Stability of nearly-integrable systems with dissipation

We study the stability of a vector field associated to a nearly-integrable Hamiltonian dynamical system to which a dissipation is added. Such a system is governed by two parameters, named the perturbing and dissipative parameters, and it depends on a drift function. Assuming that the frequency of motion satisfies some resonance assumption, we investigate the stability of the dynamics, and precisely the variation of the action variables associated to the conservative model. According to the structure of the vector field, one can find linear and exponential stability times, which are established under smallness con- ditions on the parameters. We also provide some applications to concrete examples, which exhibit a linear or exponential stability behavior.Comment: 38 page

The effect of Poynting-Robertson drag on the triangular Lagrangian points

We investigate the stability of motion close to the Lagrangian equilibrium points L4 and L5 in the framework of the spatial, elliptic, restricted three- body problem, subject to the radial component of Poynting-Robertson drag. For this reason we develop a simplified resonant model, that is based on averaging theory, i.e. averaged over the mean anomaly of the perturbing planet. We find temporary stability of particles displaying a tadpole motion in the 1:1 resonance. From the linear stability study of the averaged simplified resonant model, we find that the time of temporary stability is proportional to beta a1 n1 , where beta is the ratio of the solar radiation over the gravitational force, and a1, n1 are the semi-major axis and the mean motion of the perturbing planet, respectively. We extend previous results (Murray (1994)) on the asymmetry of the stability indices of L4 and L5 to a more realistic force model. Our analytical results are supported by means of numerical simulations. We implement our study to Jupiter-like perturbing planets, that are also found in extra-solar planetary systems.Comment: 47 pages, 8 figures

The influence of orbital dynamics, shape and tides on the obliquity of Mercury

Earth-based radar observations of the rotational dynamics of Mercury (Margot et al. 2012) combined with the determination of its gravity field by MESSENGER (Smith et al. 2012) give clues on the internal structure of Mercury, in particular its polar moment of inertia C, deduced from the obliquity (2.04 +/- 0.08) arcmin. The dynamics of the obliquity of Mercury is a very-long term motion (a few hundreds of kyrs), based on the regressional motion of Mercury's orbital ascending node. This paper, following the study of Noyelles & D'Hoedt (2012), aims at first giving initial conditions at any time and for any values of the internal structure parameters for numerical simulations, and at using them to estimate the influence of usually neglected parameters on the obliquity, like J3, the Love number k2 and the secular variations of the orbital elements. We use for that averaged representations of the orbital and rotational motions of Mercury, suitable for long-term studies. We find that J3 should alter the obliquity by 250 milli-arcsec, the tides by 100 milli-arcsec, and the secular variations of the orbital elements by 10 milli-arcsec over 20 years. The resulting value of C could be at the most changed from 0.346mR^2 to 0.345mR^2.Comment: Accepted for publication in Advances in Space Researc

Resonances in the Earth's Space Environment

We study the presence of resonances in the region of space around the Earth. We consider a massless body (e.g, a dust particle or a small space debris) subject to different forces: the gravitational attraction of the geopotential, the effects of Sun and Moon. We distinguish different types of resonances: tesseral resonances are due to a commensurability involving the revolution of the particle and the rotation of the Earth, semi-secular resonances include the rates of variation of the mean anomalies of Moon and Sun, while secular resonances just depend on the rates of variation of the arguments of perigee and the longitudes of the ascending nodes of the perturbing bodies. We characterize such resonances, giving precise statements on the regions where the resonances can be found and provide examples of some specific commensurability relations.Comment: 38 pages, 9 figures, submitted to Communications in Nonlinear Science and Numerical Simulatio

Breakdown of tori in low and high dimensional conservative and dissipative standard maps

