Perturbations des systèmes différentiels et résultats de géométrie différentielle

We use classical definitions and results of differential geometry in studying properties of transformations depending on a small parameter, acting on differential systems. Hori's and Deprit's algorithms can be defined for these systems. A lemma is given to show these algorithms are equivalent. The so-called property of covariance is merely established. The canonical systems are then considered as associated with Hamiltonian vectorfields on symplectic manifolds. The property that the infinitesimal generator of a canonical transformation is an Hamiltonian vectorfield permits to establish separately the generality of Hori's and Deprit's algorithms. We suggest that the Hamiltonian vectorfield property can be extended to the generators of transformations depending on several parameters.     

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.

     

On ‘out of plane’ equilibrium points in the elliptic restricted three-body problem with radiating and oblate primaries

We locate and examine the stability of the ‘out of plane’ equilibrium points, L _6,7 of an infinitesimal body in the field of stellar-oblate binary systems moving in elliptic orbits around their common center of mass. Their positions and stability depend on the oblateness as well as radiation coefficients of the primaries and the eccentricity of their orbits. A numerical application of this problem for the systems: Gamma Leporis and Altair are given.     

On the central configurations of the planar 1 + 3 body problem

We consider the Newtonian four-body problem in the plane with a dominat mass M. We study the planar central configurations of this problem when the remaining masses are infinitesimal. We obtain two different classes of central configurations depending on the mutual distances between the infinitesimal masses. Both classes exhibit symmetric and non-symmetric configurations. And when two infinitesimal masses are equal, with the help of extended precision arithmetics, we provide evidence that the number of central configurations varies from five to seven.     

Explicit adaptive symplectic integrators for solving Hamiltonian systems

We consider Sundman and Poincaré transformations for the long-time numerical integration of Hamiltonian systems whose evolution occurs at different time scales. The transformed systems are numerically integrated using explicit symplectic methods. The schemes we consider are explicit symplectic methods with adaptive time steps and they generalise other methods from the literature, while exhibiting a high performance. The Sundman transformation can also be used on non-Hamiltonian systems while the Poincaré transformation can be used, in some cases, with more efficient symplectic integrators. The performance of both transformations with different symplectic methods is analysed on several numerical examples.     

Spatial collinear restricted four-body problem with repulsive Manev potential

We outline some aspects of the dynamics of an infinitesimal mass under the Newtonian attraction of three point masses in a symmetric collinear relative equilibria configuration when a repulsive Manev potential (\(-1/r +e/r^{2}\)), \(e>0\), is applied to the central mass. We investigate the relative equilibria of the infinitesimal mass and their linear stability as a function of the mass parameter \(\beta \), the ratio of mass of the central body to the mass of one of two remaining bodies, and e. We also prove the nonexistence of binary collisions between the central body and the infinitesimal mass.     

Transformations between 2MASS, SDSS and BVRI photometric systems: bridging the near-infrared and optical

We present colour transformations for the conversion of the Two Micron All Sky Survey (2MASS) photometric system to the Johnson–Cousins UBVRI system and further into the Sloan Digital Sky Survey (SDSS) ugriz system. We have taken SDSS gri magnitudes of stars measured with the 2.5-m telescope from SDSS Data Release 5 (DR5), and BVRI and   JHK _s   magnitudes from Stetson's catalogue and Cutri et al., respectively. We matched thousands of stars in the three photometric systems by their coordinates and obtained a homogeneous sample of 825 stars by the following constraints, which are not used in previous transformations: (1) the data are dereddened, (2) giants are omitted and (3) the sample stars selected are of the highest quality. We give metallicity, population type and transformations dependent on two colours. The transformations provide absolute magnitude and distance determinations which can be used in space density evaluations at short distances where some or all of the SDSS ugriz magnitudes are saturated. The combination of these densities with those evaluated at larger distances using SDSS ugriz photometry will supply accurate Galactic model parameters, particularly the local space densities for each population.     

