but it's okay. So you see, this is directly with respect to the previous result, which is that with this differential equation, you know the initial condition, you will be able to predict all the future and to recover all the past. So I will come back to the stability of the semi-major axis and, in fact, do it the way Poincaré does it when he is writing in the method Nouvelle. And practically, how do we demonstrate the invariance of the semi-major axis? So you have the energy that depends on the semi-major axis on the longitude, lambda is for the longitude, the motion of the planet on the orbit. On the other orbital element, eccentricity, inclination, longitude of Périelion, longitude of Nouvelle. So these are constants in the Keplerian motion. What is the equation of motion? The equation of motion for the semi-major axis. And here I am just assuming that these two are canonical conjugate, which is not the case, but it's not important. It's something like that. What is the proof of the invariance of the semi-major axis? You see the semi-major axis are not invariant, but they are invariant in average. How do you do that? You just take a transformation that transforms the variable to a new variable, and through this transformation, you get a new Hamiltonian that is just the average of the previous one over the longitude. If this is the case, now you look to the variation of the semi-major axis. But here you have average over the longitude, so there are no longitude here. So this in the new variable is zero. So this is a demonstration of Laplace, which was first given by Euler with some error, then by Lagrange with some error. Of course, this is not completed, because what you have written here is a variation in the new variable in the average problem, and you have to put the relation between the old variable, the real variable, and the new variable here. And this will be a prime plus wA plus this is just an exponential Lw, and w is your generating function, so this is just LwA, and then this is just an exponential series. So practically, this is a work which can be made as an infinite series, and these are the series of celestial mechanics. So this is the series of celestial mechanics, and then of course you search w as w1 epsilon w1 plus epsilon square w2, et cetera, where epsilon is a small parameter. In fact, epsilon is a planetary mass which is about 1,000 with respect to the solar mass. So what is the demonstration of Laplace and Lagrange? The demonstration of Laplace and Lagrange is this one plus saying that the old variable is a new variable plus term data of order 1 with respect to the mass, which is written here. So this is Laplace and Lagrange. Then there is a work by Poisson to which Poincaré refer, it's an 80 page paper which demonstrates in one line here, you can do it in one line, that just this average is 0. But of course Poincaré does it like that, Poisson does it in 80 pages. Just because here, we are in the right formalism. And then there was a big question, the term of second order here. So you see Poincaré say that, he said, this was what's shown by Lagrange and Laplace, but Poisson went further. He wanted to study the slow change of the mean value. He demonstrated that this change can be reduced to periodic oscillation around the mean value that was submitted to variations that were 1,000 times slower. This was a step forward, but it remained only an approximation. So the first proof of the stability is just this one. Second proof, this term is 0, remains variation of order 2. There was a big question whether there were still variation there, secular variation there. And in fact, this question came to an end when RSU showed that in fact this is not 0. The average value of this term is not 0. In fact, before that, Mathieu thought he succeeded to show that it was 0, but Spirou R2 then demonstrated that it was wrong. But what is the outcome? The fact that these terms are not 0 doesn't mean it means that there will be variation in the semi-major axis, small variation. But in fact, with all these series, there will still be quasi-periodic variation. And in fact, what Poincaré says, he says he had thus more condemned the old method than demonstrated the instability of the system. The question remains entirely. In fact, one needs also to show that the eccentricities are bounded because if at here you have circular motion for the Earth and Mars, you don't change the semi-major axis, but just change the eccentricity, then you can have collision. So you need to look to the variation of the eccentricity. And that was also done by Laplace and Lagrange. In fact, Lagrange here was leading the problem. He gave the right way to do it. And the way to do it is to reduce the problem to a system of differential equation of first order with constant coefficient. So this is made by expanding this Hamiltonian here in series with respect to eccentricity inclination, keeping only the leading term. And then you end up with a linear system of differential equation. Which solution is he demonstrated then that the eigenvalue of this matrix are all distinct and positive, and not positive, but real. And thus you have a quasi-periodic motion, like this one for the Earth. The eccentricity longitudinal peri-alien of the Earth is a sum of periodic term. And I will call the frequency G1, GN, like that. And the frequency for the inclination will be called S. So when you put that together, you get some variation for the eccentricity of the planet. You get that the orbit process, you have change in eccentricity and inclination. But these change were bounded. This was computed by Lagrange. And as Poincare says, concerning the other orbital elements, as eccentricity and inclination, they can show around and near their mean value larger and slower oscillation, but