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 (IFAC -- CNR)
Daniele Serra (Università di Pisa)
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.
Giulio Baù
Giovanni Federico Gronchi
Giacomo Lari
Giacomo Tommei
Robert Jedicke (University of Hawaii)
Zoran Knežević (Astronomical Observatory Belgrade)
Erica Scantamburlo (Università di Pisa)
Federica Spoto (Center for Astrophysics | Harvard & Smithsonian)
Giovanni Valsecchi (Istituto Nazionale di Astrofisica, Osservatorio Astrofisico di Torino)
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.
Giulio Baù
Giovanni Federico Gronchi
Giacomo Lari
Claudio Bombardelli (Universidad Politécnica de Madrid)
Giordana Bucchioni (Università di Pisa)
Irene Cavallari (Università di Pisa)
Javier Hernando-Ayuso (ISpace)
Melaine Saillenfest (ICMCCE -- Observatoire de Paris)
Giovanni Valsecchi (Istituto Nazionale di Astrofisica, Osservatorio Astrofisico di Torino)
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.

Giovanni Federico Gronchi
Marco Fenucci (Near-Earth Object Coordination Centre - ESA)
Maria Giovanna Fusco (Università degli Studi dell'Aquila)
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.

Claudio Bonanno
Vincenzo Catrambone (Università di Pisa)
Andrea Scarciglia (Università di Pisa)
Gaetano Valenza (Università di Pisa)
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.

Claudio Bonanno
Carlo Carminati
Stefano Galatolo
Paolo Giulietti
Mauro Artigiani (Universidad del Rosario)
Oliver Butterley (Università degli Studi di Roma Tor Vergata)
Roberto Castorrini (SNS, Pisa)
Thomas Garrity (Williams College)
Stefano Isola (Università di Camerino)
Marco Lenci (Università di Bologna)
Davide Ravotti (Universität Wien)
Tanja Schindler (Universität Wien)
Matteo Tanzi (King's College London)
Sandro Vaienti (Centre de Physique Théorique)
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.

Gianluigi Del Magno
Stefano Galatolo
Paolo Giulietti
Jochen Broecker (University of Reading)
João Lopes Dias (ISEG - Universidade de Lisboa)
José Pedro Romana Gaivão (ISEG - Universidade de Lisboa)
Sandro Vaienti (Centre de Physique Théorique)
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.

Gianluigi Del Magno
Mario Bessa (Universidade da Beira Interior)
Paulo Duarte (Universidade de Lisboa)
João Lopes Dias (ISEG - Universidade de Lisboa)
José Pedro Romana Gaivão (ISEG - Universidade de Lisboa)
Manuel Torres (INESC TEC)


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
Former Members
Name Surname
Gianfranco Capriz
Andrea Milani Comparetti
Postdoctoral Fellows
Name Surname Email Personal Card
Clara Grassi
Roberto Paoli
Mirco Piccinini
Ph.D. Students at the University of Pisa
Name Surname Email Personal Card
Nicola Bertozzi
Anargyros Dogkas
Miriam Falletta
Francesco Geroni
Maddalena Mochi
Ph.D. Students at other institutions
Name Surname Affiliation
Andrea Scarciglia 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)
2023 Irene Cavallari Patched dynamics and perturbation theory in some problems of Celestial Mechanics Giulio Baù and Giovanni Federico Gronchi
2020 Marco Fenucci Variational methods and Hamiltonian perturbation theory applied to the N-body problem: a theoretical and computational approach Giovanni Federico Gronchi
2018 Hélène Ma On the determination of preliminary orbits and detection of possible conjunctions for space debris Giovanni Federico Gronchi and Andrea Milani Comparetti
2018 Alessio Del Vigna On Impact Monitoring of Near-Earth Asteroids Andrea Milani Comparetti
2018 Giacomo Lari The Galilean satellites' dynamics and the estimation of the Jovian system's dissipation from JUICE data Andrea Milani Comparetti
2017 Melaine Saillenfest Secular theories and orbital dynamics beyond Neptune Giacomo Tommei
2016 Daniele Serra Satellite Geodesy of other planets Andrea Milani Comparetti
2012 Stefano Cicalò On theoretical and experimental aspects of the rotation of celestial bodies Andrea Milani Comparetti
2012 Davide Farnocchia Population orbit determination: Algorithms and Simulations Andrea Milani Comparetti
2006 Giacomo Tommei Impact monitoring of Near-Earth Objects: theoretical and computational results Andrea Milani Comparetti
2002 Giovanni Federico Gronchi Theoretical and computational aspects of collision singularities in the N–body problem Andrea Milani Comparetti
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
2010 Linda Dimare Problems of Celestial Mechanics Sapienza Università di Roma Giovanni Federico Gronchi and Piero Negrini




