A well-known theorem due to Birkhoff asserts that every continuous graph invariant by a symplectic twist map of the annulus is Lipschitz. I have recently proved that these graphs are in fact more regular than Lipschitz: they are C1 on a residual set with full Lebesgue measure. The goal of my lectures is to present some analogous results for the so-called Tonelli Hamiltonians.
To do that, I will:
The final theorems that I will state concern the cotangent bundles of compact manifolds, but a lot of the notions that I use are valid in the non-compact case. Two of these theorems are:
First lecture: Basic results on Hamiltonian, Lagrangians, Lagrangian Lipschitz submanifolds, Hamilton-Jacobi equation and minimization properties
1) Hamiltonian and Lagrangian formalism
2) Lipschitz Lagrangian submanifolds, images of such manifolds and minimization properties
Second lecture: 3) Lagrangian subbunbles
An order relation on the Lagrangian subbundles which are transverse to the vertical. A notion of upper and lower semi-continuity for these bundles. In the convex case: links with the image of the vertical and the notion of conjugate points
Third lecture: 4) Green bundles
Construction of the Green bundles. A dynamical criterion. The reduced Green bundles. A characterization of hyperbolicity
Fourth lecture: 5) Regularity of Lipschitz Lagrangian invariant graphs
Generalized tangent vectors and cones. Generalized derivative and generalized tangent planes. Regularity of the invariant graphs for two degrees of freedom. More results of regularity.
Cet exposé questionne l'identité algébrique d'une pratique propre à un corpus de textes publiés sur une période antérieure à l'élaboration de théories algébriques, comme la théorie des matrices, ou de disciplines, comme l'algèbre linéaire, qui donneront à cette pratique l'identité d'une méthode de transformation d'un système linéaire par la décomposition de la forme polynomiale de l'équation caractéristique associée. Dans les années 1760-1775, Lagrange élabore une pratique algébrique spécifique à la mathématisation des problèmes mécaniques des petites oscillations de cordes chargées d'un nombre quelconque de masses ou de planètes sur leurs orbites. La spécificité de cette pratique s'affirme par opposition à la méthode des coefficients indéterminés, elle consiste à exprimer les solutions des systèmes linéaires par des factorisations polynomiales d'une équation algébrique particulière, l'équation à l'aide de laquelle on détermine les inégalités séculaires des planètes. Élaborée en un jeu sur les primes et les indices des coefficients des systèmes linéaires, la pratique de Lagrange est à l'origine d'une caractéristique des systèmes issus de la mécanique, la disposition en miroirs des coefficients de systèmes que nous désignons aujourd'hui comme symétriques. Elle est également à l'origine d'une discussion sur la nature des racines de l'équation qui lui est associée. Nous questionnerons l'identité algébrique de cette discussion qui se développera sur plus d'un siècle en étudiant les héritages, permanences et évolutions de la pratique élaborée par Lagrange au sein de différentes méthodes élaborées dans divers cadres théoriques par des auteurs comme Laplace, Cauchy, Weierstrass, Jordan et Kronecker. Nous verrons que, préalablement à l'élaboration d'une théorie des formes dont la nature, algébrique ou arithmétique, suscitera une vive controverse entre Jordan et Kronecker en 1874, le caractère algébrique de la discussion renvoie davantage à l'identité historique d'un corpus qu'à une identité théorique et s'avère indissociable d'une constante revendication de généralité. Entre 1766 et 1874, la discussion sur l'équation à l'aide de laquelle on détermine les inégalités séculaires permet de mettre en évidence différentes représentations associées à une même pratique algébrique ainsi que des évolutions dans les philosophies internes portées par les auteurs du corpus sur la généralité de l'algèbre.
For a paper in english on this subject see Algebraic generality vs. arithmetic generality in the controversy between C. Jordan and L. Kronecker (1874) at http://arxiv.org/ftp/arxiv/papers/0712/0712.2566.pdf.In this talk we briefly outline a variational proof for the existence and minimizing properties of retrograde orbits for the planar three-body problem. These solutions begin with collinear configurations and they trace out pure braids in the three-dimensional space-time on some rotating frames. When some of masses are equal, our arguments can be applied to prove the existence of some prograde orbits and part of the Hénon family.
In our Solar system there are more than 20 satellites revolving around the respective main bodies on an (apparently) exact 1:1 spin-orbit resonance (i.e. in one revolution period the satellite revolves one time around its spin-axis, like our Moon). There is only one other object observed in a spin-orbit resonance: Mercury, in a 3:2 resonance. A simple, nearly-integrable, weakly dissipative mathematical model is discussed:
We will show the proof that in the C2-topology the geodesic flow of a generic Riemmanian metric on a compact manifold contains a hyperbolic and in particular it has positive entropy.
Lecture notes:
Majority of works on perturbations of integrable Hamiltonian PDE (under periodic boundary conditions) deals with the situation when the unperturbed equation is linear and depends on a certain number of parameters. These results seem to be rather far from the problems in the celestial mechanics. Indeed, there we also often deal with Hamiltonian problems of (almost) infinite dimension, but usually the unperturbed equations are non-linear and do not depend on extra parameters. In my lecture I will discuss results and open problems for perturbations of an integrable PDE which do not depend on extra parameters.
There are many problems that lead to the analysis of Hamiltonian dynamical systems in which one can distinguish motions of two types: slow motions and fast motions. Adiabatic perturbation theory is a mathematical tool for the asymptotic description of dynamics in such systems. This theory allows to construct adiabatic invariants, which are approximate first integrals of the systems. These quantities change by small amounts on large time intervals, over which the variation of slow variables is not small. Adiabatic invariants usually arise as first integrals of the system after having been averaged over the fast dynamics.
In the lectures it is planed to consider the following topics: exponential accuracy of conservation of adiabatic invariants in analytic one-frequency systems, jumps of adiabatic invariants at separatrices, destruction of adiabatic invariance due to captures into resonances and passages through resonances, examples of adiabatic invariance in problems of celestial mechanics.
We present a phenomenological study of Area Preserving Maps, with emphasis on simple models. Paradigmatic models like the standard map, the separatrix map and a new model, the biseparatrix map, are useful to undertand the dynamics.
These paradigms can be analysed in a reasonable way. They allow to explain the main features of dynamics. As test example the conservative Hénon map will be used.
Main points of interest are the escape of points from a given region when they are not surrounded by invariant rotational curves and the measure of the set of non-escaping points. Different rates of escape have been found.
The existence of regions where the dynamics is close to a diffusion, separated by cantori which are difficult to cross, having noble numbers as rotation number, are illustrated.
Applications are also made to the stability regions around triangular points in the RTBP as a function of the mass parameter.
The talk will be closed by several open questions.
In the end of the last century Vershik introduced some dynamical systems called polymorphisms. Systems of this kind are multivalued self-maps of an interval, where (roughly speaking) each branch has some probability. By definition the standard Lebesgue measure should be invariant.
Unexpectedly polymorphisms appeared in the problem of destruction of an adiabatic invariant after a multiple passage through a separatrix.
We plan to discuss ergodic properties of polymorphisms.