for which one can compute some limits. Of course, compute some limits to first order. So it's just here to show you how series are introduced in celestial mechanics and that they are truncated in a very crude way to give one result. Just to give you a better view of that, this is the real solar system. With the real planet, you see Mercury, Venus, the Earth, and Mars. This is the time here already 200,000 years have passed. You see the change in the, you don't see much the eccentricity. You see more that the orbit is shifted and that the sun is always at the center. So in space, you see more that there is some motion. At least you see that there is some motion that the orbits are not fixed, that you have a dynamical system, and then you need to wonder about the stability of these orbits. When you see them like that, you even wonder, how can they stay? You see, here the planets are not at their position. They are put at the position of Peri-Elean. And you see here Mercury, the Peri-Elean of Mercury is just changing slowly. For the motion of the other planet, it's a more complicated motion. And so you see that things are moving. And that's what we get at the end of the 18th century. And you see that there was this big effort, big effort showing that all this term were of average, of zero average. And this took 200 years. So what is the comment of Poincare? He said, all these studies have required great effort that seems useless today. The method of Mr. Gildan and those of Mr. Lincest, indeed, so far as you push the approximation, give only periodic terms, so that all orbital elements can only experience oscillation around their mean value. The question would be solved if this expansion were convergent. Unfortunately, we know that this is not the case. And we know this is since the memoir of Oscar, the memoir on the three-body problem done by Poincare, where he showed that, as was shown in detail by Rick Mockel two days ago, that these were these very complicated features in the dynamical system because the stable and unstable manifolds did not coincide and then. And what is impressive is that also he says that he will not describe this figure. He will not draw this figure. He is able to describe the figure in a complete detail. And you cannot believe that he did not draw it at some point on a small piece of paper because I cannot understand how we will be able to say something like that. You see this intersection from a kind of lattice of fabric, an infinitely tight mesh network, each of the two curves, the stable and unstable manifold, must not intersect itself, but must fall back onto itself in a very complex way to come across an infinite number of times, all the measures of the network. So he described exactly what arises. He said that orbit there will be highly unstable. Then the problem is like that. He lost the stability of the system. He lost the possibility to prove the stability of the solar system. So at one time, he made this theorem of recurrence, which in a sense is another form of stability. This is what he called stability à la poisson, but it's a very weak form because he just says that if you start within some initial condition, after a time which can be very long, it would come back in the vicinity of the same initial condition. And obviously that was not sufficient for the search of stability because he will come back to this problem and he will search for a different form of stability. And this is what I am speaking now because he will say that this material point, the problem with this material point is a mathematical problem and we are looking for the stability of the real solar system. So and the real celestial objects are not material points and they are subject to other forces than the Newtonian attraction. The effect of these complementary forces should be to alter gradually the orbit even through the fictitious object considered by the mathematician would enjoy absolute stability. And the force he is thinking of is the tidal forces, tidal forces in the solar system. And this is because it was after the work of George Darwin in 1880 and George Darwin has been the one who has explained why the moon is always facing us because of tidal dissipation among the Earth's moon system which makes the rotation slow down until it is synchronized with the Earth. And so we must then ask whether this stability will be more quickly destroyed by the simple effect of Newtonian attraction or by these complementary forces. So that's the big question whether the tidal forces, the dissipative force are more important than the basically the diffusion due to the nonlinearity of the neglected term in the expansion here. And then he says without wanting to code figure I think this effect, the effect of these complementary forces are much larger than those of the term neglected by analysts in the latest demonstration of the stability. So this one you see the term neglected, the third order term or fourth order. And what will be the outcome of this tidal friction? So tidal friction, the earth or the moon or the sun will deform the earth here make a bulge in the direction of the of the body here, the perturbing body. But as if the planet rotates faster than the moon for example around the earth then there will be a permanent offset in this bulge. And the attraction of the moon on the bulge no longer goes through the center of the earth and tends to slow down the rotation of our globe. But here Poincaré is wrong. The problem is that the effect is true, the effect exists. So there is no doubt that there is some this effect, the tidal effect is there and what Poincaré thinks is that this effect happened as long as the orbit are not synchronized with their rotation. That after an infinite time what will happen? The solar system would