Data: 12/10/2018

Ora: 09:30:00

Descrizione:

A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary. This simple model has been first proposed by G.D. Birkhoff as a mathematical playground where "it the formal side, usually so formidable in dynamics, almost completely disappears and only the interesting qualitative questions need to be considered".Since then billiards have captured much attention in many different contexts, becoming a very popular subject of investigation. Despite their apparently simple (local) dynamics, their qualitative dynamical properties are extremely non-local. This global influence on the dynamics translates into several intriguing rigidity phenomena, which are at the basis of several unanswered questions and conjectures.In this talk I shall focus on some of these questions. In particular, I shall describe some recent results related to the classification of integrable billiards (also known as Birkhoff conjecture).ÂÂÂ This talk is based on works in collaboration with V. Kaloshin and with G. Huang-V. Kaloshin.

]]>Data: 11/10/2018

Ora: 14:30:00

Descrizione:

We prove the existence of solutions to the classical Liouville equation which blows-upalong a smooth, simple and closed curve inside a doubly connected domain.The result has been obtained in collaboration with Giusi Vaira and Mike Kowalczyk.

]]>Data: 25/06/2018

Ora: 15:00:00

Descrizione:

A simple general theorem permits to look for time-dependent Lyapunov functions for Lagrangian systems. We show three cases where the research is successful: the mechanical systems with viscous fluid resistance and the conservative and dissipative Maxwell-Bloch equations of laser dynamics. By means of these Lyapunov functions we study some dynamical features.

]]>Data: 25/06/2018

Ora: 12:00:00

Descrizione:

Using the Conley-Zehnder index theory for symplectic paths, we discuss a general characterization of elliptic stability in planar time dependent Hamiltonian systems. Applications to stable harmonic oscillations of the forced pendulum system, stability transitions for equilibrium of the restricted three body problem and elliptic stable geodesics on the two sphere are described.

]]>Data: 11/06/2018

Ora: 14:30:00

Descrizione:

Motivated by the phenomena observed before the famous Tacoma Narrows Bridge collapse in 1940, we deal with some nonlinear fourth-order differential equations related to the analysis of the dynamics of suspension bridges. Following a "structural" approach, we discuss the role of the position of intermediate piers in the stability of a hinged beam, making a comparison between different notions of stability. The analysis is carried out using both analytical and numerical tools. (Joint work with Filippo Gazzola)

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