The research areas of the Mathematical Physics group are
 Celestial Mechanics: mathematical theory and applications to real problems in astronomy and astrodynamics.
 Dynamical Systems: general theory, statistical and stochastic properties, applications to other disciplines.
More specific research topics investigated by this group are described below.
The group manages the spinoff SpaceDyS s.r.l. (Polo Tecnologico di Navacchio).
Research Topics
Mathematical models and methods for the analysis of interplanetary tracking data
The Celestial Mechanics Group is involved in three research projects, linked to the space missions BepiColombo (ESA), Juno (NASA), and Hera (ESA). The BepiColombo mission was launched in 2018 and will arrive in orbit around Mercury in 2025: it aims to study the planet’s gravity field, its composition, its state of rotation, and carry out tests on theories of gravitation. The Celestial Mechanics group will analyze the data from the radio science experiment with the dedicated software Orbit14, developed with ASI funding in 2007. Simulations and cruise data analysis are currently being carried out. The Juno mission was launched in 2011 and has been in orbit around Jupiter since 2016: it aims to study the origin and evolution of the planet. The Celestial Mechanics Group is responsible for analyzing the data from the radio science experiment with the dedicated software Orbit14 with the aim of estimating Jupiter’s physical parameters, including gravitational and tidal coefficients. These parameters are necessary to improve the internal structure models of the planet. The Hera mission will be launched in 2024 and the goal will be the binary asteroid Didymos which it will reach in 2026, a few years after the moon of Didymos, called Dimorphos, was impacted by DART probe of NASA. Hera’s purpose is to characterize the postimpact environment of the asteroid through different tools. The Celestial Mechanics group will develop software, based on Orbit14, to analyze radio science data and will also conduct an analysis of the nongravitational perturbations acting on the system (YarkovskyYORPBYORP effects).
Members
Orbit determination (OD)
The Celestial Mechanics Group (CMG) boasts more than thirty years of experience in the dynamics and orbit determination of minor bodies of the solar system, e.g. Main Belt Asteroids (MBAs) and NearEarth Asteroids (NEAs). Such work led to the realization of the online services AstDyS (it contains data about numbered and multiopposition asteroids, including for example their orbital elements with the related uncertainties and proper elements) and NEODyS (it contains data about NEAs, including close approaches information, and probabilities of impact with the Earth). Since 2006 the CMG has also been involved in research in the field of space debris. In these broad fields of research, a nonexhaustive list of subtopics is the following.
 New initial OD methods for very short arcs of observations: this problem arises from the fact that the nextgeneration surveys (e.g. PanSTARRS, LSST) will be able to gather a huge number of observations of small celestial bodies per night.
 Classification of the asteroid families in the Main Belt.
 Impact Monitoring (IM) of NEAs: the goal of IM is to establish, starting from observations, if a given NEA could have a chance to impact the Earth in the future. Such algorithms have been implemented in the CLOMON2 software (since 2002), which daily processes observational data from observers around the world by calculating the orbits of objects and their uncertainty. In the last ten years the NEOScan software has been developed: it analyzes the NEOCP data (where there are objects with very few observations) to determine the type of object and, in the case of a NEA, evaluate possible imminent impacts.
 Chaotic OD: it is the problem of determining the state and the associated uncertainty of a chaotic dynamical system from a set of observations. In particular, we want to understand the behavior of the uncertainty as the number of observations grows and how it changes if a dynamical parameter is included among the variables to be determined; such analysis is applied to celestial bodies that undergo several close encounters with planets or planets’ satellites.
 Correlation and OD for space debris.
 Collision avoidance of space debris: we study the problem of finding in a given time interval (usually a few days) which pairs of space debris orbiting around the Earth can become sufficiently close to each other, providing also the probability of this event. Due to the big number of space debris present in the catalog, filters are necessary to quickly discard many pairs of objects. One of these filters consists of a new procedure to speed up the computation of the minimum distance between two confocal comics.
Members
Collaborators
Solar system dynamics and astrodynamics
 Longterm evolution of satellites’ dynamics driven by tidal dissipation. Dissipative effects due to tidal forces between satellites and their hosting planet influence greatly the dynamics and the interior of the moons. For example, the volcanism of the moon Io is due to the coupling between tides and resonances with the other satellites of the Jovian system. We develop dynamical models that we use to investigate the orbital history of the moons of our solar system.
 Longterm evolution of planets’ obliquity. The planets of our solar system are characterized by very different inclinations of their spin axis (or obliquities). For example, Jupiter has an obliquity of just 3°, Saturn 27°, and Uranus even 98°. We develop dynamical models that we use to explain the current obliquity of the gas giants of the solar system.
 New formulations of the perturbed twobody problem based on nonsingular orbital elements: these are obtained by regularization of the equations of motion or via a purely geometrical approach. Such formulations can substantially improve the accuracy in the propagation of the state and associated uncertainty of Earth’s artificial satellites as well as of natural small bodies in the Solar system.
 Patched dynamics: it is used as a first approximation of more complex models. The solutions of the simpler problem can be used as a starting guess to compute the solutions of the more difficult ones, or sometimes they can be used to obtain the qualitative behavior of the more complex system. We apply patched dynamics to study close approaches of asteroids with the Earth and the passage through shadow for an Earth satellite.
Members
Collaborators
Variational methods to study periodic orbits of the $N$body problem
We search for new periodic orbits of the Newtonian $N$body problem as minimizers of the Lagrangian action functional in a set of symmetric loops, imposing also topological constraints. We also search for periodic motions with the same symmetries and topological constraints in the case of nonNewtonian forces, with interaction potentials of the form $1/r^\alpha$ ($\alpha>1$), where $r$ is the distance between the two bodies. The linear stability of such orbits is also investigated.