therefore tend to a limit state where the sun, all the planet and their satellite revolve with the same speed about the same axis as if they were part of the same invariant solid. So he thinks that at the end all the solar system to a system where all planets will be rotating as a solid with the sun. The problem is just the time scale. If you look, if you make a small computation you will find for Sun Mercury for example that in 300 million years the rotation of Mercury is synchronized with its orbit. But if you make the computation for example for Jupiter and you want to know in how much time the rotation of the Sun will be synchronized with the orbit of Jupiter you find that it's 210 to the power 16 gigahertz. So we have something but the time scale is not right. So that's end up and in fact this dissipative effect is much smaller than the diffusion due to the Newton interaction. But after Poincaré, so at the time here after Poincaré this is where Poincaré left it you know he said going further asserting that these elements will be not only stay for a long time now he referred to pure Newtonian problem without the dissipation but always between narrow limits this is what we cannot do. So the question of stability in infinite time the question of Leibniz you know Leibniz was saying God should have made a solar system that is stable in infinite time and this is what came back with KM theorem. So basically the question the problem is for the 18th and 19th century it was like that just pure rotation or combination of these pure rotations but then Poincaré showed that there were these complicated behavior in the vicinity of the resonance and then there was this result of Kolmogorov who demonstrated that despite all this exist there are still some initial conditions for which you have regular quasi-periodic solution that would be stable in infinite time so following Leibniz requirement. So and this rigorous result so Kolmogorov did it in a general way then Arnold showed that it applied to a two-planet planar problem one of my students Philippe Robutel showed that it was possible to extend it to spatial case and also there are some more recent results by Hermann and Feijot and Kierkepindari who showed that this was also possible for n-planet spatial case so everything is fine you can apply you can have infinitely stable quasi-periodic solution but only in a very small extremely small value of the masses and eccentricity and inclination which doesn't apply to the solar system so to get results for the solar system you need to put the equation on computer and the computer result will show the contrary it will show that it's not stable and that the first the first way to show it's not stable is just to look how diverge solution and in fact they diverge exponentially like 10 to the power t over 10 where t is in million of years which means that every 10 million years you lose one digit so if you start here with 15 meter accuracy after 10 million years you have 150 meter accuracy but after 100 million years you have 150 million kilometer error so that's the first thing it's seven words so because here you could say okay it's diverging exponentially but still if they have an infinitely precise initial condition I can go to infinity but the problem is the more you will try to be precise the more you will have to take into account feature and what we could show is that the presence of the asteroid and particularly of the largest of them Vesta and Ceres which are very teeny you know just around 22,000 of the earth mass this will strictly limit the solution to about 60 million years because these two asteroids perturb the planets and they have themselves a chaotic motion at a much shorter time scale than the planets what is 10 million years for the planets is just 50,000 years for these two bodies which means that after 400,000 years you don't know where are these two bodies so you have an error that you cannot reduce and if you want to improve a solution say for example that 15 meter error in the planet motion and orbital motion or initial condition of the asteroid give you a solution that is given valid of over 60 million years if you want to go further you will need to improve the motion of the asteroids which have a chaotic time scale of 50,000 years so if you improve everything by a factor of 1,000 you go to 15 millimeter then you will only be able to predict over 60 million and 150,000 years you say no wonder we go to micrometer level and I will let you guess how far we can go you can do it there are several ways of reasoning you can just try to understand you can just make a linear approximation and then you get the result in the same way so here we see that there is in fact a limit to Laplace-Damon and the only thing which was in fact foreseen by Poincaré is that in this case you have to look to not only to single trajectory but you have to look to the vicinity of this trajectory you have to look globally and you have to look you have to go to a statistical view you have to so you have to look more globally you I will not take now a single trajectory I will take a local vicinity of the trajectory and see the behavior of this trajectory in this case I know that after 60 million years the trajectory are not the trajectory of the solar system but I can go statistically and look to what are the outcomes so this is what we did recently with 2000 solution which we computed over 5 billion years for the whole solar system with everything in it that you can imagine it took a lot of CPU time you're about 7 million hour of CPU and what we get is this that's for the eccentricity of Mercury you see most of them for most of them the eccentricity change but doesn't this is the maximum value of the eccentricity doesn't go much beyond 0.4 but for 22 case 21 case