Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System

Bifibrations, in symplectic geometry called also dual pairs, play a relevant role in the theory of superintegrable Hamiltonian systems. We prove the existence of an analogous bifibrated geometry in dynamical systems with a symmetry group such that the reduced dynamics is periodic. The integrability of such systems has been proven by M. Field and J. Hermans with a reconstruction technique. We apply the result to the nonholonomic system of a ball rolling on a surface of revolution.Comment: This is a contribution to the Proc. of workshop on Geometric Aspects of Integrable Systems (July 17-19, 2006; Coimbra, Portugal), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Conservation of `moving' energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces

Energy is in general not conserved for mechanical nonholonomic systems with affine constraints. In this article we point out that, nevertheless, in certain cases, there is a modification of the energy that is conserved. Such a function coincides with the energy of the system relative to a different reference frame, in which the constraint is linear. After giving sufficient conditions for this to happen, we point out the role of symmetry in this mechanism. Lastly, we apply these ideas to prove that the motions of a heavy homogeneous solid sphere that rolls inside a convex surface of revolution in uniform rotation about its vertical figure axis, are (at least for certain parameter values and in open regions of the phase space) quasi-periodic on tori of dimension up to three

Moving energies as first integrals of nonholonomic systems with affine constraints

In nonholonomic mechanical systems with constraints that are affine (linear nonhomogeneous) functions of the velocities, the energy is typically not a first integral. It was shown in [Fass\`o and Sansonetto, JNLS, 26, (2016)] that, nevertheless, there exist modifications of the energy, called there moving energies, which under suitable conditions are first integrals. The first goal of this paper is to study the properties of these functions and the conditions that lead to their conservation. In particular, we enlarge the class of moving energies considered in [Fass\`o and Sansonetto, JNLS, 26, (2016)]. The second goal of the paper is to demonstrate the relevance of moving energies in nonholonomic mechanics. We show that certain first integrals of some well known systems (the affine Veselova and LR systems), which had been detected on a case-by-case way, are instances of moving energies. Moreover, we determine conserved moving energies for a class of affine systems on Lie groups that include the LR systems, for a heavy convex rigid body that rolls without slipping on a uniformly rotating plane, and for an $n$-dimensional generalization of the Chaplygin sphere problem to a uniformly rotating hyperplane.Comment: 25 pages, 1 figure. Final version prepared according to the modifications suggested by the referees of Nonlinearit

On the number of weakly Noetherian constants of motion of nonholonomic systems

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D-STEM v2: A Software for Modelling Functional Spatio-Temporal Data

Functional spatio-temporal data naturally arise in many environmental and climate applications where data are collected in a three-dimensional space over time. The MATLAB D-STEM v1 software package was first introduced for modelling multivariate space-time data and has been recently extended to D-STEM v2 to handle functional data indexed across space and over time. This paper introduces the new modelling capabilities of D-STEM v2 as well as the complexity reduction techniques required when dealing with large data sets. Model estimation, validation and dynamic kriging are demonstrated in two case studies, one related to ground-level air quality data in Beijing, China, and the other one related to atmospheric profile data collected globally through radio sounding.Comment: 29 pages, 11 figure

On the Dynamics of a Heavy Symmetric Ball that Rolls Without Sliding on a Uniformly Rotating Surface of Revolution

We study the class of nonholonomic mechanical systems formed by a heavy symmetric ball that rolls without sliding on a surface of revolution, which is either at rest or rotates about its (vertical) figure axis with uniform angular velocity Omega. The first studies of these systems go back over a century, but a comprehensive understanding of their dynamics is still missing. The system has an SO(3) x SO(2) symmetry and reduces to four dimensions. We extend in various directions, particularly from the case Omega = 0 to the case Omega not equal 0, a number of previous results and give new results. In particular, we prove that the reduced system is Hamiltonizable even if Omega not equal 0 and, exploiting the recently introduced "moving energy," we give sufficient conditions on the profile of the surface that ensure the periodicity of the reduced dynamics and hence the quasiperiodicity of the unreduced dynamics on tori of dimension up to three. Furthermore, we determine all the equilibria of the reduced system, which are classified in three distinct families, and determine their stability properties. In addition to this, we give a new form of the equations of motion of nonholonomic systems in quasi-velocities which, at variance from the well-known Hamel equations, use any set of quasi-velocities and explicitly contain the reaction forces

Conservation of energy and momenta in nonholonomic systems with affine constraints

We characterize the conditions for the conservation of the energy and of the components of the momentum maps of lifted actions, and of their `gauge-like' generalizations, in time-independent nonholonomic mechanical systems with affine constraints. These conditions involve geometrical and mechanical properties of the system, and are codified in the so-called reaction-annihilator distribution

Scenario analysis of livestock-related PM2.5 pollution based on heteroskedastic geostatistical modelling

The air in the Lombardy region, Italy, is one of the most polluted in Europe because of limited air circulation and high emissions levels. There is a large scientific consensus that the agricultural sector has a major impact on air quality. In Lombardy, livestock activities are widely acknowledged to be responsible for approximately 97% of regional ammonia emissions due to the high density of livestock. The main objective of our study is to quantify the relationship between ammonia emissions and PM2.5 concentrations in the Lombardy region and evaluate PM2.5 changes due to the reduction of ammonia emissions through scenario analysis. In particular, the study refers to the years between 2016 and 2020 inclusive. The information contained in the data is exploited using a spatiotemporal model capable of handling spatial and temporal correlation, as well as missing data. In this study, we propose a heteroskedastic extension of the Hidden Dynamic Geostatistical Model (HDGM) which is a two-level hierarchical model suitable for complex environmental processes. Scenario analysis will be carried out on high-resolution maps of the Lombardy region showing the changes in PM2.5 across the area. As a result, it is shown that a 26% reduction in NH3 emissions in the wintertime could reduce the PM2.5 average by 2.09 mg/m3 while a 50% reduction could reduce the PM2.5 average by 4.02 mg/m3 which corresponds to a reduction close to 5% and 10% respectively. Finally, results are detailed by province and land type

Agrimonia: a dataset on livestock, meteorology and air quality in the Lombardy region, Italy

The air in the Lombardy region, Italy, is one of the most polluted in Europe because of limited air circulation and high emission levels. There is a large scientific consensus that the agricultural sector has a significant impact on air quality. To support studies quantifying the role of the agricultural and livestock sectors on the Lombardy air quality, this paper presents a harmonised dataset containing daily values of air quality, weather, emissions, livestock, and land and soil use in the years 2016–2021, for the Lombardy region. The daily scale is obtained by averaging hourly data and interpolating other variables. In fact, the pollutant data come from the European Environmental Agency and the Lombardy Regional Environment Protection Agency, weather and emissions data from the European Copernicus programme, livestock data from the Italian zootechnical registry, and land and soil use data from the CORINE Land Cover project. The resulting dataset is designed to be used as is by those using air quality data for research