Hamiltonian Dynamics of a Gyrostat in the N-Body Problem: Relative Equilibria

We consider the non-canonical Hamiltonian dynamics of a gyrostat in Newtonian interaction with n spherical rigid bodies. Using the symmetries of the system we carry out two reductions. Then, working in the reduced problem, we obtain the equations of motion, a Casimir function of the system and the equations that determine the relative equilibria. Global conditions for existence of relative equilibria are given. Besides, we give the variational characterization of these equilibria and three invariant manifolds of the problem; being calculated the equations of motion in these manifolds, which are described by means of a canonical Hamiltonian system. We give some Eulerian and Lagrangian equilibria for the four body problem with a gyrostat. Finally, certain classical problems of Celestial Mechanics are generalized.     

Jacobi Dynamics And The N-Body Problem With Variable Masses

An appropriate generalization of the Jacobi equation of motion for the polar moment of inertia I is considered in order to study the N-body problem with variable masses. Two coupled ordinary differential equations governing the evolution of I and the total energy E are obtained. A regularization scheme for this system of differential equations is provided. We compute some illustrative numerical examples, and discuss an average method for obtaining approximate analytical solutions to this pair of equations. For a particular law of mass loss we also obtain exact analytical solutions. The application of these ideas to other kind of perturbed gravitational N-body systems involving drag forces or a different type of mass variation is also considered. This revised version was published online in July 2006 with corrections to the Cover Date.     

自旋致密双星后牛顿哈密顿系统轨道动力学数值研究

由中子星或黑洞构成的旋转致密双星后牛顿哈密顿系统属于相对论二体问题,该系统不但含有丰富的共振和混沌等动力学现象,而且成为探测引力波的理想天然波源.引力体的轨道动力学性质会在引力波中得到反映.因此,实际天体的混沌性既可能是对引力波探测的挑战,又可望是获得观测效应的机遇.本学位论文正是在这样的国际学术氛围下数值研究旋转致密双星后牛顿保守哈密顿动力学问题.基于最小二乘法原理我们构造了单和双标度因子等几种流形改正方法,分别对旋转致密双星后牛     

The Inverse Problem of Dynamics for Systems with Non-Stationary Lagrangian

We construct a non-stationary form of the Lagrangian of a material point with a known integral of motion and given monoparametric family of evolving orbits. An equation for non-stationary space symmetrical ‘potential’ function of such Lagrangian is given and this stands for the analog of Szebehely's (1974) equation. As an application of the problem, an integrable equation from celestial mechanics of variable mass with use of non-perturbed orbits of evolving type is constructed. On its basis adiabatic invariants of non-stationary two-body problem containing a tangential force are found. This revised version was published online in July 2006 with corrections to the Cover Date.     

Topology of the Two Fixed Centers Problem

In this paper we give a complete topological characterization of the Two Fixed Centers (TFC) problem flow. This characterization is based on the link formed by some basic periodic orbits. The Restricted Circular Three-Body (RCTB) problem is considered as a perturbation of the TFC in the case of two primaries with equal masses. The basic periodic orbits of the integrable problem can be continued in the non-integrable one.This revised version was published online in October 2005 with corrections to the Cover Date.     

The Spin-Orbit Resonant Rotation of Mercury: A Two Degree of Freedom Hamiltonian Model

The paper develops a hamiltonian formulation describing the coupled orbital and spin motions of a rigid Mercury rotation about its axis of maximum moment of inertia in the frame of a 3:2 spin orbit resonance; the (ecliptic) obliquity is not constant, the gravitational potential of mercury is developed up to the second degree terms (the only ones for which an approximate numerical value can be given) and is reduced to a two degree of freedom model in the absence of planetary perturbations. Four equilibria can be calculated, corresponding to four different values of the (ecliptic) obliquity. The present situation of Mercury corresponds to one of them, which is proved to be stable. We introduce action-angle variables in the neighborhood of this stable equilibrium, by several successive canonical transformations, so to get two constant frequencies, the first one for the free spin-orbit libration, the other one for the 1:1 resonant precession of both nodes (orbital and rotational) on the ecliptic plane. The numerical values obtained by this simplified model are in perfect agreement with those obtained by Rambaux and Bois [Astron. Astrophys. 413, 381–393]. This revised version was published online in July 2006 with corrections to the Cover Date.     

