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.     

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.     

On a relation among lie series

In this paper we discuss the relation between the structures of the series expansion for the Dragt and Finn composition of Lie transforms and for a transformation introduced by Giorgilli and Galgani. A recursive algorithm is presented which is used to generate the series expansion for the composition of Lie transforms. This algorithm strongly resembles the algorithm of Giorgilli and Galgani, and differs from it only in an ordering property. The relation with the algorithms of Kamel and Henrard for Deprit's direct and inverse series is also discussed.     

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.     

Poincaré's méthode nouvelle by skew composition

Poincaré designed the méthode nouvelle in order to build approximate integrals of Hamiltonians developed as series of a small parameter. Due to several critical deficiencies, however, the method has fallen into disuse in favor of techniques based on Lie transformations. The paper shows how to repair these shortcomings in order to give Poincaré’s méthode nouvelle the same functionality as the Lie transformations. This is done notably with two new operations over power series: a skew composition to expand series whose coefficients are themselves series, and a skew reversion to solve implicit vector equations involving power series. These operations generalize both Arbogast’s technique and Lagrange’s inversion formula to the fullest extent possible. This revised version was published online in July 2006 with corrections to the Cover Date.     

Canonical transformations depending on a small parameter

The concept of a Lie series is enlarged to encompass the cases where the generating function itself depends explicity on the small parameter. Lie transforms define naturally a class of canonical mappings in the form of power series in the small parameter. The formalism generates nonconservative as well as conservative transformations. Perturbation theories based on it offer three substantial advantages: they yield the transformation of state variables in an explicit form; in a function of the original variables, substitution of the new variables consists simply of an iterative procedure involving only explicit chains of Poisson brackets; the inverse transformation can be built the same way.     

A unified treatment of some expansion procedures in perturbation theory: Lie series,Faà di Bruno operators,and Arbogast's Rule

The problem of expanding the transform of a function under a near-identity change of coordinates is reviewed. A common derivation is given for Musen's Faà di Bruno operators and Kamel's Lie series, and both are related to Arbogast's Rule for composing power series.     

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.     

Quantitative predictions with detuned normal forms

The phase-space structure of two families of galactic potentials is approximated with a resonant detuned normal form. The normal form series is obtained by a Lie transform of the series expansion around the minimum of the original Hamiltonian. Attention is focused on the quantitative predictive ability of the normal form. We find analytical expressions for bifurcations of periodic orbits and compare them with other analytical approaches and with numerical results. The predictions are quite reliable even outside the convergence radius of the perturbation and we analyze this result using resummation techniques of asymptotic series.     

Literal solution for Hill's lunar problem

A computer program for the manipulation of power series in several variables is used to find the first order solution to Hill's lunar problem. The ratiom of the mean motion of the Sun to that of the Moon is kept as a formal parameter. The series inm are known to converge very poorly. It is shown how efficient algorithms among them the Lie transformation allow us to compute the series inm as far as they are needed. When the series are evaluated at Brown's numerical value form the results achieve or exceed his accuracy.     

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.     

Application of Hori's technique in general planetary theory

In this part we present the complete solution of the planetary canonical equations of motion by the method of G. Hori through successive changes of canonical variables using the Lie series. Thus, we can eliminate the long or critical terms of the planetary perturbing function, in our general planetary theory. In our formulas, we neglect perturbation terms of order higher than the third with respect to planetary masses.     

The Translational–Rotational Motion of an Earth Spheroid Satellite

In this paper we consider the translational–rotational motion of a spheroid satellite in the gravitational field, taking into account the asphericity of the earth. The harmonic coefficients of the earth’s gravitational field are taken up to J ₄. The equations of motion are obtained in terms of the canonical elements of Delaunay-Andoyer. A first order solution is obtained using the perturbing technique of Lie series.     

Analytical Study of The Resonance Caused by Solar Radiation Pressure on a Spacecraft

The resonance terms produced by the effect of direct solar radiation pressure on the motion of a spacecraft in the oblate field of the earth are analyzed. The spacecraft was assumed axially symmetric with a despun antenna and solar panels. A canonical transformation technique is developed, based on the Bohlin technique of expansion in fractional powers, using Lie series and transformation as well as the concept of the Delaunay anomaly. The developed technique, applied to the problem averaged over the mean anomaly, is suitable in the presence of more than one resonant vector.     

Radius of convergence of Lie series for some elliptic elements

For equatorial orbits about an oblate body, we show that the Lie series for the elliptic elementse,f,l and diverge when the oblateness exceeds a critical multiple of the transformed eccentricity constant. The use of similar truncated series expansions for such elliptic elements by Brouwer accounts for the first-order errors at low eccentricity in his derived coordinates for an artificial satellite.     

Analytical theories of satellite motion constructed with a new package of formula manipulation and compared with numerical integration

Specialized to the Lie series based perturbation method of Kirchgraber and Stiefel (1978) a new computer algebra package called ANALYTOS has been developed for constructing analytical orbital theories either in noncanonical or canonical form. We present results on the (extended) Main Problem of orbital theory of artificial earth satellites and related issues. The order of the solutions achieved is generally one order higher than those known from literature. Moreover, the analytical orbits have been checked succesfully against precise numerical ephemerides. This revised version was published online in July 2006 with corrections to the Cover Date.     

Keplerian expansions in terms of Henrard's practical variables

Formulae for the Keplerian expansions in terms of Henrard's practical variables are given. Two different methods were applied: one using the Bessel functions and one based on the Lie transforms. The former involves less series products, but the latter is more flexible and universal.     

Analytical treatment of the two-body problem with slowly varying mass

The present work is concerned with the two-body problem with varying mass in case of isotropic mass loss from both components of the binary systems. The law of mass variation used gives rise to a perturbed Keplerian problem depending on two small parameters. The problem is treated analytically in the Hamiltonian frame-work and the equations of motion are integrated using the Lie series developed and applied, separately by Delva (1984) and Hanslmeier (1984). A second order theory of the two bodies eject mass is constructed, returning the terms of the rate of change of mass up to second order in the small parameters of the problem.     

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 ideal resonance problem a comparison of two formal solutions I

Garfinkel's solution of the Ideal Resonance problem derived from a Bohlin-von Zeipel procedure, and Jupp's solution, using Poincaré's action and angle variables and an application of Lie series expansions, are compared. Two specific Hamiltonians are chosen for the comparison and both solutions are compared with the numerical solutions obtained from direct integrations of the equations of motion. It is found that in deep resonance the second-mentioned solution is generally more accurate, while in the classical limit the first solution gives excellent agreement with the numerical integrations.This article represents a summary of a much more extensive programme of research, the complete results of which will be published in a future article.