A Counterexample to a Generalized Saari's Conjecture with a Continuum of Central Configurations

In this paper we show that in the n-body problem with harmonic potential one can find a continuum of central configurations for n= 3. Moreover we show a counterexample to an interpretation of Jerry Marsden Generalized Saari's conjecture. This will help to refine our understanding and formulation of the Generalized Saari's conjecture, and in turn it might provide insight in how to solve the classical Saari's conjecture for n≥ 4. This revised version was published online in July 2006 with corrections to the Cover Date.     

基于逆散射理论的介质参数与其地震响应的尺度对应关系研究

在反演过程中进行频带控制是多尺度反演的重要策略,但数据频率成分与要反演的参数之间的尺度对应关系尚不清晰。本文利用Kormendi论文中的测井曲线模型,基于逆散射理论将地下介质参数的扰动分解为0-0-9-11 Hz、9-11-18-22 Hz、18-22-36-44 Hz和36-44-80-90 Hz共4个尺度,并利用反射率正演模拟方法生成4个尺度扰动成分的地震响应。通过对比地下介质参数不同尺度的扰动及其地震响应的频带成分,发现二者在4个扰动尺度上均具有一致性,由此得出介质参数与其地震响应具有定量尺度对应关系的结论。据此从介质扰动的尺度概念引申出反射系数的尺度概念,从逆散射理论角度探讨实验结果的理论依据,并对其在指导反演分频参数选取及薄互层解释等方面的应用展望进行讨论。     

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.     

Orbit injection error analysis for the Lageos III mission

An analysis is presented of the orbital injection errors for the Lageos III satellite mission. Several methods are introduced for the solution of the Inverse Problem in the Theory of Errors. The novelty of the present approach consists in the use of the full geopotential covariance matrix in the error propagation equations. The GEM-T1 covariance matrix is used. It is found that by properly accounting for the correlation among the even zonal harmonic coefficients the acceptable error bounds increase by an order of magnitude with respect to the case when only the variances are used. The most stringent constraint, even when using the full covariance, is on inclination, whose nominal value must be realized within approximately 0.1° for the recovery of the Lense-Thirring precession to be successful at the 3% level (accounting only for injection errors). The associated tolerance in the semimajor axis is about 30 km while that in eccentricity is approximately 0.2. However, if the errors in semimajor axis and eccentricity can be kept to the routinely achievable levels respectively of 10 km and 0.004, then the tolerance in inclination can be relaxed to 0.2°.     

Invariant rotational curves in Sitnikov's Problem

The Sitnikov's Problem is a Restricted Three-Body Problem of Celestial Mechanics depending on a parameter, the eccentricity,e. The Hamiltonian,H(z, v, t, e), does not depend ont ife=0 and we have an integrable system; ife is small the KAM Theory proves the existence of invariant rotational curves, IRC. For larger eccentricities, we show that there exist two complementary sequences of intervals of values ofe that accumulate to the maximum admissible value of the eccentricity, 1, and such that, for one of the sequences IRC around a fixed point persist. Moreover, they shrink to the planez=0 ase tends to 1.     

An implementation ofN-body chain regularization

The chain regularization method (Mikkola and Aarseth 1990) for high accuracy computation of particle motions in smallN-body systems has been reformulated. We discuss the transformation formulae, equations of motion and selection of a chain of interparticle vectors such that the critical interactions requiring regularization are included in the chain. The Kustaanheimo-Stiefel (KS) coordinate transformation and a time transformation is used to regularize the dominant terms of the equations of motion. The method has been implemented for an arbitrary number of bodies, with the option of external perturbations. This formulation has been succesfully tested in a generalN-body program for strongly interacting subsystems. An easy to use computer program, written inFortran, is available on request.     

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).     

The accelerated Kepler problem

The accelerated Kepler problem (AKP) is obtained by adding a constant acceleration to the classical two-body Kepler problem. This setting models the dynamics of a jet-sustaining accretion disk and its content of forming planets as the disk loses linear momentum through the asymmetric jet-counterjet system it powers. The dynamics of the accelerated Kepler problem is analyzed using physical as well as parabolic coordinates. The latter naturally separate the problem’s Hamiltonian into two unidimensional Hamiltonians. In particular, we identify the origin of the secular resonance in the AKP and determine analytically the radius of stability boundary of initially circular orbits that are of particular interest to the problem of radial migration in binary systems as well as to the truncation of accretion disks through stellar jet acceleration.     

Lie group variational integrators for the full body problem in orbital mechanics

Equations of motion, referred to as full body models, are developed to describe the dynamics of rigid bodies acting under their mutual gravitational potential. Continuous equations of motion and discrete equations of motion are derived using Hamilton’s principle. These equations are expressed in an inertial frame and in relative coordinates. The discrete equations of motion, referred to as a Lie group variational integrator, provide a geometrically exact and numerically efficient computational method for simulating full body dynamics in orbital mechanics; they are symplectic and momentum preserving, and they exhibit good energy behavior for exponentially long time periods. They are also efficient in only requiring a single evaluation of the gravity forces and moments per time step. The Lie group variational integrator also preserves the group structure without the use of local charts, reprojection, or constraints. Computational results are given for the dynamics of two rigid dumbbell bodies acting under their mutual gravity; these computational results demonstrate the superiority of the Lie group variational integrator compared with integrators that are not symplectic or do not preserve the Lie group structure.     

The manifolds of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem

This paper deals with the Sitnikov family of straight-line motions of the circular restricted three-body problem, viewed as generator of families of three-dimensional periodic orbits. We study the linear stability of the family, determine several new critical orbits at which families of three dimensional periodic orbits of the same or double period bifurcate and present an extensive numerical exploration of the bifurcating families. In the case of the same period bifurcations, 44 families are determined. All these families are computed for equal as well as for nearly equal primaries (μ = 0.5, μ = 0.4995). Some of the bifurcating families are determined for all values of the mass parameter μ for which they exist. Examples of families of three dimensional periodic orbits bifurcating from the Sitnikov family at double period bifurcations are also given. These are the only families of three-dimensional periodic orbits presented in the paper which do not terminate with coplanar orbits and some of them contain stable parts. By contrast, all families bifurcating at single-period bifurcations consist entirely of unstable orbits and terminate with coplanar orbits.