Global Secular Dynamics in the Planar Three-Body Problem

We use the global construction which was made in [6, 7] of the secular systems of the planar three-body problem, with regularized double inner collisions. These normal forms describe the slow deformations of the Keplerian ellipses which each of the bodies would describe if it underwent the universal attraction of only one fictitious other body. They are parametrized by the masses and the semi-major axes of the bodies and are completely integrable on a fixed transversally Cantor set of the parameter space. We study this global integrable dynamics reduced by the symmetry of rotation and determine its bifurcation diagram when the semi-major axes ratio is small enough. In particular it is shown that there are some new secular hyperbolic or elliptic singularities, some of which do not belong to the subset of aligned ellipses. The bifurcation diagram may be used to prove the existence of some new families of 2-, 3- or 4-frequency quasiperiodic motions in the planar three-body problem [7], as well as some drift orbits in the planar n-body problem [8].     

地球自转动力学中的两个问题

介绍了地球自动力学中的两个目前正在研究的问题，1．Chandler摆动的随机激发，在分析Chandler摆动各种激发的可能性后，认为随机运动是最可能的激发源，在此基础,提出了一个Chandler摆动激发的动力学模型，并从理论和数值模拟两方面对此模型做了统计分析研究，描述了今后对此问题的研究思路，2．地球内部动力学是目前国际地球动力学界的一个热点研究课题，介绍了它的现状和最近的发展动态以及准备在这方面开展研究工作的打算。     

The Expansion of the Hamiltonian of the Planetary Problem into the Poisson Series in All Keplerian Elements (Theory)

The study of the evolution of planetary systems, primarily of the Solar System, is one of the basic problems of celestial mechanics. The stability of motion of giant planets on cosmogonic time scales was established by numerical and analytical methods, but the question about the evolution of orbits of terrestrial planets and arbitrary solar-type planetary systems remained open. This work initiates a series of papers allowing one to advance in solving the problem of the evolution of the solar-type planetary systems on cosmogonic time scales by using powerful analytical tools. In the first paper of this series, we choose the optimum reference system and obtain the Poisson series expansion of the Hamiltonian of the problem in all Keplerian elements. We propose to use the integral representation of the corresponding coefficients or the Poisson processor means instead of conventionally addressing any possible special functions. This approach extremely simplifies the algorithm. The next paper of this series deals with the calculation of the expansion coefficients.     

Non-Integrability of the Generalized Two Fixed Centres Problem

We consider two fixed centres attracting a third body. Each centre is the source of a force field with potential V=−ar ⁻²ⁿ , where n is a real number. We prove that this generalization of the classical two fixed centres problem is non-integrable except when n= 0, 1/2, −1 and −2. This revised version was published online in July 2006 with corrections to the Cover Date.     

Special Families of Orbits in the Direct Problem of Dynamics

The direct problem of dynamics in two dimensions is modeled by a nonlinear second-order partial differential equation, which is therefore difficult to be solved. The task may be made easier by adding some constraints on the unknown function = f _y /f _x , where f(x, y) = c is the monoparametric family of orbits traced in the xy Cartesian plane by a material point of unit mass, under the action of a given potential V(x, y). If the function is supposed to verify a linear first-order partial differential equation, for potentials V satisfying a differential condition, can be found as a common solution of certain polynomial equations.The various situations which can appear are discussed and are then illustrated by some examples, for which the energy on the members of the family, as well as the region where the motion takes place, are determined. One example is dedicated to a Hénon—Heiles type potential, while another one gives rise to families of isothermal curves (a special case of orthogonal families). The connection between the inverse/direct problem of dynamics and the possibility of detecting integrability of a given potential is briefly discussed.This revised version was published online in October 2005 with corrections to the Cover Date.     

The third integral and the Hamiltonian

It is reiterated that any suggestion of the existence of a third integral is at variance with Poincaré's theorem on the non-existence of such integrals. Even in a purely numerical approach no form of a new integral can be constructed that is valid in every domain of the phase space; and it is devoid of meaning to use as a third integral different forms of functions in various cases.     

