Numerical integration of dynamical systems with Lie series

The integration of the equations of motion in gravitational dynamical systems—either in our Solar System or for extra-solar planetary systems—being non integrable in the global case, is usually performed by means of numerical integration. Among the different numerical techniques available for solving ordinary differential equations, the numerical integration using Lie series has shown some advantages. In its original form (Hanslmeier and Dvorak, Astron Astrophys 132, 203 1984), it was limited to the N-body problem where only gravitational interactions are taken into account. We present in this paper a generalisation of the method by deriving an expression of the Lie terms when other major forces are considered. As a matter of fact, previous studies have been done but only for objects moving under gravitational attraction. If other perturbations are added, the Lie integrator has to be re-built. In the present work we consider two cases involving position and position-velocity dependent perturbations: relativistic acceleration in the framework of General Relativity and a simplified force for the Yarkovsky effect. A general iteration procedure is applied to derive the Lie series to any order and precision. We then give an application to the integration of the equation of motions for typical Near-Earth objects and planet Mercury.     

Integration of the elliptic restricted three-body problem with Lie series

The method of Lie series is used to construct a solution for the elliptic restricted three body problem. In a synodic pulsating coordinate system, the Lie operator for the motion of the third infinitesimal body is derived as function of coordinates, velocities and true anomaly of the primaries. The terms of the Lie series for the solution are then calculated with recurrence formulae which enable a rapid successive calculation of any desired number of terms. This procedure gives a very useful analytical form for the series and allows a quick calculation of the orbit.The project is supported by the Austrian Fonds zur Förderung der wissénschaftlichen Forschung under Project No. 4471.     

Lie groups of motor integrals of generalized Kepler motion

The singularity of the Kepler motion can be eliminated by means of the spinor regularization. The extensive integrals of the Kepler motion form a Lie algebra with respect to the Poisson bracket operation. Mayer-Gürr has shown that in the caseH>0 the corresponding Lie group is the multiplicative group of all real 4×4 unimodular matrices SL(4,R). Kustaanheimo has posed the problem of the identification of the corresponding Lie groups in the elliptic and parabolic cases. We solve this problem here and use the opportunity to introduce the concept of the Clifford algebra which is needed in our solution.     

Third-order Uranus-Neptune theory

In this paper of the third order Uranus-Neptune planetary theory which is the third part of this work for the third order theory, we compute the Poisson brackets in the Lie series which is used to transform canonical variables. We apply Hori-Lie technique in this work and neglect all powers higher than the second in Poincaré variables H, K, P, Q. We restrict this work to the principal part of the disturbing function.     

The equivalence of von Zeipel mappings and Lie transforms

An order independent method is used to identify the transformation obtained from Von Zeipel's method with that obtained from Lie transforms. The correspondence between the generators is given in explicit form.     

Algebraic aspects of perturbation theories

In this paper the following problems are considered: Hori's perturbation equations, the composition of two Lie series, the elimination of geometrical (virtual) singularities in perturbation theory, the connection between the methods of Hori and Deprit. The analysis is based on an isomorphism between the Lie algebra of the non-associative algebra of vector fields and a Lie algebra of linear operators. All linear operators, however, form an associative algebra.     

Synchronization of perturbed non-linear Hamiltonians

We propose a new method based on Lie transformations for simplifying perturbed Hamiltonians in one degree of freedom. The method is most useful when the unperturbed part has solutions in non-elementary functions. A non-canonical Lie transformation is used to eliminate terms from the perturbation that are not of the same form as those in the main part. The system is thus transformed into a modified version of the principal part. In conjunction with a time transformation, the procedure synchronizes the motions of the perturbed system onto those of the unperturbed part.A specific algorithm is given for systems whose principal part consists of a kinetic energy plus an arbitrary potential which is polynomial in the coordinate; the perturbation applied to the principal part is a polynomial in the coordinate and possibly the momentum.We demonstrate the strategy by applying it in detail to a perturbed Duffing system. Our procedure allow us to avoid treating the system as a perturbed harmonic oscillator. In contrast to a canonical simplification, our method involves only polynomial manipulations in two variables. Only after the change of time do we start manipulating elliptic functions in an exhaustive discussion of the flows.     