here you had a very big increase of mercury the eccentricity when you look to one of these increase I look here to one increase it was in another simulation so it in the reverse time just to to see that you had this relatively irregular behavior and suddenly an increase to 0.8 so the first question was to understand why do you have this increase and in fact the reason is because you have you saw the perillion of mercury it was moving quite well quite regularly but in fact you didn't notice that it was about the same speed as the perillion of jupiter perillion of mercury is is 5 5.5 arc second per year perillion of jupiter is 4.25 arc second per year they are not so close but close but you have all the other interaction which make a diffusion of the value of the frequency of perillion of mercury so it will change and the problem is as soon as it goes close to the one of jupiter you will have a resonance and then you will have this big increase in the eccentricity of mercury we look more precisely to this you see here is an example you have in four million years you have this increase of the eccentricity of mercury and if you look at the same time to the semi-major axis you see that the semi-major axis has not changed it has it has not changed it really doesn't move until you go here but here it's because you have closed encounter with Venus so when you have closed encounter with Venus then you have any kind of thing so we look to how can we you know thinking of you know mathematicians they want always to be able to prove things so you want what here we have a dynamical system that is 30 degrees of freedom with phase space of dimension 60 lots of secular interaction lots of resonance but we wanted to look for a simple model to explain the eccentricity of mercury and i'm speaking of it because it went out two days ago so it's a good opportunity what we look is the minimal feature needed to explain this big change in the eccentricity of mercury and the minimal feature it's what what we do we average over the mean longitude which puts the semi-major axis constant but you saw that this was actually the case they don't move so we do things like that then we develop we expand we expand but we want to go to high eccentricity of mercury so we don't want to expand in in this eccentricity so we put it exact in eccentricity but we will use the other planet as a forcing term we take for granted the motion of the other planet so all them are just expanded up to degree one in eccentricity and even this is a lot of degree of freedom we will limit it to only one argument so you consider the motion of mercury perturbed with the other planet only in their component which is related to this resonance which we have already isolated as the leading term in that the problem is we still have to expand everything up to order 15 in order to be realistic because we want a realistic system so it's simple because you have only one degree of freedom left here one action variable and one angle so it's a one degree of freedom system quite complicated because you have all these terms that appear but they are just polynomial expression of the eccentricity just the computation is complicated the when you think of it it's just a one degree of freedom system you can just write the phase space and this is what it is you see you look over the picture of Poincaré it's a resonant you put it in resonance here and you see that you have this resonance if you have an orbit eccentricity is a distance from the center if you have an orbit which is here it will be of low eccentricity here but then it will go here to high eccentricity and when you compare the outcome here with the this is the true solution with dimension 60 phase space this is the dimension two phase space of this small smaller thing and you get exactly the same behavior with the same time scale starting with the same initial condition so it's just here that not only we get the numerical feature on the full system but we get the leading aspect of it and then you can make it a little bit more but here of course it's integrable you don't have chaos if you want to get chaos you just add a degree of freedom so we can get went to a spatial problem with just we add not a full spatial problem we just add one harmonic in the in the spatial problem because we knew it was important from other studies then you get a two degree of freedom system and then the phase space will be like that and then so you have as expected by your Poincaré and by others you have a chaotic zone around the separatrix here and then you will have intermittency between slow low eccentricity of mercury and high eccentricity excursion of mercury and when you look to that this is the kind of orbit you have you see you have diffusion of the of the eccentricity of mercury then you will go to resonance with Jupiter which will put into high eccentricity and then you may have then interaction with the other planets and coming back to the solution to this experiment we which we did I told you that we had 21 solution with high eccentricity of mercury so the one which went to resonance out of them we had six collision mercury with Venus nine collision of mercury with the Sun five rich five billion year before collision and one is interesting because there was a close encounter not a collision but a close encounter of of Mars with the with the Earth and you see this close encounter is at about 80 800 kilometer so it will be beautiful you know but again this is a this is a single trajectory this is a single trajectory you start here and just you you nearly miss it at 3.4 billion year you nearly miss it so what you can go you you go a little bit before three million year before you start here and you want to have a statistical view of it so you take a vicinity of initial