 Prvso editora, da imam svih pravidov in izglade Stefanova, si svojo začaj za to zelnostavne konferenc v toj delavne tukaj. Petroča moh dveje tebe kot 16.000 kožajstanjice na planejtari in kako sem prav, in se se vsega plikacije uporil. Mojeh, da imajo lahko nekaj delov, neko se zelo izgledaj, zato potrebno, da sem začala njega tukaj, 3,5, pa tudi na vsega, na vsega izgleda. Iga vsega, da sem začala, Pojinaj del taj, kur worodru tenemos, je neko, da reflectioni oddod lega na konce 급onana s экранnje오 in stiglave zašpevajnega. Nešto je Caseg approaching, ki se moj tudi taj našto zjad enduretna? Daj v kvalitah rev꽑nega boj v kanalju svato glavu in kemo Итакu, če so jednak Ext Ministeriali karkega sodiumenje v butterflyvara Saʻ Situation. Zvukam o aplikacijske vse, kaj je tudi svojo zvukovana teori protubacije. Zvukam, da je tudi več več vse, kaj je teori protubacije, ki je tudi za to problem, nekaj, da je pričel, nekaj je pričel, ker problem je hljev, da generuje in protubacije seori je zelo vsočen. Argumenti je podobno do vse, da Bassam zelo v zelo prijev. In tudi, da je vse pravite, da je več tehnika. Zdaj imam vsega vsega vsega vsega vsega vsega vsega vsega vsega vsega vsega vsega vsega vsega vsega vsega. Viicer sličili mi, da je vsega inzretilija nežalovicija. Tone inzretične koncijovate v Rahmanih, možemovih,lajših, inzretične, zato različi v gravični. Tova problema je hamiltonijan, je koncerativna. zna Miltonian, če je zemljanoj zelo, zelo je zelo. Planetirne probleme je, kaj je zelo, da je zelo, ki je zelo, da je m0 vzelo, je nekaj jazelji, ki so imamo so 1 son in planet. Planetirne mesi je nekaj nekaj, da je zelo, nekaj nekaj nekaj, in je nekaj nekaj, je zelo, bila parametri, ki je tudi zelo v mu. The V will denote the positions of the planets, and the U, they are conjugated momenta. With the 1 plus n particles, we shall have a system with 3 plus 3 n degrees of freedom, because we have 3 degrees of freedom for each particle. The standard, the symplactic form is the standard one. Problema is many symmetries, and the first of them, or maybe the simplest, to be eliminated is the symmetry by translations. There are many ways to eliminate the symmetry in a Hamiltonian way, and the one that I have written here is usually called the heliocentric reduction. This consists in to, the invariance by rotation is caused by the fact that, for example, the motion of the center of mass of the system is known. So it is possible to switch to a new Hamiltonian that governs just the motion of the planets. We have here new coordinates, the y and the x that are reskelled coordinates for the heliocentric relative positions of the planets. Precisely, ci times ci will be the heliocentric position of each planet, where c and the yi over ci, they are conjugated momenta. The y and the x are summarized coordinates that I have introduced here, in order to have simple formulas in the following. The Hamiltonian splits in two parts, a leading part that represents the interaction of each planet with the fixed star in the origin of the system of coordinates, and the small perturbation that is of strength mu that represents the interaction between one planet and one another. The c and the b are just constants that depend only on the masses, and our order one will respect to mu. The symplective form is analogous to the previous one, but has just 3n degrees of freedom instead of 3n. After the reduction of translation, the system has still other symmetries that are important for its symplectic description, and this is in the invariance by rotations and by reflections. The invariance by rotations means that we can change all the y and all the x by the same rotation matrix, and we shall have the Hamiltonian to be preserved. And it is also possible to change all the y by some rotation, and all the x by another rotation to have the Hamiltonian preserved, but this factor will not be used here. The invariance by rotation is ruled by the existence of three independent integrals. These are the three components of the total angular momentum of the planets. They are three independent integrals, but they do not commute. So we should expect that the number of degrees of freedom can be further reduced, but not by three units, because the three integrals do not commute. We shall see that it is possible to reduce the number of degrees of freedom at most by two other units. Then there is the invariance by reflections. We change the signs of all the y, of all the x, or both of the components of those in the same way. We shall have the Hamiltonian still preserved. Let us look at the integrable part. The integrable part is this expression. When all the imperturbated energies are negative, as we all know, the motion evolves on Keplerian ellipses. The energy levels are compact, and using the method by Jacobi, it is possible to introduce by the theorem by Ljubil Arnold and Jost, any set of classical coordinates that are action angle coordinates, such that each of the imperturbated energies is just a function of only actions. But here appears the first of the degeneracies of the problem. While at the beginning we have, each of these Hamiltonians has three degrees of freedom, after the integration, the new Hamiltonian will have just one degrees of freedom. There is just one only action coordinate in this expression that corresponds to be the square root of the semi-major axis of the ellipse. And so the integrable part of the whole system will have just an action coordinate instead of 3n. And this represents a problem if we want to apply a n theory to the planetary problem when we regard it as a perturbed problem. It looks like if here