J.-H. Lambert (1728–1777) is one of the founders of non-Euclidean geometry, but he also discovered a strange and useful property of the Keplerian motion in a Euclidean space. The time required to reach a point B from a point A with a given energy, under the Newtonian attraction of a mass at a fixed point O, will not vary if we shift A and B in such a way that AB and and OA+OB remain constant.
P. Serret (1827–1898) and W. Killing (1847–1923) introduced the Kepler problem in constant curvature spaces and listed its impressive analogies with the usual Kepler problem. Here we complete this list by proving that the time required to reach a point B from a point A, under the attraction of a mass at a fixed point O of the curved space, with given energy, does not vary if we shift A and B in such a way that d(A,B) and d(O,A)+d(O,B) remain constant, where d is the geodesic distance.
We will also discuss the case of pseudo-Riemannian spaces with constant curvature. We will mainly use the well-known formulas of variational calculus that Hamilton introduced in 1834, and a simple property of the eccentricity vector. This work benefited from many discussions with Zhao Lei, from the University of Augsburg.
b-Symplectic manifolds, manifolds where the symplectic form develops a singularity along a hypersurface of the manifold in a controlled fashion, have recently become an object of intense study. Due to the structure of these manifolds, they possess very limited symmetries and the existence symmetries of these objects have interesting local and global consequences. In particular one can construct simple local models of these structures close to orbits of Poisson group actions. The existence of global actions can imply that the b-symplectic structures is of a very particular form. Local and global aspects of the symmetries of these fascinating objects will be discussed.
References
Il peut être fructueux de considérer un problème sur les systèmes dynamiques « autonomes » (indépendants du temps) d’un point de vue non-autonome. Le conférencier a mis vingt ans à comprendre une question que lui avait posée Vaughan Jones, et qu’il avait trouvée particulièrement stupide sur le moment.
The restricted planar elliptic three body problem (RPETBP) describes the motion of a massless particle (a comet) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this work is to show the existence of orbits whose angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive orbits, that is, with a large variation of angular momentum.
The leading idea of the proof consists in analyzing parabolic motions of the comet. By a well-known result of McGehee, the union of future (resp. past) parabolic orbits is an analytic manifold. In a properly chosen coordinate system these manifolds are stable (resp. unstable) manifolds of a manifold at infinity, which turns out to be topologically equivalent to a normally hyperbolic invariant manifold (TNHIM).
On this TNHIM, it is possible to define two scattering maps, which contain the map structure of the homoclinic trajectories to it, i.e. orbits parabolic both in the future and the past. A non-canonical symplectic structure still persists close this TNHIM and extends naturally to a $b^3$-symplectic structure. Such singular structures appear also in other problems of Celestial Mechanics.
Since the inner dynamics inside the TNHIM is trivial, two different scattering maps are used. The combination of these two scattering maps permits the design of the desired diffusive pseudo-orbits. Using shadowing techniques and these pseudo orbits we show the existence of true trajectories of the RPETBP whose angular momentum varies in any predetermined fashion.
This is a joint work with Vadim Kaloshin, Abraham de la Rosa and Tere M. Seara.
This is joint work with Kai Cieliebak and Otto van Koert. The restricted three body Problem describes the movement of a massless particle attracted by two masses according to Newton's law of Gravitation. For example one could imagine a satellite attracted by the earth and the moon, the moon attracted by the earth and the sun, or a planet in a double star System. The trajectory of the massless particle is usually immersed except for collisions with one of the masses or a phenomenon which is referred to by Hill as a moon of maximal lunarity. By the theorem of Whitney-Graustein the rotation number is a complete invariant for immersed loops in the plane up to homotopy. In a generic homotopy three disasters can occur - triple intersection, inverse and direct self tangencies. Arnold's J+ invariant is unchanged under the first two disasters but is sensible to direct self tangencies. For families of periodic orbits in the restricted three body problems two additional disasters can occur - occurence of cusps in the case of a moon of maximal lunarity and collisions. We show how the theory of Arnold's J+ invariant can be modified to obtain invariants for families of periodic orbits in the restricted three body problem.