condition and you look to the outcome of this value solution here you are in a region of highly chaotic behavior because you have the plenty of encounter and this is what we did with just changes 0.15 millimeter in the semi-major axis of Mars and this is then what you get you get anything collision of mercury with the Sun collision of Mars with the Sun of Mercury with Venus Mercury with the Earth Mercury with Mars Venus with the Earth Venus with Mars Earth with Mars and five ejection of Mars because Mars went too close to Jupiter and now I will just show you all this in two minutes I'm two minutes ahead of my time but I will nevertheless let it so this is the present behavior of the planet as before the time still here 300 000 years and you see that there will be the diffusion of the of the orbit of mercury diffusion of the frequency of the perealion until here you get two resonance with Jupiter which makes the orbit much more elongated and then you have you have increased by that you have increased the image the eccentricity of Jupiter and once it's increased you can collide with Venus if you want but if you don't do it you can then exchange angular momentum with the other planet and this is what happened here so you reduce the eccentricity of mercury but all the eccentricity of the other planets are increased and in this case you have all these these behavior not that you don't necessarily have collision because this is a projection of a spatial problem and to need a collision you need that the planet be at the same time at the same moment so you will have to wait a little bit and you will see for example that interesting thing can happen like here you have Venus here you have the earth when you will have close encounter between the two then you may have exchange between the two orbits and the orbit of you see it happen just like the orbit of the earth is now inner to the orbit of Venus so you have nice climatic change on the earth at the time and if you wait enough then you will have several of these close encounter and after you have close encounter enough you have a thing like that you see collision of mass with the earth and we try to figure out what it would look like you know we didn't had so it's not a documentary it's not and and you can even have this you can have a collision of Venus with the earth so you see and this was just by changing by 0.15 millimeters the position of mass so I think I can conclude on this sentence of Poincare that it can happen that small difference in the initial condition lead to very big change in the final result and you see big change it would be for us in this case thanks are there any questions or comments so because of the collisions the motion is not reversible so I would like to know if you made these kind of simulations in the past rather than the future yes I made in fact the first work I did like that was in 15 years ago maybe even more the more it goes the more it's before you know it was 1994 and I did it at the beginning using average equation average equation like doing this analytical averaging like that and I showed at the time that Mercury's eccentricity could go to very high value and could eventually come into collision with Venus but I could not demonstrate that it would go completely to collision because when you make perturbation series the when you go close to collision there is divergence of the series you even that means a collision in the past yes but just tell me I am just going to answer the question the problem is at the time I send the paper showing collision in the past and I even integrated you know I was too much influenced by mathematician and I thought that everybody would be the same and I was making an integration of the solar system 10 billion years in the past and showing that collision is possible six billion years in the past and that was too much for the editor you know that's because if you are doing that basically you have a dynamical system in this case a chaotic system like that if you integrate in the future or in the past it's exactly the same thing apart from the small dissipative term but it's exactly the same thing you have diffusion when you look at it you have diffusion the two in the two way so now after this experience I prepare to integrate in the future because of the did ethical I know that it's a more understandable for common people but of course you you can have the interesting thing here is when you look to the probability of collision or probability to have a very high eccentricity of Mercury it's about one percent so it means that over five billion years so it means that the probability that everything went smooth and a little bit like it is now is 99 percent so we have we have a very coherent view which means that you you would not be able to because the problem is if you if you had found that the probability to lose Mercury was 50 percent or 60 percent it would have been a problem you know how can you explain that mercury is still here so does it answer your question no no no I didn't have an answer if you take your film and you run it in the negative time it will look strange yes because the problem is to get the initial condition for the if I if I take if I take this are there accidents if I take this and I leave it like that I cannot go back I'm okay I understand that I don't have the initial condition to go back is it what you mean not really I mean you are suggesting that some collisions are possible in the future and some kind of argument you could argue that there are some accidents in the past that's right but I want to understand what kind of accidents would it be it could be for example at the end at the end of the formation of the solar system you what I you see here is that the system is unstable but it's marginally