On Szebehely's problem for holonomic systems involving generalized potential functions

We consider the problem of finding the generalized potential function V = U _i(q ¹, q ²,..., q ⁿ)q ⁱ + U(q ¹, q ²,...;q ⁿ) compatible with prescribed dynamical trajectories of a holonomic system. We obtain conditions necessary for the existence of solutions to the problem: these can be cast into a system of n – 1 first order nonlinear partial differential equations in the unknown functions U ₁, U ₂,...;, U _n, U. In particular we study dynamical systems with two degrees of freedom. Using adapted coordinates on the configuration manifold M ₂ we obtain, for potential function U(q ¹, q ²), a classic first kind of Abel ordinary differential equation. Moreover, we show that, in special cases of dynamical interest, such an equation can be solved by quadrature. In particular we establish, for ordinary potential functions, a classical formula obtained in different way by Joukowsky for a particle moving on a surface.Work performed with the support of the Gruppo Nazionale di Fisica Matematica (G.N.F.M.) of the Italian National Research Council.     

Separable and partially separable systems in the light of the inverse problem of dynamics

Using a generalization of Joukovsky's formula, we determine three-dimensional families of curves that are orbits only in separable potentials and we note the importance of iso-energetic families of orbits. We also obtain more general families that are orbits of partially separable systems and we examine from this point of view the classical curvilinear coordinate systems.     

On the adelphic potentials compatible with a set of planar orbits

Given a planar potentialB=B(x, y), compatible with a monoparametric family of planar orbitsf(x, y)=c, we face the problem of producing potentialsA=A(x, y), adelphic toB(x, y), i.e. nontrivial potentials which have in common withB(x, y) the given set of orbits. We establish a linear, second order partial differential equation for a functionP(x, y) and we prove that, to any definite positive solution of this equation, there corresponds a potentialA(x, y) adelphic toB(x, y).     

On the number of isolating integrals in systems with three degrees of freedom

Dynamical systems with three degrees of freedom can be reduced to the study of a fourdimensional mapping. We consider here, as a model problem, the mapping given by the following equations: $$\left\{ \begin{gathered} x_1 = x_0 + a_1 {\text{ sin (}}x_0 {\text{ + }}y_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{)}} \hfill \\ y_1 = x_0 {\text{ + }}y_0 \hfill \\ z_1 = z_0 + a_2 {\text{ sin (}}z_0 {\text{ + }}t_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{) (mod 2}}\pi {\text{)}} \hfill \\ t_1 = z_0 {\text{ + }}t_0 \hfill \\ \end{gathered} \right.$$ We have found that as soon asb≠0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).     

On the number of isolating integrals in resonant systems with 3 degrees of freedom

The 221-resonance case for a potential problem with three degrees of freedom is characterized by the existence of two isolating approximate integrals apart from the energy. This result completes a statement by Gustavson concerning the number of formal integrals in resonant Hamiltonian systems.     

Explicit solutions of the three-dimensional inverse problem of dynamics,using the frenet reference frame

Given a two-parameter of three-dimensional orbits, we construct the unit tangent vector, the normal and the binormal which define the Frenet reference frame. In this frame, by writing that the force is conservative, we explicitly obtain the potential as a function of the energy along the trajectories and of its derivatives.Observatoire de Besançon     

On the disappearance of isolating integrals in dynamical systems with more than two degrees of freedom

We continue to study the number of isolating integrals in dynamical systems with three and four degrees of freedom, using as models the measure preserving mappingsT already introduced in preceding papers (Froeschlé, 1973; Froeschlé and Scheidecker, 1973a).Thus, we use here a new numerical method which enables us to take as indicator of stochasticity the variation withn of the two (respectively three) largest eigenvalues-in absolute magnitude-of the linear tangential mappingT ⁿ * ofT ⁿ . This variation appears to be a very good tool for studying the diffusion process which occurs during the disappearance of the isolating integrals, already shown in a previous paper (Froeschlé, 1971). In the case of systems with three degrees of freedom, we define and give an estimation of the diffusion time, and show that the gambler's ruin model is an approximation of this diffusion process.     

Main features of dynamical escape from three-dimensional triple systems

The dynamical evolution of triple systems with equal and unequal-mass components and different initial velocities is studied. It is shown that, in general, the statistical results for the planar and three-dimensional triple systems do not differ significantly. Most (about 85%) of the systems disrupt; the escape of one component occurs after a triple approach of the components. In a system with unequal masses, the escaping body usually has the smallest mass. A small fraction (about 15%) of stable or long-lived systems is formed if the angular momentum is non-zero. Averages, distributions and coefficients of correlations of evolutionary characteristics are presented: the life-time, angular momentum, numbers of wide and close triple approaches of bodies, relative energy of escapers, minimum perimeter during the last triple approach resulting in escape, elements of orbits of the final binary and escaper.     