Normalization of Hamiltonian in the Generalized Photogravitational Restricted Three Body Problem with Poynting–Robertson Drag

We have performed normalization of Hamiltonian in the generalized photogravitational restricted three body problem with Poynting–Robertson drag. In this problem we have taken bigger primary as source of radiation and smaller primary as an oblate spheroid. Whittaker’s method is used to transform the second order part of the Hamiltonian into the normal form.      

The Expansion of the Hamiltonian of the Two-Planetary Problem into a Poisson Series in All Elements: Estimation and Direct Calculation of Coefficients

This is the second paper in a series of articles devoted to one of the basic problems of celestial mechanics: the evolution of solar-type planetary systems. In the first paper (Kholshevnikov et al., 2001), we reviewed the history and the current state of the issue, outlined the scheme of the study, introduced Jacobi coordinates and related osculating elements, and indicated the form of the Hamiltonian expansion into a Poisson series in all elements. In this paper, the expansion coefficients are found according to a simple algorithm that is reduced to the calculation of multiple integrals of elementary functions. At the first stage, we restricted our analysis to the two-planetary problem (Sun–Jupiter–Saturn). The general case will be investigated in a forthcoming paper.     

Energy and stability in the Full Two Body Problem

The conditions for relative equilibria and their stability in the Full Two Body Problem are derived for an ellipsoid–sphere system. Under constant angular momentum it is found that at most two solutions exist for the long-axis solutions with the closer solution being unstable while the other one is stable. As the non-equilibrium problem is more common in nature, we look at periodic orbits in the F2BP close to the relative equilibrium conditions. Families of periodic orbits can be computed where the minimum energy state of one family is the relative equilibrium state. We give results on the relative equilibria, periodic orbits and dynamics that may allow transition from the unstable configuration to a stable one via energy dissipation.      

On the Dynamics and Topology of the Elliptic Rectilinear Restricted 3-Body Problem

We study a symmetric collinear restricted 3-body problem, where the equal mass primaries perform elliptic collisions, while a third massless body moves in the line between the primaries, during the time between two consecutive elliptic collisions. After desingularizing binary and triple collisions, we prove the existence of a transversal heteroclinic orbit beginning and ending in triple collision. This orbit is the unique homothetic orbit that the problem possess. Finally, we describe the topology of the compact extended phase space. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamics of Two Planets in the 2/1 Mean-Motion Resonance

The dynamics of two planets near a first-order mean-motion resonance is modeled in the domain of the general three-body planar problem. The system studied is the pair Uranus-Neptune (2/1 resonance). The phase space of the resonance and near-resonance regions is studied by means of surfaces of section and spectral analysis techniques. After a thorough investigation of the topology of the phase space, we find that several regimes of motion are possible for the Uranus-Neptune system, and the regions of transition between the regimes of motion are the seats of chaotic motion. This revised version was published online in July 2006 with corrections to the Cover Date.     

The contact transformation groups of the Extended Hamiltonian System

The 1-parameter transformation groups (otherwise known as infinitesimal transformations) admitted by a system of differential equations are fundamental to the study of its properties. In this paper we first of all consider 1-parameter groups of contact transformations. Then, by generalizing Noether's theorem, we show how they are fundamental to what I call the Extended Hamiltonian System. Finally, this is illustrated by the extendedN-Body problem.

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Resume Les groupes de transformations à 1 paramètre (appelés aussi transformations infinitésimales) admis par un systeme d'équations différentielles sont fondamentaux dans l'étude de ses propriétés. Dans cet article, nous considérons d'abord les groupes à 1 paramètre de transformations de contact. Ensuite, par la généralisation du théorème de Noether, nous montrons qu'ils sont fondamentaux dans l'étude de ce que j'appelle le Système Hamiltonien Etendu. Enfin ceci est illustré par le problème étendu desN-Corps.

     

Generalized Problem of Two and Four Newtonian Centers

We consider integrable spherical analog of the Darboux potential, which appear in the problem (and its generalizations) of the planar motion of a particle in the field of two and four fixed Newtonian centers. The obtained results can be useful when constructing a theory of motion of satellites in the field of an oblate spheroid in constant curvature spaces.