the integrable part does not depend on all the i, but only on a part of them. And so this effect is due, as it is well known, to the fact that the two-body problem has too many integrals, it is super integrable. And if the canonical elliptic elements after the Jacobi methods are usually called the Deloné coordinates, and besides the action L, capital L, the remaining five elements are represented in this picture. The small L is conjugated to the capital L, and this is an angle that is proportional to the area that is panned on the ellipse. Then there are two actions and two angles that are here denoted as capital theta, capital G, small theta and small g. The two angles, G, the theta is the longitude of the line that is the intersection of the plane of the orbit with the prefixed plane. This is defined when the inclination of this plane with respect to the prefixed plane is not zero. And the small g denotes the argument of the perihelion of the ellipse, and this is defined when this ellipse has an eccentricity different from zero, when this is not a circle. The two actions, G and theta, are two components of the angular momentum of the single planet. The capital theta is the third component, while G is its Euclidean length. So, we all are used to the fact that after integration, all the angles disappear from the Hamiltonian, but we are not used to the fact that also the action disappear. And here, in the two body problem, this is a remarkable fact that happens because also the two angles that are conjugated to those two actions, they also are integrals in the amperturbed problem, but they will be not integrals, of course, in the whole problem. The deloné coordinates are canonical coordinates for each planet. So, if we collect them all together, we shall have a system of coordinates for the whole system. In the end of the 19th century, Harry Poincaré introduced a new set of coordinates named after him, that in order to describe motions when the eccentricities may vanish or also the inclinations may vanish, those configurations are singular for the deloné coordinates. This is the definition. Instead of the coordinates theta and G, here we have new coordinates p and q, they are not action angle, but they are rectangular coordinates and are defined in a bowl around the origin. When the first couple of p and q vanish, this corresponds to the vanishing of the eccentricity. When the second couple of p and q vanish, this corresponds to the vanishing of the inclination. The coordinates are again canonical and the remarkable fact is that the transformation from Poincaré coordinates to, for example, the y and x is smooth. It is even analytical. The importance of Poincaré coordinates is here that if we consider the average of the perturbing function with respect to the angle l, italics l, these l are more or less the same as the mean anomalies, those l here. Those l here just shifted by a constant. If we consider this average, this is the average in the sense of Lagrange, or from the point of view of dynamical system is a sort of time average. By the symmetries of the systems, by rotations and refraction, what happens is that the origin, so the configuration with all the motions on circles in a single plane, this point is an elliptic equilibrium point for the averaged Hamiltonian. So having an elliptic equilibrium point, the very natural attempt would be to try to construct a bulk of normal form for the averaged system. So having an overall old system with the integrable part that depends just on one part of the action. But the perturbing function having an average may be in a bulk of normal form of some order. Arnold in the 70s formulated an abstract theorem where he that if applied to the Hamiltonian of the planetary problem might lead to state the existence of quasi periodic motions for the majority of initial conditions. For a set of initial conditions having a positive Lebesgue measure. At least if we are in a zone of the phase space where the eccentricities and the inclinations are sufficiently small. So in a smaller region around the elliptic equilibrium point of the system. The proof of this theorem took more was more difficult than expected because in fact Arnold proved it in 63 just for the first non-trivial case the planar three body problem using just one correct coordinated. And the difficulty of the expression to the other cases was caused by the stronger the generacies of the problem. The problem for the theorem for the spatial three body problem was proved just 30 years later by Laskar and Rubjutel. And they used a new system of coordinate in order to bypass the generacies. The theorem was completely proved by Arman and Fezios in 2004, but they used abstract arguments in order to reduce the generacies. Their proof is indirect. And finally in 2011 in a joint work with Luigi Kierke we proved this theorem finding a new system of coordinate that allowed to overcome all the generacies of the problem and we also obtained a sharp estimate on the measure of the invariant set. The Arnold theory that is used by Arnold, Laskar and Rubjutel and Luigi and myself is basically settled out by Arnold in the 60s and it requires the following that the average the system is in bulk of normal form up to the order two and that the matrix of the second order term is non-singular. This condition could be applied directly to the system by us. The theory that was applied by Arman and Fezios is weaker and also Basam talked about it is and it requires non-planarity for the frequency map and it just looks only the first order in variance, the coefficients here in the first order of this function