Formal solution of the asteroid motion near the 2:1 commensurability

In a previous paper (Ammar in Proc. Math. Phys. 77:99, 2002) the statement of the problem was formulated and the basic equations of motion were formed in terms of variables suitable for the applications in the problem of asteroid motion close to 2:1 commensurability. The short period terms has been eliminated up to the first order in masses O(μ), using a perturbation approach based on the Lie series, the problem is reduced to that of secular resonance one. In the present work the extended Delaunay method has been applied to develop the Hamiltonian and the generator as a power series in rather than the power of ε, where ε is a small parameter of order of the relative mass of the perturber. Hamilton–Jacobi method were used as a method of integration of the equations of the dynamical system in order to build a formal solution for the resonant problem of the type 2:1 with one degree of freedom.     

The elimination of the parallax in satellite theory

When the perturbation affecting a Keplerian motion is proportional tor ^–n (n3), a canonical transformation of Lie type will convert the system into one in which the perturbation is proportional tor ^–2. Because it removes parallactic factors, the transformation is called the elimination of the parallax.In the main problem for the theory of artificial satellites, the elimination of the parallax has been conducted by computer to order 4. The first order in the reduced system may now be integrated in closed form, thereby revealing the fundamental property of the first-order intermediary orbits in line with Newton's Propositio XLIV.Extension beyond order 1 leads to identify a new class of intermediaries for the main problem in nodal coordinates, namely the radial intermediaries.The technique of smoothing a perturbation prior to normalizing the perturbed Keplerian system, of which the elimination of the parallax is an instance, is applied to derive the intermediaries in nodal coordinates proposed by Sterne, Garfinkel, Cid-Palacios and Aksnes, and to find the canonical diffeomorphisms which relate them to one another and to the radial intermediaries.     

New approach of studying electromagnetic mode coupling in inhomogeneous magnetized plasma

In this paper we use a new approach to derive the system of first-order coupled equations governing the propagation of electromagnetic waves in an inhomogeneous magnetized plasma for normal incidence. In this new approach we employ a step model and use Maxwell's equations indirectly. The method we present here possesses simplicity in mathematical manipulation and gives a clearer physical picture of the mechanism of mode-coupling. The variablesE _i used in the coupled equations are directly related to experimental measurements. Our result is shown to be equivalent to that obtained by Budden and Clemmow (1957).     

The generalized canonical form of Hori's method for non-canonical systems

Any dynamical system can be put in generalized canonical form through the introduction of a set of auxiliary ‘conjugate’ variables or momenta and solved by perturbation theory based on Lie series. The application of Hori's method for generalized canonical system leads to a new canonical transformation — the Mathieu transformation — defined by the solution of the Hori auxiliary system. This new transformation simplifies the algorithm since the inversion of the solution of the Hori auxiliary system is no longer necessary. In this paper, we wish to show some peculiarities of this technique.     

The Use of Lie Series in the Construction of a Perturbation Theory and Some Recent Results in the Theory of the Motion of Hyperion

This paper begins with a brief review of a form of the Lie series transformation, and then reports some new results in the study, using Lie series methods, of the orbit of Saturn's satellite Hyperion. In particular, improved expressions are given for the long-period perturbations of the orbital elements which describe the motion in the orbit plane, and also first results for expressions for the short-period perturbations in the apse longitude, derived from the Lie series generating function. This revised version was published online in July 2006 with corrections to the Cover Date.     

Non-linear stability in photogravitational non-planar restricted three body problem with oblate smaller primary

We have discussed non-linear stability in photogravitational non-planar restricted three body problem with oblate smaller primary. By photogravitational we mean that both primaries are radiating. We normalized the Hamiltonian using Lie transform as in Coppola and Rand (Celest. Mech. 45:103, 1989). We transformed the system into Birkhoff’s normal form. Lie transforms reduce the system to an equivalent simpler system which is immediately solvable. Applying Arnold’s theorem, we have found non-linear stability criteria. We conclude that L ₆ is stable. We plotted graphs for (ω ₁,D ₂). They are rectangular hyperbola.     

Multiple-parameter Lie transform

A method is developed to construct a Lie transform in the multiple-parameter case; a generalization of Deprit's, Henrard's, and Varadi's methods. The details of the theoretical developments are given for the case of three different parameters. A simple approach is suggested to reduce the case ofn parameters to that of the one parameter case.     