Members
Complexity of time series
The methods of nonlinear dynamical systems have been applied to the study of time series since the 90s. These applications have included time series of different origins, biomedical, economic, physical, etc. In particular, it is by now clear that one useful method is the classification of the time series in terms of some notion of entropy and complexity. We collaborate with Centro Piaggio of the University of Pisa to study the complexity of biomedical time series and the effects of noise on their classification.
Members
Statistical properties of dynamical systems
Our group is interested in the study of the statistical properties of discrete and continuous group actions describing physical phenomena at different scales. We are mostly interested in spectral methods, which have interesting applications in other contexts. The dynamical systems we study are divided into three classes: uniformly hyperbolic systems; nonuniformly hyperbolic and infinitemeasure systems; parabolic systems.
Uniformly hyperbolic systems. We are interested in the statistical properties of single and coupled systems. The main properties we study are the existence of rare events and their frequency, and the robustness of the system to small perturbations.
Nonuniformly hyperbolic and infinitemeasure systems. In this case, many of the classical results fail. We are interested in studying the asymptotic behaviour of Birkhoff sums (i.e. the sum of an observable along the orbits of the system), the notion of mixing (i.e. the speed of the loss of memory along the orbits of the system), the analytical properties of the transfer operator and the dynamical zeta function (tools of the socalled thermodynamic formalism approach). In particular, we consider dynamical systems related to regular and multidimensional continued fractions algorithms, and to the geodesic and horocycle flows on modular surfaces.
Parabolic systems. These systems are characterised by slow convergence for the Birkhoff sums of the observables along the orbits of the system. The main idea is to accelerate the system or decompose it by looking at observables in suitable anisotropic spaces of functions. By these techniques, one obtains information on the speed of convergence to the equilibrium state and identifies eventual deviations from the typical behaviour or obstructions to it.
Members
Stochastic dynamical systems and applications
Stochastic Dynamical Systems for Climate Studies. We are interested in applying transfer operator techniques and results obtained in the case of abstract dynamical systems to certain climate models. Once a model flow has been constructed, it is possible to construct operators close in spirit to the transfer operator associated to the timeone map of the flow. The statistical properties of such a system can then be investigated through the functional analytic properties of the operator. We are particularly interested in three aspects of such a study. First, when we interpret the response of the system to the perturbations, we investigate how the invariant measure changes, i.e., the socalled “linear response” of the system (or its absence). Second, we compare our abstract asymptotic results with the existence of appropriate time scales, we do so by scouting for metastable observables, i.e., for example, the system might not be in equilibrium but might stay very close to a periodic cycle for a very long time. Last, we study the appearance and distribution of extreme climatic events. This can be done by exploring recurrence properties to special sets of the phase space on which chosen observables are above/below certain thresholds, which encode the extreme event.
Thermodynamic Formalism, Stochastic Filter and Stochastic Billiards. We study the ergodic properties of stochastic dynamical systems generated by transformations with hyperbolic behaviour on average through the analysis of the spectrum of the transfer operator (thermodynamic formalism). This approach has interesting applications to the problem of the stability of the stochastic filter in probability theory. The problem consists in obtaining an optimal estimate of the state of the system starting from a sequence of observations affected by noise and showing that such an estimate loses memory of its initial condition. We are also interested in the study of stochastic billiards that are obtained from a deterministic billiard by replacing the law of specular reflection with a stochastic reflection law. An orbit of a stochastic billiard table is a Markov chain with transition probability depending on the geometry of the billiard table and the law of reflection. The goal is to show the existence of stationary measures for a large family of billiards with stochastic reflection. Of particular interest is the comparison between the statistical properties of a stochastic billiard and those of the corresponding deterministic billiard.
Members
Collaborators
Dynamical systems with singularities and billiards
Hyperbolic Systems with Singularities. A hyperbolic dynamical system is characterized by the presence of expanding and contracting directions. This property produces a complex behavior that in many ways appears stochastic despite the deterministic nature of the system. We are interested in hyperbolic dynamical systems with piecewise regular dynamics (systems with singularities). They represent natural models for mechanical systems with collisions such as billiards. The main aim of the project is the study of the ergodic properties of hyperbolic systems with singularities. In particular, we want to study the existence and properties of special invariant measures called SRB that describe the statistical properties of most of the system’s orbits. We are also interested in the construction of hyperbolic billiards and the analysis of their ergodic properties. Conditions on the geometry of the billiard table and on the law of reflection that guarantee the hyperbolicity of billiard tables in the plane are well known. The goal is to weaken these conditions and extend them to billiards in Euclidean spaces of any dimension.
Generic Properties of Convex Billiards. We investigate generic properties of billiards in convex regions of any size with a regular border (Birkhoff billiards). A property of particular interest is the positivity of the topological entropy which guarantees the existence of a nontrivial hyperbolic set (Smale’s horseshoe). The genericity of this property is known for Birkhoff billiards in the plane. The extension of this result to Birkhoff billiards in arbitrary dimensions is the main objective of this project.
Members
Collaborators
People
Faculty
Name  Surname  Personal Card  

