On the stability of periodic N-body motions with the symmetry of Platonic polyhedra

In (Fusco et. al., 2011) several periodic orbits of the Newtonian N-body problem have been found as minimizers of the Lagrangian action in suitable sets of T-periodic loops, for a given T>0. Each of them share the symmetry of one Platonic polyhedron. In this paper we first present an algorithm to enumerate all the orbits that can be found following the proof in (Fusco et. al., 2011). Then we describe a procedure aimed to compute them and study their stability. Our computations suggest that all these periodic orbits are unstable. For some cases we produce a computer-assisted proof of their instability using multiple precision interval arithmetic

The low surface thermal inertia of the rapidly rotating near-Earth asteroid 2016 GE1

Asteroids smaller than about 100 meters are observed to rotate very fast, with periods often much shorter than the critical limit of 2.2 h. Some of these super-fast rotators can also achieve a very large semi-major axis drift induced by the Yarkovsky effect, that in turn, is determined by internal and surface physical properties. We consider the small super-fast rotating near-Earth asteroid 2016 GE1. This object rotates in just 34 seconds, and a large Yarkovsky effect has been determined from astrometry. Here we aim to constrain the thermal inertia of the surface of this extreme object. We used a recently developed statistical method to determine the thermal properties of near-Earth asteroids. The method is based on the comparison between the observed and the modelled Yarkovsky effect, and the thermal conductivity (inertia) is determined by a Monte Carlo approach. Parameters of the Yarkovsky effect model are either fixed if their uncertainty is negligible, modelled with a Gaussian distribution of the errors if they are measured, or deduced from general properties of the population of near-Earth asteroids when they are unknown. Using a well-established orbit determination procedure, we determined the Yarkovsky effect on 2016 GE1, and verified a significant semi-major axis drift rate. Using a statistical method, we showed that this semi-major axis drift rate could be explained only by low thermal inertia values below 100 J m$^{-2}$ K$^{-1}$ s$^{-1/2}$: namely, 90\% of the probability density function of the model outcomes is contained at values smaller than 100 J m$^{-2}$ K$^{-1}$ s$^{-1/2}$. We propose two possible interpretations for the extremely low values: a high porosity or a cracked surface, or a thin layer of fine regolith on the surface. Though this seems unexpected in either case, it opens up the possibility of a subclass of low thermal inertia, super-fast rotating asteroids.Comment: Accepted for publication on A&

Riflessioni sullo stato di guerra

War is traditionally regarded as the ultimate expression of the emergency. Typically, each legal order has constitutional rules dedicated to the warfare. Waging war allows you to disregard some constitutional rules. Nowadays in most constitutional legal orders the war decision is officially committed to the Parliament while the Government has the real war power

On the linear stability of some periodic orbits of the N-body problem with the simmetry of Platonic polyhedra

In the last few years, many interesting periodic motions of the classical Newtonian N-body problem have been discovered as minimizers of the Lagrangian action functional on a particular subset of T-periodic loops. The interest in this classical problem was revived by the numerical discovery of the now famous figure eight solution of the three-body problem, by C. Moore in 1993. In 2000 A. Chenciner and R. Montgomery rediscover this particular orbit, giving a formal proof of its existence that uses the direct method of Calculus of Variations. The figure eight is a first example of a N-body choreography, that is a solution of the classical N-body problem in which N equal masses chase each other around a fixed closed curve, equally spaced in phase along the curve: since 2000 many other choreographies that present a strong symmetrical structure have been found. Moreover, in 2007 T. Kapela and C. Simò proved the linear stability of the figure eight: this fact is quite surprising, since that no other stable choreographies are known. In this thesis we prove the existence of a number of periodic motions of the classical N-body problem which, up to relabeling of the N particles, are invariant under the rotation group R of one of the five Platonic polyhedra. The number N coincides with the order of the rotation group R and the particles have all the same mass. We use again variational techniques to minimize the Lagrangian action A on a suitable subset K of the H^1 T-periodic maps. The set K is a cone and is determined by imposing on $u$ both topological and symmetry constraints which are defined in terms of the rotation group R. For a certain number of cones K, using level estimates and local perturbations, we show that minimizers are free of collisions and therefore they are classical solutions on the N-body problem. A natural question that comes out in presence of a periodic orbit is whether is it stable or not. To perform a study of the linear stability we use numerical methods, since our problem is not integrable. In fact we know only that periodic orbits with the previous symmetries exist, but we do not have their analytic expression. These particular solutions were found numerically, using a method described by C. Moore and called relaxation dynamics. The numerical implementation of this method boils down to a gradient search of the minima in some finite-dimensional approximation of the path space: in short, it is a numerical implementation of a direct method of Calculus of Variations. Starting from these numerical solutions, we can propagate numerically the variational equation in order to produce an approximation of the monodromy matrix, from which we can determine the linear stability studying its spectral properties: this is a first method that we develop. However, because of the convergence of the gradient search is slow, especially when the orbit presents some close approaches, this method could result inefficient. An alternative approach is to find an initial condition of the periodic orbit and then propagate numerically the equation of motion and the variational equation coupled together. A classical method to find initial conditions is the well known multiple shooting method. This method has been successfully used by T. Kapela and C. Simò to find initial conditions for the figure eight and some other non-symmetric choreographies. However, since this is an iterative method too, it could fail to converge and this typically happens when the orbit passes close to a collisions. Therefore, it is clear that the problem of close approaches must be treated with more care. The thesis is structured as follows: Chapter 1. It contains results on the existence of periodic orbits of the classical N-body problem with the symmetries of Platonic polyhedra. Chapter 2. In this chapter we try to develop an automatic procedure in order to find all the periodic orbits described in Chapter 1. Chapter 3. We present the classical theory of linear stability for periodic solutions of autonomous systems. In particular, we introduce here the monodromy matrix, the Floquet multipliers and the Poincaré map. Chapter 4. It is the heart of the work, in which we develop the two different numerical methods to study the linear stability of periodic orbits found in Chapters 1 and 2. Tests of the software written are reported at the end of the chapter. Chapter 5. In this chapter we list all the results obtained with our software, from which we can get some conclusions. At the end we suggest some improvements of the methods of Chapter 4 and of the software, which could represent a continuation of the present work, in order to produce a true computer assisted proof of stability or instability of these orbits

