Data: 07/04/2017

Ora: 13:30:00

Descrizione:

The $n$-center problem of celestial mechanics describes the motion

of a particle in the force field of n fixed gravitational centers.

For $n\le 2$ this is integrable. For $n\ge 3$ and positive energy,

a rather complete understanding of the dynamics has been achieved,

using geodesic motion on surfaces of negative curvature ($d=2$ dimensions),

and also for $d=3$.

In an ongoing collaboration with J. Fejoz (Paris Dauphine) and

R. Montgomery (UC Santa Cruz), we try to derive related results

for the $n$-body problem.

Data: 07/04/2017

Ora: 14:40:00

Descrizione:

We consider the orbit computation problem for a Solar system body observed from a point on the surface of the Earth. In our case the available data are two elements of the tangent bundle of the celestial sphere at two different epochs, and the unknowns are the radial distances and velocities $\rho_1$, $\rho_2$, $\dot{\rho}_1$,$\dot{\rho}_2$ of the observed body at these epochs. Using the first integrals of Kepler's motion we can write algebraic equations for this problem, which can be put in polynomial form. From these we obtain a univariate polynomial equation of degree 9 in one of the radial distances. Using Groebner bases theory we show that this equation has the minimum degree among the univariate polynomial equations in $\rho_1$or $\rho_2$ that are consequence of the conservation laws of Kepler's problem, provided that we drop the dependence between the inverse of the heliocentric distance $1/|r|$ of the observed body and the unknown radial distance.

]]>Data: 27/03/2017

Ora: 15:30:00

Descrizione:

From pseudodifferential operators to Lagrangian distributions

I will give a talk on a joint project with Marco Cappiello (Turin University) and Patrik Wahlberg (Växjö University), in which we study the microlocal properties of distributions in the framework of the Shubin pseudodifferential calculus. In particular, we review the theory of pseudodifferential and Fourier integral operators in this setting and provide an adapted notion of conormal and Lagrangian distributions. One of our main tools is to employ an integral transform of FBI-type and therefore our theory may be viewed as a phase space analysis of Shubin-type Lagrangian distributions. It turns out that the use of these techniques simply the proofs of theorems that are very hard to obtain by classical methods. As an application, we apply our methods to prove a geometric propagation result for singularities of Schrödinger equations.

The talk is designed for a non-specialized audience and will feature an extensive introduction.

]]>Data: 27/03/2017

Ora: 14:30:00

Descrizione:

We consider radial periodic perturbations of a central force field, and prove the existence of rotating periodic solutions, whose orbits are nearly circular. Our result applies, in particular, to the classical Kepler problem.

]]>Data: 24/03/2017

Ora: 14:00:00

Descrizione:

]]>For the N-body problem with equal masses, there exists a special type of periodic solutions called simple choreographies, where all the masses chase each other on a single loop. Well known examples including the rotating N-gon, the Figure-Eight solution of three body and the Super-Eight solution of four body. Simple choreographies with different shapes have been found numerically. However for many of them, rigorous proofs are still unavailable. In this talk, we give a proof of a special family of simple choreographies called "linear chain", which looks like a sequence of consecutive bubbles along a given line.