Giulio  Baù  giulio.bau@unipi.it  
Claudio  Bonanno  claudio.bonanno@unipi.it  
Gianluigi  Del Magno  gianluigi.delmagno@unipi.it  
Paolo  Giulietti  paolo.giulietti@unipi.it  
Giovanni Federico  Gronchi  giovanni.federico.gronchi@unipi.it  
Giacomo  Lari  giacomo.lari@unipi.it  
Cyrill  Muratov  cyrill.muratov@unipi.it  
Giacomo  Tommei  giacomo.tommei@unipi.it 
Affiliate Members
Name  Surname  Personal Card  

Giovanni  Cimatti  giovanni.cimatti@unipi.it  
Alessio  Del Vigna  alessio.delvigna@dm.unipi.it  
Daniele  Serra  daniele.serra.ds@gmail.com 
Postdoctoral Fellows
Name  Surname  Personal Card  

Roberto  Castorrini  roberto.castorrini@gmail.com  
Roberto  Paoli  roberto.paoli@dm.unipi.it 
Ph.D. Students at the University of Pisa
Name  Surname  Personal Card  

Nicola  Bertozzi  nicola.bertozzi@phd.unipi.it  
Miriam  Falletta  miriam.falletta@phd.unipi.it  
Clara  Grassi  clara.grassi@phd.unipi.it  
Maddalena  Mochi  maddalena.mochi@phd.unipi.it 
Ph.D. Students at other institutions
Name  Surname  Affiliation  

Andrea  Scarciglia  Department of Information Engineering (DII), Università di Pisa 
Ph.D. Theses supervised by members of the group
awarded by the University of Pisa
Year  Name  Surname  Title of the Thesis  Supervisor(s) 

2017  Melaine  Saillenfest  Secular theories and orbital dynamics beyond Neptune  Giacomo Tommei 
awarded by another institution
Year  Name  Surname  Title of the Thesis  Institution  Supervisor(s) 

2017  Davide  Amato  Advanced orbit propagation methods applied to asteroids and space debris  Universidad Politécnica de Madrid  Giulio Baù and Claudio Bombardelli 
2015  Imen  Chouari  Opérateurs de transfert en Théorie Ergodique associés à la transformation de Farey  Université de Tunis El Manar  Claudio Bonanno 
Grants
Current

Monitoraggio Asteroidale (MONitoring ASTERoids) (ASI Project)
Principal Investigator: Giacomo Tommei
Project period: Dec 05, 2022 – Dec 04, 2025

Attività scientifiche per la Missione HERA (ASI Project)
Principal Investigator: Paolo Tortora
Coordinator of the Research Unit: Giacomo Tommei
Project period: Apr 06, 2022 – Apr 05, 2025

Radio scienza: attività scientifiche per la fase E della missione Bepi Colombo e per la missione estesa Juno (ASI Project)
Principal Investigator: Luciano Iess
Coordinator of the Research Unit: Giacomo Tommei
Project period: Aug 02, 2022 – Aug 01, 2024

ICELMECH "New frontiers of Celestial Mechanics: theory and applications" (Prin 2017)
Principal Investigator: Massimiliano Guzzo
Coordinator of the Research Unit: Giulio Baù
Project period: Aug 01, 2019 – Aug 19, 2023

Regular and stochastic behaviour in dynamical systems (Prin 2017)
Principal Investigator: Carlangelo Liverani
Coordinator of the Research Unit: Claudio Bonanno
Project period: Aug 01, 2019 – Aug 19, 2023
Past