Local integrals and the plane motion of a point in rotating systems

This paper deals with the plane motion of a star in the gravitational field of a system which is in a steady state and rotates with a constant angular velocity. For these systems a class of potentials permitting a local integral, linear with respect to the velocity components, has been found. The concept of the local integral itself was introduced by one of the authors of the present paper (Antonov, 1981). A detailed model has been constructed. The corresponding domain of the particle motion and the form of the trajectory coils have been determined. The result is compared with the motion in a more realistic potential.     

利用双视角消除太阳矢量磁场观测中的180◦不确定性

随着"环日轨道器"(Solar Orbiter, SO)的在轨运行,太阳磁场观测进入了双视角遥测的时代.对利用太阳磁场的双视角观测改正矢量磁图中存在的横场(垂直于视线方向的磁场分量) 180°不确定性进行了模拟,首先模拟了对解析解得到磁图的双视角观测,然后利用"日震学和磁学成像仪"(Helioseismic and Magnetic Imager, HMI)在不同时间观测到的一个老化黑子的磁图模拟了双视角观测.发现要改正一个磁图中横场方向的180°不确定性,在观测上只需要另外一个平行于视线方向的磁场即纵向磁场观测的协助.利用HMI的磁场观测模拟,估算显示30°的张角能够改正50 Gs磁场中的180°不确定性.更大的张角虽然更有利于更弱磁场的改正,但是考虑到投影效应的不利影响, 30°左右的张角应该是未来空间设备进行多视角观测太阳磁场的最佳张角.     

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.     

On the emptiness of voids

The observability of a galaxy population inside of voids is estimated by assuming a void population similar to the one of nearby field galaxies in density as well as in morphological mixture. Obviously an extension to apparent magnitudes beyond m = 22 for a complete sample of galaxies in a sufficient large field is needed to get reliable information on a void population.     

Distribution of minima of the current size of the triple systems and their disruption

The dynamical evolution of triple systems has been studied by computer simulations. A function (t) has been defined, where p is the maximum distance of the components from their centre of inertia, and t is the time. The value of is used to indicate the current size of the triple system. The minima of have been followed during the course of evolution of the triples. A distribution of f(_min) has been obtained, which is described by the following statistical parameters: the mode is equal to 0.65d, the mean value _min= 0.750d, r.m.s. is 0.477d, the asymmetry is 0.218, the excess is 2.04 where d is the mean harmonic distance between the bodies in the equilibrium state of the triple system. As a rule, escapes from triples occur only after close three-body approaches.     

On a property of zero-velocity curves in N-body ring-type systems

Zero-velocity curves are a useful tool in the investigation of various aspects of a dynamical system. These curves that distinguish the regions where the motion of a particle is permissible from the regions where this motion is not permitted, present some basic properties. In this paper, we prove that in symmetric ring-type systems where a small particle moves under the resultant gravitational field of N coplanar big bodies, of which ν=N−1 are arranged at equal distances among them on the periphery of a circle, a new property concerning these curves, exists. All the zero-velocity curves drawn in the space of the initial conditions (x₀,C) and concerning configurations with the same number of peripheral primaries but various mass parameters, pass through two different focal points, the position of which does not depend on the value of the mass parameter.     

疏散星团NGC6530的成员研究和空间运动

利用上海天文台40厘米折射望远镜照相观测资料所得出的恒星自行,按最大似然原理确定了疏散星团NGC6530天区内恒星的成员概率．分析表明星团成员确定是非常有效的,成员概率p≥0．9的恒星计有250颗．最后,确定了星团的空间运动,并作了简单的讨论．     

Coronal expansion and the transfer of solar angular momentum to the interplanetary space

We discuss the question of loss of angular momentum through coronal expansion. From a large volume of data on Type-1 cometary tails we have confirmed the presence of a tangential component in the coronal expansion, which has not only a stochastic component but also a constant component of 9.8 km/s. Through coronal expansion the Sun has lost 80% of its angular momentum since it evolved on to the main sequence and the angular velocity of the Sun is decreasing exponentially. This result should have a large effect on the dynamical evolution of the Sun.     

On the two-dimensional inverse problem of dynamics

The authors extend the deduction of the equations satisfied by the force fields from inertial to rotating frames, when the curves of a certain family are known to be solutions for the equations of motion. Then Drǎmbâ's equation is obtained as a consequence of this result. The works of Hadamard and Moiseev are proved to be closely related to the inverse problem of dynamics.     

On the global solution of the N-body problem

In connection with the publication (Wang Qiu-Dong, 1991) the Poincaré type methods of obtaining the maximal solution of differential equations are discussed. In particular, it is shown that the Wang Qiu-Dong'sglobal solution of the N-body problem has been obtained in Babadzanjanz (1979). First the more general results on differential equations have been published in Babadzanjanz (1978).     

On the stability of particular solutions of the singly averaged Hill problem

We consider the particular solutions of the evolutionary system of equations in elements that correspond to planar and spatial circular orbits of the singly averaged Hill problem. We analyze the stability of planar and spatial circular orbits to inclination and eccentricity, respectively. We construct the instability regions of both particular solutions in the plane of parameters of the problem.