Resolution methods of perturbed differential equations,using tools of differential geometry

We use classical definitions and results of differential geometry in studying properties of transformations depending on a small parameter, acting on differential systems.Notions of one-parameter Lie's group of transformations, of bracket of vector fields (Lie's derivative) ard used. In the same way, the notion of symplectic manifold and of transformations which keep invariant a 2-form are useful.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.     

Effects of triaxiality of primaries on oblate infinitesimal in elliptical restricted three body problem

This paper examines the effects of triaxiality of both the primaries on the position and stability of the oblate infinitesimal mass in the neighborhood of triangular equilibrium points in the framework of Elliptical restricted three body problem. We have found the solutions for the locations of triangular equilibrium points. We have investigated the stability of infinitesimal mass around the triangular equilibrium points.It is observed that the infinitesimal motion around triangular equilibrium points are stable under certain condition with respect to triaxiality of primaries. We have applied the method of averaging used by Grebenivok, throughout the analysis of the stability of the infinitesimal mass around the triangular equilibrium points. We have exploited simulation technique using MATLAB 15 to analyze the stability of the system. The critical mass ratio depends on the triaxiality, oblateness, semi- major axis and eccentricity of the elliptical orbits.     

The General Method for Fixing the Gauges of Relativistic Astronomical Reference Systems

This article applies a new scheme of the first post-Newtonian theory (Damour et al., 1991–1994) to the problem of gauge in relativistic reference systems. Choosing and fixing gauge are necessary when the precision of time measurement and application needs to reach the 2PN level (10⁻¹⁶ or better). We present a general method for fixing the gauges of both the global and local coordinate systems, and for determining the expressions of gravitational potentials and coordinate transformations. The results relevant are consistent with the newest IAU resolutions, therefore they can be applied to astronomical practice.     

A general timing condition for consecutive collision orbits in the limiting case µ = 0 of the elliptic restricted problem

Consecutive collision orbits in the limiting case µ = 0 of the elliptic restricted three-body problem are investigated. in particular those in which the infinitesimal mass collides twice with the smaller (massless) primary. A timing condition is presented that allows the extension of previous results to the case of arbitrary relative orientation of the orbits of the infinitesimal mass and the smaller primary. The timing condition is expressed in two general forms - in terms of orbit parameters and eccentric (or hyperbolic) anomalies at the times of collision - for the specific cases of elliptic. parabolic or hyperbolic orbits of the infinitesimal mass. Some families of solutions are presented.     

Analytical potential–density pairs from complex-shifted Kuzmin–Toomre discs

The complex-shift method is applied to the Kuzmin–Toomre family of discs to generate a family of non-axisymmetric flat distributions of matter. These are then superposed to construct non-axisymmetric flat rings. We also consider triaxial potential–density pairs obtained from these non-axisymmetric flat systems by means of suitable transformations. The use of the imaginary part of complex-shifted potential–density pairs is also discussed.     

Families of Asymmetric Periodic Orbits in the Restricted Three-body Problem

This paper studies the asymmetric solutions of the restricted planar problem of three bodies, two of which are finite, moving in circular orbits around their center of masses, while the third is infinitesimal. We explore, numerically, the families of asymmetric simple-periodic orbits which bifurcate from the basic families of symmetric periodic solutions f, g, h, i, l and m, as well as the asymmetric ones associated with the families c, a and b which emanate from the collinear equilibrium points L ₁, L ₂ and L ₃ correspondingly. The evolution of these asymmetric families covering the entire range of the mass parameter of the problem is presented. We found that some symmetric families have only one bifurcating asymmetric family, others have infinity number of asymmetric families associated with them and others have not branching asymmetric families at all, as the mass parameter varies. The network of the symmetric families and the branching asymmetric families from them when the primaries are equal, when the left primary body is three times bigger than the right one and for the Earth–Moon case, is presented. Minimum and maximum values of the mass parameter of the series of critical symmetric periodic orbits are given. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, we regularize the equations of motion of the problem using the Levi-Civita transformations.     