Sistemi dinamici in logica, geometria, fisica matematica e scienza delle costruzioni (Progetti di Ricerca di Ateneo (PRA) 2020  2021)
Principal Investigator: Giacomo Tommei
Project period: Jul 07, 2020 – Dec 31, 2022

Stardust Reloaded (MSCAITN2018  Innovative Training Networks)
Coordinator of the Research Unit: Giovanni Federico Gronchi
Project period: Nov 21, 2018 – Dec 31, 2022

Deterministic and stochastic dynamical systems for climate studies (Progetto Giovani GNFM)
Principal Investigator: Paolo Giulietti
Project period: Feb 01, 2021 – Jul 31, 2022

Radio scienza per BepiColombo e Juno: prosecuzione attività scientifiche (ASI Project)
Principal Investigator: Luciano Iess
Coordinator of the Research Unit: Giacomo Tommei
Project period: Apr 21, 2020 – Apr 20, 2022

Radio scienza per BepiColombo e Juno: prosecuzione attività scientifiche (ASI Project)
Principal Investigator: Luciano Iess
Coordinator of the Research Unit: Giacomo Tommei
Project period: Dec 21, 2017 – Mar 20, 2020

Sistemi dinamici in analisi, geometria, logica e meccanica celeste (Progetti di Ricerca di Ateneo (PRA) 2017  2018)
Principal Investigator: Marco Abate
Project period: Apr 10, 2017 – Apr 09, 2019

Radioscienza per BepiColombo e Juno  fasi B2/C/D  Attività scientifiche (ASI Project)
Principal Investigator: Luciano Iess
Coordinator of the Research Unit: Andrea Milani Comparetti
Project period: Dec 23, 2009 – Jul 20, 2017

ITN Stardust  Workpackage CloseRange Navigation and Manipulation of Space Debris and Asteroids (FP7 Marie Curie Initial Training Networks (ITN) scheme)
Coordinator of the Research Unit: Giovanni Federico Gronchi
Project period: Feb 01, 2013 – Jan 31, 2017

Radioscienza per BepiColombo e Juno  fasi B2/C/D  Attività scientifiche (ASI Project)
Principal Investigator: Luciano Iess
Coordinator of the Research Unit: Andrea Milani Comparetti
Project period: Jan 12, 2007 – Jan 11, 2009
Visitors
Prospective
First Name  Last Name  Affiliation 

Ana  Sanches Rivero  Universidad de Sevilla 
Current
First Name  Last Name  Affiliation  Building  Floor  Office 

Simon  Theresa  Building A  2 
Grouped by year
2023
First Name  Last Name  Affiliation  From  To 

Giovanni  Panti  Università di Udine  Feb 14, 2023  Feb 17, 2023 
Davide  Ravotti  University of Vienna  Feb 13, 2023  Feb 18, 2023 
Ana  Sanches Rivero  Universidad de Sevilla  Sep 15, 2023  Dec 14, 2024 
Matteo  Tanzi  CNRS and Université de ParisSorbonne Université  Jan 25, 2023  Jan 28, 2023 
Simon  Theresa  Mar 27, 2023  Apr 02, 2023 
2022
First Name  Last Name  Affiliation  From  To 

Claudio  Bombardelli  Jul 24, 2022  Aug 17, 2022  
Giovanni  Canestrari  Università di Roma Tor Vergata  Nov 09, 2022  Nov 09, 2022 
Alessandro  Della Corte  Università di Camerino  Nov 30, 2022  Nov 30, 2022 
José Pedro  Gaivão  ISEG, Universidade de Lisboa  Oct 27, 2022  Oct 28, 2022 
Mikael  Granvik  University of Helsinki  Nov 07, 2022  Nov 11, 2022 
Robert  Jedicke  University of Hawaii  Oct 31, 2022  Nov 04, 2022 
Alexey  Korepanov  LPSM, Sorbonne Université, Université Paris Cité, CNRS  Dec 12, 2022  Dec 17, 2022 
Philipp  Kunde  Universität Hamburg  Dec 12, 2022  Dec 17, 2022 
Carlangelo  Liverani  Università di Roma Tor Vergata  Dec 12, 2022  Dec 17, 2022 
João  Lopes Dias  ISEG, Universidade de Lisboa  Oct 27, 2022  Oct 28, 2022 
João  Lopes Dias  ISEG, Universidade de Lisboa  Oct 08, 2022  Oct 19, 2022 
Melaine  Saillenfest  IMCCE, Observatoire de Paris  Dec 01, 2022  Dec 03, 2022 
Tanja  Schindler  University of Vienna  May 27, 2022  Jul 02, 2022 
Paolo  Tortora  Università di Bologna  Nov 07, 2022  Nov 11, 2022 