# Mathematical Physics

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 spin-off 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 post-impact 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 non-gravitational perturbations acting on the system (Yarkovsky-YORP-BYORP effects).

Giulia Schettino
##### 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 Near-Earth Asteroids (NEAs). Such work led to the realization of the online services AstDyS (it contains data about numbered and multi-opposition 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 non-exhaustive list of sub-topics is the following.

• New initial OD methods for very short arcs of observations: this problem arises from the fact that the next-generation surveys (e.g. Pan-STARRS, 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
Giulio Baù
giulio.bau@unipi.it
Giovanni Federico Gronchi
giovanni.federico.gronchi@unipi.it
Giacomo Lari
giacomo.lari@unipi.it
Giacomo Tommei
giacomo.tommei@unipi.it
###### Collaborators
Robert Jedicke
jedicke@hawaii.edu
Zoran Knežević
zoran@aob.rs
Erica Scantamburlo
erica.scantamburlo.93@gmail.com
Giovanni Valsecchi
Giovanni.Valsecchi@iaps.inaf.it
##### Solar system dynamics and astrodynamics
• Long-term 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.
• Long-term 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 two-body problem based on non-singular 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
Giulio Baù
giulio.bau@unipi.it
Giovanni Federico Gronchi
giovanni.federico.gronchi@unipi.it
Giacomo Lari
giacomo.lari@unipi.it
###### Collaborators
Claudio Bombardelli
claudio.bombardelli@upm.es
Giordana Bucchioni
giordana.bucchioni@ing.unipi.it
Irene Cavallari
irene.cavallari@dm.unipi.it
Javier Hernando-Ayuso
j-hernando@ispace-inc.com
Melaine Saillenfest
melaine.saillenfest@obspm.fr
Giovanni Valsecchi
Giovanni.Valsecchi@iaps.inaf.it
##### 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 non-Newtonian 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
Giovanni Federico Gronchi
giovanni.federico.gronchi@unipi.it
###### Collaborators
Marco Fenucci
Maria Giovanna Fusco
##### 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
Claudio Bonanno
claudio.bonanno@unipi.it
Andrea Scarciglia
###### Collaborators
Vincenzo Catrambone
Gaetano Valenza
##### 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; non-uniformly hyperbolic and infinite-measure 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.

Non-uniformly hyperbolic and infinite-measure 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 so-called thermodynamic formalism approach). In particular, we consider dynamical systems related to regular and multi-dimensional 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
Claudio Bonanno
claudio.bonanno@unipi.it
Carlo Carminati
carlo.carminati@unipi.it
Roberto Castorrini
roberto.castorrini@gmail.com
Stefano Galatolo
stefano.galatolo@unipi.it
Paolo Giulietti
paolo.giulietti@unipi.it
Mauro Artigiani
Oliver Butterley
Thomas Garrity
Stefano Isola
Marco Lenci
Davide Ravotti
Tanja Schindler
Matteo Tanzi
Sandro Vaienti
##### 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 time-one 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 so-called “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
Gianluigi Del Magno
gianluigi.delmagno@unipi.it
Stefano Galatolo
stefano.galatolo@unipi.it
Paolo Giulietti
paolo.giulietti@unipi.it
###### Collaborators
Jochen Broecker
José Pedro Gaivão
jpgaivao@iseg.ulisboa.pt
João Lopes Dias
jldias@iseg.ulisboa.pt
Sandro Vaienti
##### 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 non-trivial 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
Gianluigi Del Magno
gianluigi.delmagno@unipi.it
###### Collaborators
Mario Bessa
Paulo Duarte
José Pedro Gaivão
jpgaivao@iseg.ulisboa.pt
João Lopes Dias
jldias@iseg.ulisboa.pt
Manuel Torres

#### People

##### Faculty
Name Surname Email Personal Card
Giulio Baù
Claudio Bonanno
Gianluigi Del Magno
Paolo Giulietti
Giovanni Federico Gronchi
Giacomo Lari
Cyrill Muratov
Giacomo Tommei
##### Affiliate Members
Name Surname Email Personal Card
Giovanni Cimatti
Alessio Del Vigna
Daniele Serra
##### Postdoctoral Fellows
Name Surname Email Personal Card
Roberto Castorrini
Roberto Paoli
##### Ph.D. Students at the University of Pisa
Name Surname Email Personal Card
Nicola Bertozzi
Miriam Falletta
Clara Grassi
##### Ph.D. Students at other institutions
Name Surname Affiliation Email
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

• ##### 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

• ##### I-CELMECH "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

• ##### 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 (MSCA-ITN-2018 - 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 Close-Range 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 Paris-Sorbonne 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