1,721,001 research outputs found
Secular orbital dynamics of the innermost exoplanet of the υ -Andromedæ system
Full text linkWe introduce a quasi-periodic restricted Hamiltonian to describe the secular motion of a small-mass planet in a multi-planetary system. In particular, we refer to the motion of υ -And b which is the innermost planet among those discovered in the extrasolar system orbiting around the υ -Andromedæ A star. We preassign the orbits of the Super-Jupiter exoplanets υ -And c and υ -And d in a stable configuration. The Fourier decompositions of their secular motions are reconstructed by using the well-known technique of the (so-called) frequency analysis and are injected in the equations describing the orbital dynamics of υ -And b under the gravitational effects exerted by those two external exoplanets (that are expected to be major ones in such an extrasolar system). Therefore, we end up with a Hamiltonian model having 2 + 3 / 2 degrees of freedom; its validity is confirmed by the comparison with several numerical integrations of the complete four-body problem. Furthermore, the model is enriched by taking into account also the effects due to the relativistic corrections on the secular motion of the innermost exoplanet. We focus on the problem of the stability of υ -And b as a function of the parameters that mostly impact on its orbit, that are the initial values of its inclination and the longitude of its node (as they are measured with respect to the plane of the sky). In particular, we study the evolution of its eccentricity, which is crucial to exclude orbital configurations with high probability of (quasi)collision with the central star in the long-time evolution of the system. Moreover, we also introduce a normal form approach, that is based on the complete average of our restricted model with respect to the angles describing the secular motions of the major exoplanets. Therefore, our Hamiltonian model is further reduced to a system with 2 degrees of freedom, which is integrable because it admits a constant of motion that is related to the total angular momentum. This allows us to very quickly preselect the domains of stability for υ -And b , with respect to the set of the initial orbital configurations that are compatible with the observations
Hamiltonian control of magnetic field lines: Computer assisted results proving the existence of KAM barriers
No full textWe reconsider a control theory for Hamiltonian systems, that was introduced on the basis of KAM theory and applied to a model of magnetic field in previous articles. By a combination of Frequency Analysis and of a rigorous (Computer Assisted) KAM algorithm we prove that in the phase space of the magnetic field, due to the control term, a set of invariant tori appear, and it acts as a transport barrier. Our analysis, which is common (but often also limited) to Celestial Mechanics, is based on a normal form approach; it is also quite general and can be applied to quasi-integrable Hamiltonian systems satisfying a few additional mild assumptions. As a novelty with respect to the works that in the last two decades applied Computer Assisted Proofs into the framework of KAM theory, we provide all the codes allowing to produce our results. They are collected in a software package that is publicly available from the Mendeley Data repository. All these codes are designed in such a way to be easy-to-use, also for what concerns eventual adaptations for applications to similar problems
Areali visioni
No full textLa realtà percettivo-visiva viene considerata sulla base della chiave di lettura "Areale", secondo i principi teorici del ricercatore visuale Ugo Locatelli
Computer-assisted proofs of existence of KAM tori in planetary dynamical models of υ-And b
No full textWe reconsider the problem of the orbital dynamics of the innermost exoplanet of the υ-Andromedæsystem (i.e., υ-And b) into the framework of a Secular Quasi-Periodic Restricted Hamiltonian model. This means that we preassign the orbits of the planets that are expected to be the biggest ones in that extrasolar system (namely, υ-And c and υ-And d). The Fourier decompositions of their secular motions are injected in the equations describing the orbital dynamics of υ-And b under the gravitational effects exerted by those two exoplanets. By a computer-assisted procedure, we prove the existence of KAM tori corresponding to orbital motions that we consider to be very robust configurations, according to the analysis and the numerical explorations made in our previous article. The computer-assisted assisted proofs are successfully performed for two variants of the Secular Quasi-Periodic Restricted Hamiltonian model, which differs for what concerns the effects of the relativistic corrections on the orbital motion of υ-And b, depending on whether they are considered or not
A classical self-consistent proof of Kolmogorov's theorem on invariant tori
No full textThe celebrated theorem of Kolmogorov on persistence of
invariant tori of a nearly integrable Hamiltonian system is revisited
in the light of classical perturbation algorithm. It is shown that the
original Kolmogorov's algorithm can be given the form of a
constructive scheme based on expansion in a parameter. A careful
analysis of the accumulation of the small divisors shows
that it can be controlled geometrically. As a consequence, the proof
of convergence is based essentially on Cauchy's majorant's method,
with no use of the so called quadratic method. A short comparison with
Lindstedt's series is included
Canonical perturbation theory for nearly integrable systems
No full textThese lectures are devoted to the main results of classical
perturbation theory. We start by recalling the methods of
Hamiltonian dynamics, the problem of small divisors, the series of
Lindstedt and the method of normal form. Then we discuss the theorem
of Kolmogorov with an application to the Sun--Jupiter--Saturn problem
in Celestial Mechanics. Finally we discuss the problem of long--time
stability, by discussing the concept of exponential stability as
introduced by Moser and Littlewood and fully exploited by
Nekhoroshev. The phenomenon of superexponential stability is also
recalled
Quasi-periodic motions in a special class of dynamical equations with dissipative effects: a pair of detection methods
No full textWe consider a particular class of equations of motion, generalizing to n degrees of freedom the "dissipative spin-orbit problem", commonly studied in Celestial Mechanics. Those equations are formulated in a pseudo-Hamiltonian framework with action-angle coordinates; they contain a quasi-integrable conservative part and
friction terms, assumed to be linear and isotropic with respect to the action variables. In such a context, we transfer two methods determining quasi-periodic solutions, which were originally designed to analyze purely Hamiltonian quasi-integrable problems.
