The symplectic property of matrizants in Störmer's problem

The matrizants of periodic solutions in Störmer's problem, because of their symplectic character, can be transformed by means of multiplication with constant matrices into symmetric ones. As a result the six bilinear relations between their elements, existing on account of the symplectic property, are replaced by 14 linear and simple forms. This fact is very useful in numerical integrations where these relations are used as criteria of accuracy.     

The symplectic character of periodic orbits in the three dipole problem

In this work we reveal for the first time that in the three dipole problem only asymmetric periodic orbits exist.For these periodic orbits — planar and three dimensional — of a charged particle moving under the influence of the electromagnetic field of the three dipoles we give their symplectic relations using the Hamiltonian formulation which is related to the symplectic matrix. Also we study the properties of the symplectic matrix and we give the relations there are among the variations of a periodic solution. These relations have been used to check the accuracy of numerical integration of equations of first order variations.     

Planar Symmetric Periodic Orbits in Four Dipole Problem

We have extend Stormer’s problem considering four magnetic dipoles in motion trying to justify the phenomena of extreme “orderlines” such as the ones observed in the rings of Saturn; the aim is to account the strength of the Lorentz forces estimating that the Lorentz field, co-acting with the gravity field of the planet, will limit the motion of all charged particles and small size grains with surface charges inside a layer of about 200 m thickness as that which is observed in the rings of Saturn. For this purpose our interest feast in the motion of charged particles with neglected mass where only electromagnetic forces accounted in comparison to the weakness of the Newtonian fields. This study is particularly difficult because in the regions we investigate these motions there is enormous three dimensional instability. Following the Poincare’s hypothesis that periodic solutions are ‘dense’ in the set of all solutions in Hamiltonian systems we try to calculate many families of periodic solutions and to study their stability. In this work we prove that in this environment charged particles can trace planar symmetric periodic orbits. We discuss these orbits in details and we give their symplectic relations using the Hamiltonian formulation which is related to the symplectic matrix. We apply numerical procedures to find families of these orbits and to study their stability. Moreover we give the bifurcations of these families with families of planar asymmetric periodic orbits and families of three dimensional symmetric periodic orbits.     

A Tenth Order Symplectic Runge–Kutta–Nyström Method

A tenth order explicit symmetric and in consequence symplectic Runge–Kutta–Nyström method is presented here. We derive the order conditions needed and solve them for the parameters of the method. Numerical results indicate the superiority of the new method compared to the other high order symplectic methods appeared in the literature until now.     

Symplectic methods and their application to the motion of small bodies in the Solar System

Symplectic methods have been widely used in Solar System dynamics. This paper discusses both single step and multistep symplectic methods. For single step methods we point out that the modified algorithm (Wisdom et al., 1991, Kinoshita et. al., 1991) can be executed in the mass center coordinate system and in the Jacobian coordinate system. For multistep methods we describe the connections between symmetric and symplectic methods.     

Symplectic methods and their application to the motion of small bodies in the Solar System

Symplectic methods have been widely used in Solar System dynamics. This paper discusses both single step and multistep symplectic methods. For single step methods we point out that the modified algorithm (Wisdom et al., 1991, Kinoshita et. al., 1991) can be executed in the mass center coordinate system and in the Jacobian coordinate system. For multistep methods we describe the connections between symmetric and symplectic methods.     

Hamilton系统数值计算的新方法

系统地介绍了近年来对Ｈａｍｉｌｔｏｎ系统数值计算新建立的辛算法和线性对称多步法，并对它们在动力天文中的应用作了一简要回顾。     

On the Numerical Stability of Some Symplectic Integrators

In this paper, we analyze the linear stabilities of several symplectic integrators, such as the first-order implicit Euler scheme, the second-order implicit mid-point Euler difference scheme, the first-order explicit Euler scheme, the second-order explicit leapfrog scheme and some of their combinations. For a linear Hamiltonian system, we find the stable regions of each scheme by theoretical analysis and check them by numerical tests. When the Hamiltonian is real symmetric quadratic, a diagonalizing by a similar transformation is suggested so that the theoretical analysis of the linear stability of the numerical method would be simplified. A Hamiltonian may be separated into a main part and a perturbation, or it may be spontaneously separated into kinetic and potential energy parts, but the former separation generally is much more charming because it has a much larger maximum step size for the symplectic being stable, no matter this Hamiltonian is linear or nonlinear.     

Several fourth-order force gradient symplectic algorithms

By adding force gradient operators to symmetric compositions, we build a set of explicit fourth-order force gradient symplectic algorithms, including those of Chin and coworkers, for a separable Hamiltonian system with quadratic kinetic energy T and potential energy V . They are extended to solve a gravitational n-body Hamiltonian system that can be split into a Keplerian part H 0 and a perturbation part H 1 in Jacobi coordinates. It is found that the accuracy of each gradient scheme is greatly superior to ...     