and not the second one. It is a weaker KN theory, but it cannot be applied to the theorem directly because of the generacies of the problem. What are the generacies of the problem? In general bulk of normal form, the construction of bulk of normal form requires a condition of no resonances on the frequencies on the first order terms on the frequencies in sigma one. While for the planetary problem these no resonance conditions are not verified, there are two resonances that are verified identically for any value of the L. The first one was known to Arnold and he knew that this is due to the variance of the problem by rotation and the second one has been noticed in its full generalities by Michel Arman in the nineties and this prevents the two identities, prevents the construction of bulk of normal form and also they prevent to check Hermann's condition. Hermann's condition is not very, this is just the country, Hermann's condition is not verified. The rotational degeneracy is even worse than that because if we look at the coefficients at any order of the bulk of normal form, even formally, if we try to construct it formally, we discover that all those coefficients when one of the indices is equal to two and vanish identically. For example, if we look at the torsion, we shall have one row and one column identically vanishing and determinant would be different from zero. So this violates Arnold conditions. The idea is that the generality of the problem allows for more freedom in the choice of coordinates because the Keplerium ellipse is an object that we can regard as described as by four quantities, the two L that we want to remain fixed, not to change the part of the problem and then the angular momentum of each planet and its perihelium. The angular momentum has three independent components. The perihelium is to be perpendicular to the angular momentum, this is the perihelium direction, so it's Euclidean length equal to one, it's just introduced one other quantity, independent quantity. So we have four independent quantity that have to be described. And all what we need is just a system of coordinates that here I denote as L, L, Y and X, where the capital and small L are just what I said before, while the Y and X are two and couple of coordinates that are needed to describe the planets perihelium and their angular momenta. And moreover, we require that the system of coordinates is canonical, this means that the standard two form is preserved. System of coordinates of these are the Delonante and Koraipo ordinace, as I told about, they are widely used in the literature, their application is countless in celestial mechanics. And for the two body problem in the 19th century, there is another system of coordinates that was developed by Jacobi and Rado, that reduce the number of degrees of freedom by two units, they reduce completely the invariance but since they are available just for two planets, but since they are coordinates not on the whole phase space, but just on a manifold, on a lower dimensional manifold on the phase space, for more than one century, it has been very difficult to extend this coordinates to the case of more than two planets, and this is the heart of the difficulty of the extension of Arnold's theorem to the case of more than two planets. Another system of coordinates that has been quite forgotten for many years is a system of coordinates by Boge and the pree, they were developed in the 80s by Francis Boge for three planets and André de Plis in general, their application up to nowadays are unknown, the only application that are known to me are by two researchers, two Spanish researchers in the case n equal to 2, but in that case, the coordinates by the pree and Boge coincide with the one of Jacobi and Rado, so they are not, this case is not really exhaustive. This coordinates by Boge and the pree were worked out even about 10 years previously to the work by Hermann on the planetary problem and to the result by Hermann 20 years before the result by Hermann and Peugeot. And they were by fortune rediscovered by myself during my PhD, while I was working to the theorem, to the Arnold's theorem, and starting with them it is possible to produce a new system of coordinates that with my advisor Luigi Kertia, we call it the RPS regular planetary and sympletic that are analogous to Poincaré coordinates, but allow as an advantage with respect to Poincaré coordinates allow to bypass the degeneracies of the problem, and so allow for a direct proof of Arnold's theorem. The last system of coordinates that is of this year is what I call the perihile reduction that gives the title to my talk. Just a quick look to how the Boge and the pree coordinates, they are action angle coordinates that are denoted as L gamma Psi, small L, small gamma, small Psi, the two L are the same as before, the gamas are more or less the same as the Deloné coordinates, the most important coordinates are the two Psi, they are very different from the theta of the Deloné, and above all there are two couples that are very important, those are two couples of coordinates such that the two angles capital Z, small z and small g are both cyclic into the Hamiltonian, and also the capital Z is a cyclic coordinate, so here we have three cyclic coordinates, and the number of degrees of freedom is so lowered by two units. They work in the following way, we consider the angular momentum of each planet, and the so-called partial angular momentum as J, these are the sums from one to J of the C i, the s n, the total angular momentum is an integral, the three components of it are all integrals of the system. The first