Requisitos formais e requisitos sustanciais das Constituições

A análise da perspectiva constitucional da lei não pode ser estudada sem uma cuidadosa consideração do conceito de Constituição, o qual não reúne apenas o objeto, mas também os princípios constitucionais. Primeiramente, a Constituição é analisada como uma especial fonte do direito, a essência. Esta é a lei fundamental, cujas provisões resultam em efeitos imediatos sem intermediários legislativos. Ademais, estudos são necessários para definir a Constituição como Lei. Tendo realizado a distinção entre a lei formal e a essencial, os principais requisitos da lei enfatizados são o vigor do típico significado da efetividade da lei e o “valor” que representa a totalidade do regime da fonte legislativa. Até mesmo para a Constituição, cuja lei, no sentido técnico, os requisitos formais, para uma Constituição inflexível são a força e o valor da lei constitucional, prevalecendo acima dos requisitos essenciais. Entretanto, se o núcleo rígido da Constituição é, como se costuma dizer, o destaque dos ideais fundamentais e dos princípios supremos inexpugnáveis mesmo pela comunidade da lei. Estes requisitos essenciais fazem a função histórica e filosófica da Constituição e o conseqüente reflexo cultural, social e cultural-religioso da população. Sua importância é apreciada caso se considere o nascimento da Constituição e as diferentes fases pelas quais passou desde que foi elaborada. Emblematicamente, as palavras são ditas por um eminente constituinte, Piero Calamandrei, na assembléia constituinte, em 4 de março de 1947, como testemunha da tensão moral que acompanhou o nascimento da nova Constituição republicana. A força inovadora dos princípios fundamentais foi inicialmente contrastada por uma interpretação conservadora que a considerava como normas programáticas. O agradecimento ao Tribunal Constitucional pela lógica interpretativa tem sido subvertido, de um lado conferindo aos princípios e valores supremos completa efetividade e de outro lado confirmando sua natureza de parâmetros de legitimidade constitucional da lei e de similares. Tradução: Juliana Salvetti

Targeted killings in the U.S. policy

Targeted killings are the planned killing of alleged terrorists by the government if their capture is impossible, carried out mainly by unmanned aircrafts armed with missiles and remotely controlled, the so-called drones. Targeted killings are performed not necessarily in a war contest and sometimes without the approval of the sovereign State where the action is carried out. The decision to deal with the struggle against international terrorism as war emergency, endorsed by the U.S. administration with both Bush and Obama, had an impact on fundamental rights, the protection of which was limited just because of war emergency

A Real Time Distributed Approach to Collision Avoidance for Industrial Manipulators

Robot interaction with the surrounding environment is an important and newsworthy problem in the context of industrial and service robotics. Collision avoidance gives the robot the ability to avoid contacts with objects around it, but most of the industrial controls implementing collision avoidance checks only the robot Tool Center Point (TCP) over the objects in the cell, without taking into account the shape of the tool, mounted on the robot flange. In this paper a novel approach is proposed, based on an accurate 3D simulation of the robotic cell. A distributed real time computing approach has been chosen to avoid any overloading of the robot controller. The simulator and the client application are implemented in a personal computer, connected via a TCP-IP socket to the robot controller, which hosts and manages the anti-collision policies, based on a proper speed override control. The real time effectiveness of the proposed approach has been confirmed by experimental tests, carried out for a real industrial setup in two different scenarios

A Distributed, Passivity-Based Control of Autonomous Mobile Sensors in an Underwater Acoustic Network

This paper presents a cooperative and distributed control law for multiple Autonomous Underwater Vehicles (AUVs) executing a mission while meeting mutual communication constraints. Virtual couplings define interaction control forces between neighbouring vehicles. Moreover, the couplings are designed to enforce a desired vehicle-vehicle and vehicle-target spacing. The whole network is modelled in the passive, energy-based, port-Hamiltonian framework. Such framework allows to prove closed-loop stability using the whole system kinetic and virtual potential energy by constructing a suitable Lyapunov function. Furthermore, the robustness to communication delays is also demonstrated. Simulation results are given to illustrate the effectiveness of the proposed approach

Maps of secular resonances in the NEO region

Context. From numerical simulations, it is known that some secular resonances may affect the motion of near-Earth objects (NEOs). However, the specific location of the secular resonance inside the NEO region is not fully known, because the methods previously used to predict their location can not be used for highly eccentric orbits and when the NEOs cross the orbits of the planets. Aims. In this paper, we aim to map the secular resonances with the planets from Venus to Saturn in the NEO region, even for high values of the eccentricity. Methods. We used an averaged semi-analytical model that can deal with orbit crossing singularities for the computation of the secular dynamics of NEOs, from which we can obtain suitable proper elements and proper frequencies. Then, we computed the proper frequencies over a uniform grid in the proper elements space. Secular resonances are thus located by the level curves corresponding to the proper frequencies of the planets. Results. We determined the location of the secular resonances with the planets from Venus to Saturn, showing that they appear well inside the NEO region. By using full numerical N-body simulations we also showed that the location predicted by our method is fairly accurate. Finally, we provided some indications about possible dynamical paths inside the NEO region, due to the presence of secular resonances.Comment: Accepted for publication in A&