Global applicability of the symplectic integrator method of hamiltonian systems

The global validity of the symplectic integration method or mapping approach is discussed in this paper. The results show that in the regions of phase space where symplectic integration schemes and the Hamiltonian system possess the same topology, they are effective; but in the regions where the schemes possess some other fixed points than those of the Hamiltonian system, their topologies are different from that of the actual system, thus the symplectic integration method or mapping approach is not effective globally.Supported by the National Natural Science Foundation of China and a grant from the Ph.D. Foundation.     

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.     

Symplectic integrators with potential derivatives to third order

An operator associated with third-order potential derivatives and a force gradient operator corresponding to second-order potential derivatives are used together to design a number of new fourth-order explicit symplectic integrators for the natural splitting of a Hamiltonian into both the kinetic energy with a quadratic form of momenta and the potential energy as a function of position coordinates.Numerical simulations show that some new optimal symplectic algorithms are much better than their non-optimal c...     

New methods for large dynamic range problems in planetary formation

Modern N -body techniques for planetary dynamics are generally based on symplectic algorithms specially adapted to the Kepler problem. These methods have proven very useful in studying planet formation, but typically require the time-step for all objects to be set to a small fraction of the orbital period of the innermost body. This computational expense can be prohibitive for even moderate particle number for many physically interesting scenarios, such as recent models of the formation of hot exoplanets, in which the semimajor axis of possible progenitors can vary by orders of magnitude. We present new methods which retain most of the benefits of the standard symplectic integrators but allow for radial zones with distinct time-steps. These approaches should make simulations of planetary accretion with large dynamic range tractable. As proof-of-concept, we present preliminary science results from an implementation of the algorithm as applied to an oligarchic migration scenario for forming hot Neptunes.     

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.     

QYMSYM: A GPU-accelerated hybrid symplectic integrator that permits close encounters

We describe a parallel hybrid symplectic integrator for planetary system integration that runs on a graphics processing unit (GPU). The integrator identifies close approaches between particles and switches from symplectic to Hermite algorithms for particles that require higher resolution integrations. The integrator is approximately as accurate as other hybrid symplectic integrators but is GPU accelerated.     

Existence of formal integrals of symplectic integrators

A recurrent method of solving the formal integrals of symplectic integrators is given. The special examples show that there are no long-term variations in all integrals of the Hamiltonian system in addition to the energy one when symplectic integrators are used in the numerical studies of the system. As an application of the formal integrals, the relation between them and the linear stability of symplectic integrators is discussed.     

Canonical forms for symplectic and Hamiltonian matrices

This paper gives a constructive method for finding canonical forms for symplectic and Hamiltonian matrices. No restrictions are made on the eigen values or their multiplicity. Real canonical forms are treated in detail.     

Numerical experiments on the efficiency of second-order mixed-variable symplectic integrators for N-body problems

We discuss the efficiency of the so-called mixed-variable symplectic integrators for N-body problems. By performing numerical experiments, we first show that the evolution of the mean error in action-like variables is strongly dependent on the initial configuration of the system. Then we study the effect of changing the stepsize when dealing with problems including close encounters between a particle and a planet. Considering a previous study of the slow encounter between comet P/Oterma and Jupiter, we show that the overall orbital patterns can be reproduced, but this depends on the chosen value of the maximum integration stepsize. Moreover the Jacobi constant in a restricted three-body problem is not conserved anymore when the stepsize is changed frequently: over a 10⁵ year time span, to keep a relative error in this integral of motion of the same order as that given by a Bulirsch-Stoer integrator requires a very small integration stepsize and much more computing time. However, an integration of a sample including 10⁴ particles close to Neptune shows that the distributions of the variation of the elements over one orbital period of the particles obtained by the Bulirsch-Stoer integrator and the symplectic integrator up to a certain integration stepsize are rather similar. Therefore, mixed-variable symplectic integrators are efficient either for N-body problems which do not include close encounters or for statistical investigations on a big sample of particles.     

On the variational equations associated with a Lagrangian

In a previous publication, Broucke [1] has studied the symplectic properties of the variational equations of a Lagrangian of a very particular form, withconstant coefficients. In this article, we generalize his results to the case of an arbitrary Lagrangian. We show that the characteristic exponents of a periodic solution can be computed in Lagrangian formulation as well as in the more usual Hamiltonian formulation.     

Variational and symplectic integrators for satellite relative orbit propagation including drag

Orbit propagation algorithms for satellite relative motion relying on Runge–Kutta integrators are non-symplectic—a situation that leads to incorrect global behavior and degraded accuracy. Thus, attempts have been made to apply symplectic methods to integrate satellite relative motion. However, so far all these symplectic propagation schemes have not taken into account the effect of atmospheric drag. In this paper, drag-generalized symplectic and variational algorithms for satellite relative orbit propagation are developed in different reference frames, and numerical simulations with and without the effect of atmospheric drag are presented. It is also shown that high-order versions of the newly-developed variational and symplectic propagators are more accurate and are significantly faster than Runge–Kutta-based integrators, even in the presence of atmospheric drag.