step is to put the system in the so-called invariable frame, this is the frame with the third axis in the direction of the total angular momentum, there are two coordinates, the capital Z and small z that describe this rotation, the capital Z is the third component of the total angular momentum, the small z is the longitude of the plane by some reason shifter by pi over 2, by some historical reason, is the longitude of the plane that passes through the third axis of a prefixed free system and the total angular momentum. Those two quantities are two conjugated coordinates and they also are integrals, they do not move, so they both disappear from the Hamiltonian. The disappearance of these two conjugated coordinates is just the origin of the degeneracy by rotation that I mentioned at the beginning, the vanishing of one of the first order eigenvalues of the torsion and so on. Then here we consider the total angular momentum of the system, the sum of the first n minus one and the last one and the triangle, the plane that passes through this triangle is its position, its position in this frame is ruled by the angle small g, that is again a cyclic angle because it is conjugated to the capital g, that is the ucalian length of the angular momentum. So, neglecting this small g would correspond geometrically to fix a system of coordinates that rotates around the angular momentum. And when n is equal to two, here we are just in the situation of the Jacobi reduction of the nodes, but the power of the precordinate is that this construction can be iterated. In this case we introduce some new coordinates, the psi and the small psi. The precordinates are not defined when the inclinations are equal to zero or also as in the case of the when the ellipses are circles, when the eccentricities are zero, because when the inclinations are equal to zero, the s and the c are parallel, one to the other, and those angles are not defined. So, in order to regularize these coordinates, I introduce the new system of coordinates at the top of one karelite, they are even denoted with the same letters. And the advantage is that two couples of coordinates are cyclic, they disappear into the Hamiltonian, and the degeneracy by rotation is so removed. With the system of coordinates, the system has a 3n minus 1 degrees of freedom, not 3n minus 2. There is an extra degrees of freedom that is the price that we pay due to the fact that we want to regularize all vanishing eccentricities and inclination. This leads to the complete proof of the Arnold's theorem, because now the overaged perturbing function can be put in Birkhoff normal form, and the Birkhoff normal form has a nontrivial twist. What are the remarks? These coordinates are not defined when the problem is planar, and this is very relevant, the planar problem is very relevant for physics, because our solar, all the planets in our solar system have eccentricities almost vanishing, and inclinations almost vanishing. There are not parities associated in general to the symmetry by reflection, except in the fact that n equal to 2, where there are symmetries, and these coordinates are not defined, as I say, the RPS coordinates are defined for the planar problem, but as I said before, there is just an extra degrees of freedom. Another question concerning Arnold's theorem is that, with the applying Arnold's theorem, we have we proved the existence of quasi of full measure in the sense of leg bag, of an almost full measure in the side of leg bag, of quasi periodic motions for the problem around elliptic equilibrium point. This means from the physical point of view that we have to take motions closer and closer to circle on a plane in order to find more and more preserved the terrain. And we could ask what happens in the remaining regions of phase space, are there periodic quasi periodic motions, and which is their measure. With the theorem by Arnold, the measure goes to zero with eccentricities and inclination. Is it possible to prove existence of quasi periodic motion with a measure that goes, that does not go to zero with eccentricities, with a measure that does not go, sorry, to one with eccentricities and inclination. And the result is that, the recent result is that, if we fix the two arbitrary values of the eccentricity up to this threshold value. And if we choose the semi-major axis is separated in this way, and the mass is really smaller with respect to the smaller of this one. It is positive to find a positive measure set of 3 n minus 2 dimensional quasi periodic motions in the problem. The measure of which goes to one with respect to the separation among the semi-axis. And I am particularly proud of this result. Such set tends to the corresponding set of the planar problem when the inclination goes to zero. With the previous set of coordinates, it was not, it is not possible to prove this one because as I said before, those coordinates are singular when eccentricity is equal to zero. A related result has been announced by Jacques Fechauss since 2013. The new system of coordinates I want to present is based on the partial angular momenta, as in the case of the pre, and instead of the angular momenta of the single planets on the perihilia. And this is why I call it perihilia reduction. We consider these chains of vectors. The first is the third axis of a fixed frame, the total angular momentum, the last of the perihilia, the last but one partial angular momentum, the perihila Pn minus 1, until we arrive to S1 and P1. Ič ring of this chain is regarded as a third axis of a fixed frame in the configuration space. The first step is the same as in the precoordinates. We