K. Siegel asked if there is an open set of initial conditions for a Three Body Problem which has a dense subset of collision orbits, i.e. initial conditions whose orbits hit a collision. Consider the Restricted Circular Planar 3-Body Problem with mass ratio of the primaries μ. We prove the existence of an open set in phase space where collision orbits form a O(μ1/20) dense set as μ tends to 0. This is a joint work with V. Kaloshin and J. Zhang.
On any symplectic manifold (M,ω), there always exist almost complex structures which are compatible with the symplectic structure and positive. Such a structure is integrable if and only if the manifold is Kähler. We review different possibilities to try and select some of these structures beyond the Kähler case. In particular, we exhibit geometrical structures associated to non integrable almost complex structures. (This is joint ongoing work with Michel Cahen and Maxime Gérard.)
Both elliptic and logarithmic symplectic structures are examples of Poisson structures which are generically symplectic, but degenerate on a submanifold, in the former case of codimension-2, in the latter case of codimension-1. Certain elliptic symplectic manifolds can be shown to be examples of generalised complex manifolds. I will describe how to generalise a range of symplectic techniques to these manifolds, how to relate elliptic symplectic and stable generalised complex structures to log symplectic structures, and describe their natural generically Lagrangian submanifolds and how these intersect the degeneracy locus. I will discuss local deformations of Lagrangians and how one might go about constructing a Fukaya category for log or elliptic symplectic manifolds.
In this talk we will prove the existence of Gevrey symplectic diffeomorphisms of $\R^6$ which admit an elliptic fixed point whose linear part is a non-resonant rotation, and an orbit which converges to the fixed point. This is joint work with Bassam Fayad and David Sauzin.
The study of singular symplectic manifolds was initiated by the work of Radko, who classified stable Poisson structures on surfaces. It was observed by Guillemin—Miranda—Pires that stable Poisson structures can be treated as a generalization of symplectic geometry by extending the deRham complex. Since then, a lot has been done to study the geometry, dynamics and topology of those manifolds.
We review the known results and talk about the first steps towards a variational approach for singular symplectic manifolds.
This is work in progress with Roisin Braddel, Jacques Fejoz, Eva Miranda, Michael Orieux and Qun Wang.
We review the geometric quantization scheme introduced by Kostant. We focus on explaining the case in which a real polarization of the symplectic manifold is used. We explain some tools used to compute this quantization for some classes of symplectic manifolds. We compute the quantization on toric and semitoric integrable systems and more general almost toric manifolds. Finally, we apply the computation to the semitoric system associated to a K3 surface and to the one associated to the spherical pendulum.
The Kirchhoff problem, also known as the N-vortex problem, is a Hamiltonian system that describes interactions of vortices in the plane. It finds application in various phenomena in physics. This system is in general not integrable when the number of vortices is more than 3. It turns out that searching periodic solutions might shed some lights on properties of such dynamics.
In this talk, we will review some results on the existence of (relative) periodic solution for the N-vortex problem,while taking the N-body problem as an analogy. In particular, we will report some recent progress on symmetric relative periodic orbits by using some global methods based on calculus of variation.
This is a joint work with Jacques Féjoz and Éric Séré.
We review geometric quantization in the symplectic case, and show how the program of formal geometric quantization can be extended to certain classes of Poisson manifolds equipped with appropriate Hamiltonian group actions. These include b-symplectic manifolds, where the quantization turns out to be finite dimensional, as well as more singular examples (bk-symplectic manifolds) where the quantization is finite dimensional for odd k and infinite dimensional, with a very simple asymptotic behavior, where k is even. If time permits we will also discuss some preliminary results for pseudoconvex domains. This is a joint work with Victor Guillemin and Eva Miranda.