First, we show how the frequency map analysis can be adapted to this kind of dissipative models. Our approach is based on a key remark: the method can work as usual, by studying the behavior of the angular velocities of the motions as a function of the so called "external frequencies", instead of the actions.
Moreover, we explicitly implement the Kolmogorov's normalization algorithm for the dissipative systems considered here. In a previous article, we proved a theoretical result: such a constructing procedure is convergent under the hypotheses usually assumed in KAM theory. In the present work, we show that it can be translated to a
code making algebraic manipulations on a computer, so to calculate effectively quasi-periodic solutions on invariant tori (and the attracting dynamics in their neighborhoods).
Both the methods are carefully tested, by checking that their predictions are in agreement, in the case of the so called ``dissipative forced pendulum''. Furthermore, the results obtained by applying our adaptation of the frequency analysis method to the dissipative standard map are compared with some existing ones in the literature
Existence proof of librational invariant tori in an averaged model of HD60532 planetary system
Full text linkWe investigate the long-term dynamics of HD60532, an extrasolar system hosting two giant planets orbiting in a 3:1 mean motion resonance. We consider an average approximation at order one in the masses which results (after the reduction in the constants of motion) in a resonant Hamiltonian with two libration angles. In this framework, the usual algorithms constructing the Kolmogorov normal form approach do not easily apply and we need to perform some untrivial preliminary operations, in order to adapt the method to this kind of problems. First, we perform an average over the fast angle of libration which provides an integrable approximation of the Hamiltonian. Then, we introduce action-angle variables that are adapted to such an integrable approximation. This sequence of preliminary operations brings the Hamiltonian in a suitable form to successfully start the Kolmogorov normalization scheme. The convergence of the KAM algorithm is proved by applying a technique based on a computer-assisted proof. This allows us to reconstruct the quasi-periodic motion of the system, with initial conditions that are compatible with the observations
New Hamiltonian expansions adapted to the Trojan problem
No full textA number of studies, referring to the observed Trojan asteroids of various planets in our Solar System, or to hypothetical Trojan bodies in extrasolar planetary systems, have emphasized the importance of so-called secondary resonances in the problem of the long term stability of Trojan motions. Such resonances describe commensurabilities between the fast, synodic, and secular frequency of the Trojan body, and, possibly, additional slow frequencies produced by more than one perturbing bodies. The presence of secondary resonances sculpts the dynamical structure of the phase space. Hence, identifying their location is a relevant task for theoretical studies. In the present paper we combine the methods introduced in two recent papers (Páez and Efthymiopoulos in Celest Mech Dyn Astron 121(2):139, 2015; Páez and Locatelli in MNRAS 453(2):2177, 2015) in order to analytically predict the location of secondary resonances in the Trojan problem. In Páez and Efthymiopoulos (2015), the motion of a Trojan body was studied in the context of the planar Elliptic Restricted Three Body or the planar Restricted Multi-Planet Problem. It was shown that the Hamiltonian admits a generic decomposition H= Hb+ Hsec. The term Hb, called the basic Hamiltonian, is a model of two degrees of freedom characterizing the short-period and synodic motions of a Trojan body. Also, it yields a constant ‘proper eccentricity’ allowing to define a third secular frequency connected to the body’s perihelion precession. Hsec contains all remaining secular perturbations due to the primary or to additional perturbing bodies. Here, we first investigate up to what extent the decomposition H= Hb+ Hsec provides a meaningful model. To this end, we produce numerical examples of surfaces of section under Hb and compare with those of the full model. We also discuss how secular perturbations alter the dynamics under Hb. Secondly, we explore the normal form approach introduced in Páez and Locatelli (2015) in order to find an ‘averaged over the fast angle’ model derived from Hb, circumventing the problem of the series’ limited convergence due to the collision singularity at the 1:1 MMR. Finally, using this averaged model, we compute semi-analytically the position of the most important secondary resonances and compare the results with those found by numerical stability maps in specific examples. We find a very good agreement between semi-analytical and numerical results in a domain whose border coincides with the transition to large-scale chaotic Trojan motions
On the break-down threshold of invariant tori in four dimensional maps
No full textWe investigate the break-down of invariant tori in a four dimensional standard mapping for different values of the coupling parameter. We select various two-dimensional frequency vectors, having eventually one or both components close to a rational value. The dynamics of this model is very reach and depends on two parameters, the perturbing and coupling parameters. Several techniques are introduced to determine the analyticity domain (in the complex perturbing parameter plane) and to compute the break-down threshold of the invariant tori. In particular, the analyticity domain is recovered by means of a suitable implementation of Pade approximants. The break-down threshold is computed through a suitable extension of Greene's method to four dimensional systems. Frequency analysis is implemented and compared with the previous techniques
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