change a frame in order to overlap our initial frame to the invariable frame with the third axis parallel to the total angular momentum. And the other two pieces of this chain are represented by this picture. In this picture we have the third axis parallel to Si, when i is equal to n, this is the total angular momentum of the system. The first axis is basically arbitrary because the initial frame is arbitrary. Then we consider the last of the perihilia with i equal to n. The plane that passes through these two vectors, its longitude is the angle phi i and the euclidean length of Si is the action phi psi i. When i is equal to n, this action is an integral and this angle is a cyclic. So with the system of coordinate we fix the motion of the horizontal projection of the last of the angular momentum. When the planar is problem we fix the motion of the last, sorry, of the last of the perihilia. When the planar is problem we fix the motion of the last of the perihilia. Then we switch to a system that has Pi as the third axis and this line and i minus 1, that is perpendicular to the plane through Si and Pi as the first axis and here we consider Si minus 1. Its third component is the action capital theta and its longitude is the angle small theta. Those coordinates are singular when Si minus 1 is parallel to Pi or Pi is parallel to Si, but this configuration is quite not physical meaning. So that it is simply we can exclude it from the phase space in order to describe the system. These coordinates are defined also in the planar problem because in the planar problem Si and Pi are perpendicular one to the other and the capital theta is equal to zero and it is easy to see that the small theta is equal to Pi. The advantage, there is another advantage, there is an involution that is associated to the reflection symmetries. If we change the two theta capital theta and small theta by the sign, we have a symmetry into the amiltonian, this does into the whole amiltonian, not only they over age the one and this involution generates the existence of two and to the minus one planar equilibria that corresponds and this planar equilibria corresponds to planar configuration with the different choices of the signs of the partial angular momenta and the try of the theorem one are obtained as before kating from the equilibrium with all the s parallel one to the other. Of course, one could study also the other equilibria. The next challenge will be to prove the coexistence of stable and unstable motion in the planetary problem. This is a result by Jeffries and Moser in 1963. They proved the existence of a three dimensional counter family of hyperbolic tori in the system provided that the planets in three cities are small and the mutual inclinations between the two planets is sufficiently large satisfies their selection. This tori disappear for small inclination because for a small inclination a suitable hyperbolic feature of the system beforkates to be elliptic and so this unstable tori disappear. But using my new coordinates I can prove that when the inclination is small a hyperbolic tori three dimensional hyperbolic tori really still exist provided the mutual inclination is small the two planets revolve in opposite versus one with respect to the other and suitable relation between the angular momenta and the semi major axis are satisfied. In this region of the phase space, I forgot to mention, Laskarn Robbutel proved the existence of full of full dimensional Kolmogorov tori in 1995 and Biasko Kjerkev and Valdinov, she proved the existence of two dimensional elliptic tori. So with the CRM that is still working progress, I proved the existence of stable and unstable motion for the planetary problem. And the last result that I want to mention is also is work in progress is that in the case of the free body problem all the motions where no mean motion resonances mean motion resonances are resonances for the leading part for the that occur only for the Keplerian frequencies and if the eccentricity of the inner planets is sufficiently small are stable in the sense of Nekoroshev. Nekoroshev in 77 proved stability only for the actions that are related to the semi major axis. With this CRM I proved and the result was later improved by Hermann in 1996. With this result I proved the stability for whole actions of the problem and in particular for the eccentricities and inclination and this may have some consequence that goes in a very asymptotic point of view in the direction of the proof of the stability is a small result that goes in the direction of the proof of the stability of the solar system. What are they mean? I have no time to explain the proof, but just to give a rough idea. In the case of the two body problem the Hamiltonian that is obtained using the Jacobi and Rado coordinates is here. It is for some reason it is integrable. It is depends by just one angle, the angle G1 and is represented here. If we switch to Poincare coordinates we shall have an Hamiltonian, a series in terms of eccentricity coordinates. While using my new coordinates I have an analog expression that still depends by just one angle but it is now a series in the inclination coordinates and the advantage is that the point with the capital theta equal to zero and the small theta equal to zero to pi are equilibria so we can integrate this part by constructing a convergent bit of normal form associated to it. When the number of planets is more than two it is possible to apply a procedure of integration mixed with the over aging planet by planet and this leads to the proof of theorem one. So this is the end of my talk. Thank you.