Perturbed Gylden Systems and Time-Dependent Delaunay-Like Transformations

We study the possibilities and limitations of the application of generalized Delaunay-like transformations (in the 6-dimensional phase space) and TR-like mappings (in the 8-dimensional, extended phase space) to perturbed two-body problems with a time-varying Keplerian parameter μ(t), that is, to Gylden-type systems. For the sake of theoretical completeness, both negative- and positive-energy motion (with nonstationary coupling parameter) are, in principle, considered. Our developments are intended to introduce canonical variables parallelling the classical ones of Delaunay and the Delaunay-Similar variables of Scheifele. This revised version was published online in August 2006 with corrections to the Cover Date.     

Periodic orbits and bifurcations in the Sitnikov four-body problem when all primaries are oblate

We study the motions of an infinitesimal mass in the Sitnikov four-body problem in which three equal oblate spheroids (called primaries) symmetrical in all respect, are placed at the vertices of an equilateral triangle. These primaries are moving in circular orbits around their common center of mass. The fourth infinitesimal mass is moving along a line perpendicular to the plane of motion of the primaries and passing through the center of mass of the primaries. A relation between the oblateness-parameter ‘A’ and the increased sides ‘ε’ of the equilateral triangle during the motion is established. We confine our attention to one particular value of oblateness-parameter A=0.003. Only one stability region and 12 critical periodic orbits are found from which new three-dimensional families of symmetric periodic orbits bifurcate. 3-D families of symmetric periodic orbits, bifurcating from the 12 corresponding critical periodic orbits are determined. For A=0.005, observation shows that the stability region is wider than for A=0.003.     

Existence of libration points in the generalised photogravitational restricted problem of three bodies

In this paper we have proved the existence of libration points for the generalised photogravitational restricted problem of three bodies. We have assumed the infinitesimal mass of the shape of an oblate spheroid and both of the finite masses to be radiating bodies and the effect of their radiation pressure on the motion of the infinitesimal mass has also been taken into account. It is seen that there is a possibility of nine libration points for small values of oblateness, three collinear, four coplanar and two triangular.     

Über die Kontinuitätsmethode zur Auffindung periodischer Lösungen

POINCARÉ's conditions for the existence of periodic solutions of systems of differential equations are simplified by use of transformations of coordinates.     

Robe’s circular restricted three-body problem with a Roche ellipsoid-triaxial versus oblate system

This paper investigates Robe’s circular restricted three-body problem for two cases: with a Roche ellipsoid-triaxial system and with a Roche ellipsoid-oblate system. Without ignoring any component in both problems, a full treatment is given of the buoyancy force. The relevant equations of motion are established, and the special case where the density of the fluid and that of the infinitesimal mass are equal (D=0) is discussed. The location of the libration point and its stability when the infinitesimal mass is denser than the medium (D>0) are studied and it is found that the point (0,0,0) is the only libration point and this point is stable.     

Equations of motion of elliptically-restricted problem of a body with variable mass and two tri-axial bodies

The differential equations of motion of the elliptic restricted problem of three bodies, an infinitesimal spherical body with decreasing mass and two tri-axial bodies are derived. We have applied Jeans's law and the space-time of Meshcherskii in the special case whenn=1,k=0,q=1/2. Also Nechvíle's transformation for the elliptic problem be applied for this case.     

Effect of Moon perturbation on the energy curves and equilibrium points in the Sun–Earth–Moon system

In this paper, we have considered that the Moon motion around the Earth is a source of a perturbation for the infinitesimal body motion in the Sun–Earth system. The perturbation effect is analyzed by using the Sun–Earth–Moon bi–circular model (BCM). We have determined the effect of this perturbation on the Lagrangian points and zero velocity curves. We have obtained the motion of infinitesimal body in the neighborhood of the equivalent equilibria of the triangular equilibrium points. Moreover, to know the nature of the trajectory, we have estimated the first order Lyapunov characteristic exponents of the trajectory emanating from the vicinity of the triangular equilibrium point in the proposed system. It is noticed that due to the generated perturbation by the Moon motion, the results are affected significantly, and the Jacobian constant is fluctuated periodically as the Moon is moving around the Earth. Finally, we emphasize that this model could be applicable to send either satellite or telescope for deep space exploration.