Stability in Hamiltonian Dynamics and Beyond

This is the official web site of the PRIN Project "Stability in Hamiltonian Dynamics and Beyond", funded by the Ministry of Education, University and Scientific Research of Italy with Project code 2022FPZEES. It will serve as the central hub for all communications related to the project, including descriptions of the research and its researchers, results, events, etc.

The core of the PRIN Project is concerned with some of the most important theoretical aspects in the study of Hamiltonian systems and their stability properties, as well as with the urgent and timely quest for extending this analysis and the available tools beyond the current range of applicability.

In particular, we focus on four cutting-edge lines of research:

Transition from integrability.
Billiards as prototypical models.
Singular Hamiltonian systems.
Beyond the Hamiltonian realm.

The connection among these problems is twofold. On the one hand, they are all motivated by the quest for a better understanding of the dynamical and stability properties of Hamiltonian systems (and beyond), a subject that has a long-established tradition and has been recently boosted by important breakthroughs, including some by members of the team. On the other hand, the methods that we plan to apply provide natural links among them.

This PRIN Project consolidate a network of 4 internationally recognized Italian research groups in Hamiltonian dynamics, dynamical systems, and geometric mechanics, each contributing with their different and complementary expertise. Due to the interdisciplinary nature, in fact, advances in these subjects require the synergy among different points of view and the combination of diverse techniques, conditions that are hardly available at the level of a single research unit.

The PRIN Project is active from 2023 to 2026.
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Transition from integrability:

Quantify the measure of initial data corresponding to stable motions (KAM tori) and the temporal scale for the appearance of possible macroscopic effects of instability (Nekhoroshev theory).

Billiards as prototypical models:

Explore three significant settings in which the study of billiards and billiard-like systems can be fruitfully exploited to tackle other problems; namely: Birkhoff conjecture and integrability of geodesic flows; refractive billiards and celestial mechanics; billiards and symplectic topology.

Singular Hamiltonian systems:

Investigate the dynamics and the stability properties of Hamiltonian dynamical systems governed by singular potentials, with a particular focus on the N-body problem and the N-centre problem in celestial mechanics, and the N-vortex problem in fluid dynamics and geophysics.

Beyond the Hamiltonian realm:

Extend the stability analysis developed in the framework of Hamiltonian systems to a broader class of systems that are important for real-world applications. We propose to focus on: conformally symplectic and almost symplectic systems, dissipative systems, and nonholonomic dynamics.