# 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 is 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).

##### Members:
• Giacomo Lari
• Orlando Luongo
• Giacomo Tommei
##### 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
• Clara Grassi
• Giovanni Federico Gronchi
• Giacomo Lari
• Óscar Rodríguez Del Río
• Erica Scantamburlo
• Giacomo Tommei
##### Collaborators:
• R. Jedicke (Univ. Hawaii, US)
• Z. Knežević (Univ. Belgrade, Serbia)
• F. Spoto (MPC-Harvard, USA)
• G.B. Valsecchi (INAF-IAPS)
##### Solar system dynamics and astrodynamics
• 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°. From the formation theories, we know that probably gas giants formed with an almost null obliquity; it is then necessary to understand what mechanisms allowed planets to reach the values observed nowadays. So far, the most promising theories predicted that planets obtained their current obliquities shortly after their formation, in a period where the orbits of the planets were still moving (more than 4 billion years ago), for example through capture into secular spin-orbit resonances (Saturn) or giant impacts (Uranus). However, the fast migration of satellites around gas giants provides a new and fascinating scenario. Through suitable averaged dynamical models, we want to investigate the long-term evolution of the spin-axis of the gas giants of our solar system. Taking into account the migration of satellites due to tidal dissipation, the precession frequency of the spin-axis of the planet is continuously changing, allowing the capture of the planet into secular spin-orbit resonances and the consequent increase of obliquity much later than the end of the late planetary migration, differently from what was thought so far.
• 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. A clear example is Io, which presents strong volcanism all over its surface due to the tides raised by Jupiter. Apart from dissipating energy in form of heat within celestial bodies, tidal dissipation of planet and satellites produce a significant variation of the angular momentum and orbital energy of the moons, with a consequent effect on their semi-major axis and then on their distance from the planet. The migration of satellites allows the crossing of mean motion resonances, from which the moons can escape or be captured, shaping the orbital configuration of the system. Through suitable averaged dynamical models, we want to investigate the long-term evolution of the satellites of our solar system, with a special focus on the Galilean satellites of Jupiter. Indeed, the Jovian system presents a strong tidal dissipation and a three-body chain of resonances, known as Laplace resonance, whose origin is still an open question in celestial mechanics.
• 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
• Giordana Bucchioni
• Irene Cavallari
• Clara Grassi
• Giovanni Federico Gronchi
• Giacomo Lari
##### Collaborators:
• C. Bombardelli (UPM Madrid, Spain)
• G. Boué (Observatoire de Paris)
• J. Hernando-Ayuso (ispace, Japan)
• M. Saillenfest (Observatoire de Paris)
• Giovanni B. Valsecchi (INAF-IAPS)
##### 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
##### Collaborators:
• G. Fusco (Univ. L’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.

##### Members:
• Claudio Bonanno
##### 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
• Carlo Carminati
• Roberto Castorrini
• Stefano Galatolo
• Paolo Giulietti
##### 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
• Stefano Galatolo
• Paolo Giulietti
##### Collaborators:
• J. Bröcker (Univ. Reading, UK)
• J. Gaivão, J. Lopes Dias (ISEG – Univ. Lisbon, Portugal)
• S. Vaienti (Centre de Physique Théorique – Marseille, Francia)
##### 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
##### Collaborators:
• M. Bessa (Univ. Beira Interior, Portugal)
• P. Duarte (Univ. Lisbon, Portugal)
• J. Gaivão, J. Lopes Dias (ISEG – Univ. Lisbon, Portugal)
• M. Torres (Univ. Minho, Portugal)

