I will discuss the existence of maximal KAM tori which are exponentially close to whiskered secular tori appearing generically at simple resonances in nearly integrable mechanical systems. Joint work with Luca Biasco.
Motivated by the need of preserving the operational orbital regions around the Earth, natural perturbations can be exploited to lead the satellites towards an atmospheric reentry at the end of life. In this way, it is possible to dilute the collision probability in the long term and reduce the disposal cost. In the case of the Medium Earth Orbit (MEO) region, home of the navigation satellites...
The modern asteroid surveys are producing very large databases of optical observations. Linking very short arcs (VSAs) of such observations we can compute the asteroid orbits. Searching for
efficient linkage algorithms is an interesting mathematical problem.
We show how this problem can be faced using the first integrals of Kepler's problem. ...
(with Holger Dullin).
In any motion of the n-body problem, seen in the Galilean frame of the center of mass, the energy H and the angular momentum vector L are constant.
The integral manifolds are the submanifolds of the phase space defined by fixing H and L. If we fix L and let H vary, the submanifold will change topology at values which we call singular. The critical values are...
In this talk, we aim to present an alternative formulation of classical KAM theory via paradifferential calculus. A powerful tool introduced by J.-M. Bony during the 80s, paradifferential calculus is widely used in delicate analysis of nonlinearities. Based on a tailored definition of paraproduct operators, it turns out that dynamical conjugacy problems involving "small denominators" can be...
In this talk I will consider a class of linear Schrödinger equations, whose Hamiltonian is given by bounded, time periodic perturbations of the 2-dimensional Harmonic oscillator. Under complete resonance assumptions, I will show that a generic class of perturbations admits solutions whose Sobolev norms undergo infinite growth as time tends to infinity.
The proof combines...
We consider two problems relying on perturbative methods in Celestial Mechanics: the computation of proper elements for the space debris problem and effective stability estimates close to resonances in rotational dynamics.
We show that perturbative methods can be integrated with Machine Learning techniques, specifically to investigate the dynamics of groups of objects for the classification...
Nekhoroshev theorem in its original form ensures stability over exponentially long times of perturbations of integrable Hamiltonian systems under a generic nondegeneracy condition introduced by Nekhoroshev and called Steepness.
Here we consider the case of a perturbation which depends quasiperiodically on time and prove that if the frequencies of the forcing are Diophantine and the...
In a previous paper written with Ezequiel Maderna, we proved that in the newtonian N-body problem, given a starting configuration x_0 and a limit shape a without collisions denoted as a, there always exists a hyperbolic solution x(t) of the N-body problem such that x(0)=x_0 and x(t)/t converges to a as t tends to +\infty. We will say that x(t) is a hyperbolic motion with limit shape a.
We...
In the three-body problem, for all values of the masses, the vertical eigenvectors of the Lagrange relative equilibrium give rise to a family of spatial quasi-periodic orbits; these orbits are periodic in a rotating frame.
For the equal masses' cases, this is the P12 family with its order-12 symmetry group, which ends at the figure Eight solution (studied in Chenciner-Fejoz-Montgomery...
This talk explores spectral rigidity phenomena appearing in billiard dynamics, with a focus on convex planar billiard-like systems, which include Birkhoff, Symplectic, Outer and Outer-length billiards.
I will present some new rigidity results for the associated action spectrum, which are linked to pointwise isoperimetric-type inequalities of Mather’s beta function. Notably, we will observe...
In the study of close to integrable Hamiltonian PDEs, a
fundamental question is to understand the behaviour of “typical” solutions. With this in mind it is natural to study the persistence of almost-periodic solutions and infinite dimensional invariant tori, which are in fact typical in the integrable case.
In this talk, I shall consider a family of NLS equations parametrized by a smooth...
Since the birth of KAM theory, planetary three-body problems were indicated as a natural benchmark to study its applicability to realistic systems of physical interest. It is well known that KAM statements proved in a purely analytical way completely fail such a challenging purpose. This the reason why, since the last two decades of the previous century Computer-Assisted Proofs (hereafter...
Quadratic Hénon maps are polynomial automorphism of $\mathbb{C}^2$ of the form $h:(x,y)\mapsto (\lambda^{1/2}(x^2+c)-\lambda y,x)$. They have constant Jacobian equal to $\lambda$ and they admit two fixed points. If $\lambda$ is on the unit circle (one says the map $h$ is conservative) these fixed points can be elliptic or hyperbolic. In the elliptic case, a simple application of Siegel...
We prove that any real-analytic Hamiltonian with a locally integrable, non-degenerate and not locally convex elliptic equilibrium, can be perturbed within the real analytic category, while keeping the Birkhoff normal form at the equilibrium unchanged up to any arbitrary order, so that the equilibrium becomes Lyapunov unstable.
Joint work with B. Fayad, M. Saprykina and T.M. Seara.
A few years ago, I realized that a single average of the inverse distance of two Keplerian bodies admits as a first integral an intriguing combination of the angular momentum and perihelion anomaly of the averaged planet.
Almost at the same time, J. Jauslin and G. Gallavotti discovered a first integral in a version of the Boltzmann problem, a model of statistical mechanics proposed by...
This work studies conformally symplectic diffeomorphisms defined in a symplectic manifold, that is, diffeomorphisms which transform a symplectic form into a multiple of itself, through a conformal factor.
We focus on Normally hyperbolic invariant manifolds (NHIM) and their (un)stable manifolds, which are important landmarks that organize long-term dynamical behaviorb
We prove that...