Direct perturbations of the planets on the Moon's motion

The aim of the present paper is to present the theoretical background of a method to compute the planetary perturbations on the Moon's motion. We formulate an algorithm based upon the Lie transform method and well-suited to the particular problem at hand.This algorithm is being implemented using Henrard's Semi-Analytical Lunar Ephemeris (SALE) as solution of the Main Problem and Bretagnon's planetary theory. The accuracy of the solution is intended to be about 0".001 for terms of period up to 2000 years.To illustrate the interest of our approach, we comment on some preliminary results obtained about the direct perturbations due to Venus on the Moon's longitude. The final results will be the subject of another paper.     

Decomposition of functions for elliptic orbits

In the algebraA of functions periodic in the mean anomaly we relate the problem of integrating over the mean anomaly with that of decomposing an element ofA as the direct sum of two functions, one in the kernel of the Lie derivative in the Keplerian flow and one in the image of this Lie derivative. We propose recursive rules amenable to general purpose symbolic processors for accomplishing such decomposition in a wide subclass ofA. We introduce the dilogarithmic function to express in exact terms quadratures involving the equation of the center.     

Propagation of the Bastille Day 2000 CME Shock in the Outer Heliosphere

The Bastille Day (14 July) 2000 CME is a fast, halo coronal mass ejection event headed earthward. The ejection reached Earth on 15 July 2000 and produced a very significant magnetic storm and widespread aurora. At 1 AU the Wind spacecraft recorded a strong forward shock with a speed jump from ∼ 600 to over 1000 km s⁻¹. About 6 months later, this CME-driven shock arrived at Voyager 2 (∼ 63 AU) on 12 January 2001 with a speed jump of ∼ 60 km s⁻¹. This provides a good opportunity to study the shock propagation in the outer heliosphere. In this study, we employ a 2.5-D MHD numerical model, which takes the interaction of solar wind protons and interstellar neutrals into account, to investigate the shock propagation in detail and compare the model predictions with the Voyager 2 observations. The Bastille Day CME shock undergoes a dramatic change in character from the inner to outer heliosphere. Its strength and propagation speed decay significantly with distance. The model results at the location of Voyager 2 are in good agreement with in-situ observations. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1014293527951     

Completely Integrable Systems Connected with Lie Algebras

In a recent paper Ballersteros and Ragnisco (1998) have proposed a new method of constructing integrable Hamiltonian systems. A new class of integrable systems may be devised using the following sequence: , where A is a Lie algebra is a Lie–Poisson structure on R ₃, C is a Casimir for is a reduced Poisson bracket and (A, ▵) is a bialgebra. We study the relation between a Lie-Poisson stucture Λ and a reduced Poisson bracket , which is a key element in using the Lie algebra A to constructing this sequence. New examples of Lie algebras and their related integrable Hamiltonian systems are given. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Variable synchrotron emission from BL Lacertae objects. I. Shock-in-jet scenario

This paper presents a modeling of the variable synchrotron emission in the BL Lacertae sources (BLLs). Flux variability is assumed to be a result of the interaction between a relativistic shock wave with a magnetized jet material. Long-term flares (of months to years durations) are modeled via the propagation of a plane relativistic shock wave though the emission zone of a cylindrical form with the radius R and length H. As for short-term bursts (lasting from days to weeks), they may result from shock passage through the jet inhomogeneities such as a shell of enhanced density downstream to a Mach disc, originated due to pressure imbalance between the jet and its ambient medium. Emitting particles (electrons) gain the energies, sufficient to produce synchrotron photons at optical—X-ray frequencies, via the first-order Fermi mechanism. Observation’s frequency is the main parameter determining a rate of the increase/ decay of the emission via the characteristic decay time of emitting electrons. The magnetic field, assumed to be turbulent with an average field constant throughout the entire emission zone, is another key parameter determining the slope of a lightcurve corresponding to the flare—the higher strength the magnetic field has, the steeper the lightcurve is. The rest input parameters (shock speed, jet viewing angle, maximum/minimum energies of the electrons, particles’ density etc.), as well the strength of average magnetic field, influence the energy output from a flare.