We study the breakdown of rotational invariant tori by implementing three different methods. First, we analyze the domains of analyticity of a torus with given frequency through the computation of the Lindstedt series expansions of the embedding of the torus and the drift term. The Pad\'e and log-Pad\'e approximants provide the shape of the analyticity domains by plotting the poles of the polynomial at the denominator of the Pad\'e approximants. Then, we implement a Newton's method to construct the embedding of the torus; the breakdown threshold is then computed by looking at the blow-up of the Sobolev's norms of the embedding. Finally, according to Greene's method, we estimate the breakdown threshold of an invariant torus with irrational frequency by looking at the stability of the periodic orbits with periods approximating the frequency of the torus. We apply these methods to 2-dimensional and 4-dimensional standard maps. The 2-dimensional maps can either be conservative (symplectic) or dissipative (precisely, conformally symplectic, namely a dissipative map with the geometric property to transform the symplectic form into a multiple of itself). The 4-dimensional maps are obtained coupling $(i)$ two symplectic standard maps, or $(ii)$ two conformally symplectic standard maps, or $(iii)$ a symplectic and a conformally symplectic standard map. The conformally symplectic maps depend on a dissipative parameter and a drift term, which is needed to get the existence of invariant attractors. While Pad\'e and Newton's methods perform quite well and provide reliable results, when applying Greene's method, the computation of the periodic orbits in higher dimensional, dissipative maps is particularly complex

Nekhoroshev stability in the elliptic restricted three body problem

Die Frage nach der Langzeitstabilität unseres Sonnensystems kann nach wie vor nicht völlig beantwortet werden. Die Untersuchung dieser Frage hat dennoch, in den letzten Jahrhunderten, die moderne Formulierung heutiger Wissenschaftsdisziplinen maßgeblich beeinflusst (z.B. Hamilton, Lagrange, ...). Bahnbrechende Entdeckungen, wie das KAM und Nekhoroshev Theorem, sind im Umfeld des Spezialgebiets Dynamischer Systeme auf diesem Wege entstanden. Basierend auf Zweiterem, soll in dieser Arbeit gezeigt werden, das es auch um die equilateralen Gleichgewichtspunkte des elliptischen eingeschränkten Dreikörperproblems einen Nekhoroshev-stabilen Bereich gibt, der für das Zeitalter unseres Planetensystems physikalisch relevant ist. Da im Sonnensystem die Massen der Asteroiden im Vergleich zu den Massen der Planeten vernachlässigt werden können folgt aus der Stabilität von Testteilchen im eingeschränkten Dreikörperproblem ebenso die Stabilität von Asteroiden in diesem (in seiner vereinfachten Darstellung). Die Beobachtung von Asteroiden um die Lagrangepunkte, z.B. Jupiters, zusammen mit einer rein mathematischen Theorie, dem Nekhoroshev Theorem, geben Einsicht in die dynamische Entwicklung unseres Sonnensystems. Die vorliegende Studie belegt die Möglichkeit der Langzeit-Stabilität von Asteroiden im Rahmen des elliptisch eingeschränkten Dreikörperproblems. Sie erweitert somit Aussagen vorangegangener Arbeiten, welche auf dem kreisförmigen eingeschränkten Dreikörperproblem basieren. Der Nekhoroshev Bereich im Sonne-Jupiter Fall wurde auf analytischem Wege explizit bestimmt, die stabile Region stimmt bzgl. der Librationsbewegungen mit den Beobachtungen gut überein. Die Stabilität der Trojaner des Sonne-Jupiter Systems konnte für Exzentrizitäten ep<0.01 und Librationsbewegungen Dp<10° für das Zeitalter des Sonnensystems gezeigt werden.The question of the long-term stability of the Solar system is still a partly unsolved problem even nowadays. Nevertheless, the treatment of this problem by various scientists during the last centuries can be seen as the origin of the modern formulation of science, from which i.e. the KAM and the Nekhoroshev theorem originated in the field of dynamical systems. Based on the results of the latter theorem, the goal of the present thesis is to show the existence of a physically relevant Nekhoroshev stable region around the equilateral fixed points of the elliptic restricted 3-body problem for times comparable to the life-time of a planetary system. In the case of our Solar system the masses of the asteroids can be neglected compared to the masses of the planets. A stability result stated in the restricted problem therefore directly translates into a stability result in the Solar system. The observation of asteroids around the Lagrangian equilibrium points on the one hand and the existence of a Nekhoroshev-type stability region around the equilibria on the other hand, validates the assumption. As a conclusion, the observations together with a mathematical theorem gives insight into the dynamical evolution of the Solar system. The present study indicates the possibility of long-term stability of asteroids in the framework of the elliptic restricted three body problem. In the case of Jupiter´s Trojan asteroids, a domain of stability could be derived by analytical means. It is quite realistic with respect to proper librations of the asteroids but limited with respect to the proper eccentricities due to the limited convergence of the series expansion approach. Trojan asteroids in the Sun-Jupiter system are found to be stable for the age of the Solar system for proper eccentricities ep<0.01 and proper librations Dp<10°