#### Members

##### Staff
GianluigiDel Magno[Mathscinet] [Orcid]
Giovanni FedericoGronchi[Mathscinet] [Orcid]
##### Postdoctoral Fellows
GiordanaBucchioni[Orcid]
RobertoCastorrini
IreneCavallari[Mathscinet] [Orcid]
OrlandoLuongo[Mathscinet]
ÓscarRodríguez Del Río[Mathscinet] [Orcid]
EricaScantamburlo
##### Ph.D. Students
IreneCavallari[Mathscinet] [Orcid]
ClaraGrassi
##### External Collaborators
GianfrancoCapriz[Mathscinet]
AlessioDel Vigna[Mathscinet] [Orcid]
StefanoMarò[Mathscinet] [Orcid]

#### Grants

• ##### Radioscienza per BepiColombo e Juno: prosecuzione attività scientifiche (ASI Addendum Contract)

Principal Investigator: Luciano Iess (Università di Roma La Sapienza) | Coordinator of the Research Unit: Giacomo Tommei

Members of the Research Unit: Giacomo Lari, Orlando Luongo, Giulia Schettino, Daniele Serra

• ##### Stardust – Reloaded (H2020-EU.1.3.1. - Excellent Science - Marie Skłodowska-Curie Actions)

Coordinator of the Research Unit: Giovanni F. Gronchi

Members of the Research Unit: G. Baù, I. Cavallari, O. Rodriguez

• ##### Attività scientifiche per la Missione Hera (Missione HERA - ASI- Università di Bologna)

Principal Investigator: Paolo Tortora (Università di Bologna) | Coordinator of the Research Unit: Giacomo Tommei

Members of the Research Unit: G. Lari

• ##### New frontiers of Celestial Mechanics: theory and applications (Prin 2017)

Principal Investigator: Alessandra Celetti (Università di Roma Tor Vergata) | Coordinator of the Research Unit: Giulio Baù

Members of the Research Unit: C. Grassi, G.F. Gronchi

• ##### Regular and stochastic behaviour in dynamical systems (Prin 2017)

Principal Investigator: Carlangelo Liverani (Università di Roma Tor Vergata) | Coordinator of the Research Unit: Claudio Bonanno

Members of the Research Unit: G. Del Magno, S. Marò

• ##### Sistemi dinamici in logica, geometria, fisica matematica e scienza delle costruzioni (Progetti di Ateneo)

Coordinator of the Research Unit: Giacomo Tommei

Members of the Research Unit: Marco Abate, Riccardo Barsotti, Giulio Baù, Alessandro Berarducci, Gianluigi Del Magno, Mauro Di Nasso, Giovanni Federico Gronchi, Stefano Marò, Giacomo Lari, Daniele Serra, David Antonio Riccardo Mustaro

• ##### Radioscienza per BepiColombo e Juno – fasi B2/C/D – Attività scientifiche (ASI Contract)

Principal Investigator: Luciano Iess (Università di Roma La Sapienza) | Coordinator of the Research Unit: Andrea Milani

Members of the Research Unit: Giacomo Tommei

• ##### Radioscienza per BepiColombo e Juno – fasi B2/C/D – Attività scientifiche (ASI Contract)

Principal Investigator: Luciano Iess (Università di Roma La Sapienza) | Coordinator of the Research Unit: Andrea Milani

Members of the Research Unit: Giacomo Tommei

• ##### Radioscienza per BepiColombo e Juno: prosecuzione attività scientifiche (ASI Contract)

Principal Investigator: Luciano Iess (Università di Roma La Sapienza) | Coordinator of the Research Unit: Giacomo Tommei

Members of the Research Unit: Giacomo Lari, Giulia Schettino, Daniele Serra

• ##### 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 F. Gronchi

Members of the Research Unit: H. Ma, S. Marò

• ##### Sistemi dinamici in analisi, geometria, logica e meccanica celeste (Progetti di Ateneo)

Coordinator of the Research Unit: Marco Abate

Members of the Research Unit: Marco Abate, Claudio Bonanno, Mauro Di Nasso, Stefano Galatolo, Alessandro Berarducci, Giovanni Federico Gronchi, Carlo Carminati, Giacomo Tommei