unstable I would say that it collision are possible but they are difficult you need about the time of the life of the system but at the end of the formation of the solar system you may have had additional planets and then the system was was much highly unstable any if you had anything there it would be highly unstable and for example you could have an additional planet around the around mass then you will have a collision and in fact we had the evidence that the moon was most probably formed by such a collision so and once you get a collision in fact the system after and I did the experiment with that is more stable because basically to get this collision you had to put all eccentricity in one of the body when you put the collision when you get the collision then you have removed a part basically a part of the nonlinearity of the system in this collision and you have a new system which is much more stable than the next one than the previous one other question because at the time scale you are working considering the sun as a stable object is really not realistic because you have a small small dissipation the loss of the mass of the sun it's 10 minus 14 per year so it would be here all already I may I put a dissipative effect which is the which is larger in fact which is the dissipation in the earth's moon system it's make a small dissipation in the system and but I didn't put the the loss of the mass of the sun because it was very it's very small it's still it won't change anything here but it makes when you do this kind of computation you want to be able that you are making the computation right so it was important to be able to track precisely the energy of the system and I didn't want to have complications due to the loss of the mass of the sun here all the energy is conserved up to 10 minus 10 at the end of the 55 billion year so that's very tricky and in some cases you you don't want to you don't want to add any additional problem but I don't think I could say you can do it again but you you have to waste seven million years of cpu just to verify that it's the same thing I come back to the question of 80 angis if you run the equations backwards yes which I do currently we did it forward and backward you get the same result no do you get a hint of the fact that the the earth and the moon at some point where the same object or just no because no because that's this will be you get in some sense a hint but that's when when you look you don't need to integrate over five billion year you need to integrate over a lower time then you get you basically input the present tidal dissipation in the earth's moon system but if you take the present tidal dissipation in the earth's moon system which is measured by your laser ranging from the earth to the moon you get very precise measurement that's three point the moon is going out that's three point eight centimeter per year you put this tidal dissipation term but if you run backwards with that with the the present model we are for tidal dissipation then you get a hit of the moon with the earth 1.5 billion year ago which is obviously wrong so the usual way to overcome this problem is to say that this we are in a presently in a particularly strong tidal dissipation because it depends on the organization of the ocean it's depend on the repetition of the sea so the the general assumption is that in average it was smaller in the past but you you have to make all this kind of assumption to to basically get a factor of three which would put back the the heat at 4.5 billion year but this is all unknown you know there is a lot of uncertainty we are so so you don't take this into consideration in this problem here this is why also making the integration in the future you don't have this problem because the more you go the less you have this tidal dissipation and you can you can modelize it in a very simple way the problem i would say in general is the question was what the problem is there is some dark matter the problem from the beginning is how good is the dynamical model that is used here that that the question and that's the question which we we wanted to be very sure of so i show you computation that are made of a billion of years but in order to get the model and in order to get the initial condition we also build a complete the most accurate model you could think of which means a model that take into account all possible observation which means so it's a model that is run for for for space research and for astronomical reduction of observation this model is directly compared to observation which means we observation are for example there is a spacecraft around mars which gives signal to the earth and you get the distance from mars to the earth at precision that is below five meters for the venus it's the same for the moon we have a few centimeters due to the laser ranging so we put all these together something like 50 000 observations we have all this in the model all the comparison to all this and from this we cannot discriminate we cannot put any additional change in the we don't need any additional change for the gravitational model for example we could show that some alternate gravitational model like a mond model are not possible because of the constraint given by the dynamics in the solar system so so that that's the point at the level of the solar system we don't have the space for any additional model and that's by comparing directly to observation so the end end this high accurate model is used is integrated for one million year and we use it as a start to fit this long-term model so they compare the two of them over one million year compare very precisely so it's not it's not a toy model it's a model that is the in best agreement with the best observation of the planet that we can have now