Practical Symplectic Methods with Time Transformation for the Few-Body Problem

The use of the extended phase space and time transformations for constructing efficient symplectic algorithms for the investigation of long term behavior of hierarchical few-body systems is discussed. Numerical experiments suggest that the time-transformed generalized leap-frog, combined with symplectic correctors, is one of the most efficient methods for such studies. Applications extend from perturbed two-body motion to hierarchical many-body systems with large eccentricities. This revised version was published online in July 2006 with corrections to the Cover Date.     

Integration rules from the equivalent differential equation method applied to equations of motion

The numerical integration of equations of motion necessarily implies the presence of errors that depend on initial conditions as well as the different physical parameters under consideration. More particularly, dumping or dissipative terms can appear and it is especially interesting to determine its causes. The equivalent differential equation method may allow the errors from a certain numerical scheme to be analyzed and, together with other considerations, can help us to eliminate or reduce them.     

Mass-Weighted Symplectic Forms for the N-Body Problem

Mass-weighted symplectic forms provide a unified framework for the treatment of both finite and vanishingly small masses in the N-body problem. These forms are introduced, compared to previous approaches, and their properties are discussed. Applications to symplectic mappings, the definition of action-angle variables for the Kepler problem, and Hamiltonian perturbation theory are outlined This revised version was published online in July 2006 with corrections to the Cover Date.     

Simple derivation of Symplectic Integrators with First Order Correctors

In this paper we consider almost integrable systems for which we show that there is a direct connection between symplectic methods and conventional numerical integration schemes. This enables us to construct several symplectic schemes of varying order. We further show that the symplectic correctors, which formally remove all errors of first order in the perturbation, are directly related to the Euler—McLaurin summation formula. Thus we can construct correctors for these higher order symplectic schemes. Using this formalism we derive the Wisdom—Holman midpoint scheme with corrector and correctors for higher order schemes. We then show that for the same amount of computation we can devise a scheme which is of order O(h ⁶)+(² h ²), where is the order of perturbation and h the stepsize. Inclusion of a modified potential further reduces the error to O(h ⁶)+(² h ⁴).This revised version was published online in October 2005 with corrections to the Cover Date.     

Perturbations of the Kepler Problem in Global Coordinates

The usual action-angle variables for the Kepler Problem (the Delaunay variables) are not globally defined, leaving out some orbits (circular orbits or those lying on the xy-plane). Moreover they are trascendental functions of the physical variables, making it quite difficult to write the perturbed Hamiltonian. The way-out proposed here is to pass to a 8-dimensional rank-6 Poisson manifold, that is, to parametrize the state of the Kepler Problem with two 4-dimensional vectors mutually orthogonal and of equal norm. This revised version was published online in July 2006 with corrections to the Cover Date.     

Perturbations of the Kepler Problem in Global Coordinates: A Program

The present paper is a brief report of how, along the lines of a previous paper, the implementation of the program KEPLER¹, for the numerical integration of the perturbations of the Kepler problem, has been carried out.     

保持Runge-Lenz向量的数值方法

对孤立积分和能够保持Runge-Lenz向量的梯形公式进行详尽讨论．孤立积分就是限制粒子运动区域的不变量，具有n个自由度的自治可积哈密顿系统且只有n个互相对合的独立孤立积分，并且其他孤立积分的存在对粒子的运动是有意义的，Kepler二体系统存在能量积分、角动量积分和Runge-Lenz向量．对于平面运动情况，这三类积分中只有3个独立孤立积分；而对于三维空间情形，该三类积分仅有5个是独立的．就前者而言，Kepler二体平面运动积分构成该系统中的对称群SO（3），经过Levi-Civita变换，它可以转化为二维各向同性谐振子系统中的对称群，而该对称群能够被梯形公式准确保持，另一方面，对于后者梯形公式对这三类积分的严格保持还可以在5个Kepler轨道根数n、e、i、Ω和w上得到体现。     

Periodic Solutions for any Planar Symmetric Perturbation of the Kepler Problem

We consider perturbations of the Kepler problem that are symmetric with respect to the origin and admit a first integral of motion which is also symmetric with respect to the origin. It has been proved that each circular solution of the unperturbed problem gives rise to a periodic solution of the perturbed system.     

Improved Leap—Frog Symplectic Integrators for Orbits of Small Eccentricity in the Perturbed Kepler Problem

I have improved the precision of the leap–frog symplectic integrators for perturbed Kepler problems at small eccentricities, without significant loss of CPU time. The integration scheme proposed is competitive, in some situations, with the so-called mixed variable integrators.     

Non-Existence of New Meromorphic First Integrals in the Planar Three-Body Problem

We consider the planar three-body problem and prove that, apart from some exceptional cases, there is no additional first integral meromorphic with respect to positions, mutual distances and momenta.     

Complete Zonal Problem of the Artificial Satellite: Generic Compact Analytic First Order in Closed Form

This paper is a contribution to the Theory of the Artificial Satellite, within the frame of the Lie Transform as canonical perturbation technique (elimination of the short period terms). We consider the perturbation by any zonal harmonic J _n (n ≥ 2) of the primary on the satellite, what we call here the complete zonal problem of the artificial satellite. This is quite useful for primaries with symmetry of revolution. We give an analytical formula to compute directly the first order averaged Hamiltonian. The computation is carried out in closed form for all terms, avoiding therefore tedious expansions in the eccentricity or in any anomaly; this feature makes the averaging process, not only valid for all kind of elliptic trajectories but at the same time it yields the averaged Hamiltonian in a very short and compact way. The formula allows us to now skip the averaging process, which means an asymptotic gain of a factor 3n/2 regarding the computational cost of the n ^th zonal. Our analytical formulae have been widely checked, by comparison on one hand with published works (Brouwer, 1959) (which contained results for particular zonal harmonics, let’s say typically from J ₂ to J ₈), and on the other hand with the results of 3 symbolic manipulation software, among which the MM (standing for ‘Moon’s series Manipulator’), which has already been used and described in (De Saedeleer B., 2004). Additionally, the first order generator associated with this transformation is given into the same closed form, and has also been validated.     

The Caledonian Symmetrical Double Binary Four-Body Problem I: Surfaces of Zero-Velocity Using the Energy Integral

The Caledonian four-body problem introduced in a recent paper by the authors is reduced to its simplest form, namely the symmetrical, four body double binary problem, by employing all possible symmetries. The problem is three-dimensional and involves initially two binaries, each binary having unequal masses but the same two masses as the other binary. It is shown that the simplicity of the model enables zero-velocity surfaces to be found from the energy integral and expressed in a three dimensional space in terms of three distances r ₁, r ₂, and r ₁₂, where r ₁ and r ₂ are the distances of two bodies which form an initial binary from the four body systems centre of mass andr ₁₂ is the separation between the two bodies.