First Name Last Name Affiliation
David Vokroulicky Charles University - Univerzita Karlova
First Name Last Name Affiliation Building Floor Office
Ana Sanches Rivero Universidad de Sevilla Building A Ground floor
Grouped by year
First Name Last Name Affiliation From To
Anna Miriam Benini Università di Parma Jan 16, 2024 Jan 17, 2024
Ana Sanches Rivero Universidad de Sevilla Sep 15, 2023 Dec 14, 2024
Theresa Simon Universität Münster Feb 22, 2024 Feb 27, 2024
Cinzia Soresina Università degli Studi di Trento Feb 21, 2024 Feb 23, 2024
David Vokroulicky Charles University - Univerzita Karlova Jun 18, 2024 Jun 27, 2024
First Name Last Name Affiliation From To
Mauro Artigiani Universidad del Rosario Oct 16, 2023 Oct 18, 2023
Adriano Barra Università del Salento May 31, 2023 Jun 01, 2023
Tien-Cuong Dinh National University of Singapore Oct 23, 2023 Oct 23, 2023
Giovanni Panti Università degli Studi di Udine Feb 14, 2023 Feb 17, 2023
Davide Ravotti Universität Wien Feb 13, 2023 Feb 18, 2023
Davide Ravotti Universität Wien Sep 03, 2023 Sep 09, 2023
Ana Sanches Rivero Universidad de Sevilla Sep 15, 2023 Dec 14, 2024
Theresa Simon Universität Münster Mar 27, 2023 Apr 02, 2023
Matteo Tanzi CNRS, Sorbonne Université Jan 25, 2023 Jan 28, 2023
Matteo Tanzi Laboratoire de Probabilités, Statistique et Modélisation Dec 19, 2023 Dec 21, 2023
Paulo Varandas Universidade Federal da Bahia Oct 04, 2023 Oct 12, 2023
First Name Last Name Affiliation From To
Claudio Bombardelli Universidad Politécnica de Madrid Jul 24, 2022 Aug 17, 2022
Giovanni Canestrari Università degli Studi di Roma Tor Vergata Nov 09, 2022 Nov 09, 2022
Alessandro Della Corte Università di Camerino Nov 30, 2022 Nov 30, 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 Laboratoire de Probabilités, Statistique et Modélisation Dec 12, 2022 Dec 17, 2022
Philipp Kunde Universität Hamburg Dec 12, 2022 Dec 17, 2022
Carlangelo Liverani Università degli Studi di Roma Tor Vergata Dec 12, 2022 Dec 17, 2022
João Lopes Dias ISEG - Universidade de Lisboa Oct 08, 2022 Oct 19, 2022
João Lopes Dias ISEG - Universidade de Lisboa Oct 27, 2022 Oct 28, 2022
José Pedro Romana Gaivão ISEG - Universidade de Lisboa Oct 27, 2022 Oct 28, 2022
Melaine Saillenfest ICMCCE -- Observatoire de Paris Dec 01, 2022 Dec 03, 2022
Tanja Schindler Universität Wien May 27, 2022 Jul 02, 2022
Paolo Tortora Università di Bologna Nov 07, 2022 Nov 11, 2022
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