Delaunay Variables Research Articles

The restricted two-body problem in constant curvature spaces

We perform the bifurcation analysis of the Kepler problem on $S^3$ and $L^3$. An analogue of the Delaunay variables is introduced. We investigate the motion of a point mass in the field of the Newtonian center moving along a geodesic on $S^2$ and $L^2$ (the restricted two-body problem). When the curvature is small, the pericenter shift is computed using the perturbation theory. We also present the results of the numerical analysis based on the analogy with the motion of rigid body.

Corrigendum to: Elliptic Two-Dimensional Invariant Tori for the Planetary Three-Body Problem

The main result of the paper [BCV] Elliptic two-dimensional invariant tori for the planetary three-body problem by the authors, published in Arch. Ration. Mech. Anal. 170, 91–135 (2003), namely, Theorem 1.2, exploits a classical result (Theorem 1.1) essentially due to Delaunay and Poincare. In Appendix C of [BCV] a detailed proof of Theorem 1.1 is discussed. However, this proof contains a flaw. In fact, the statement following equation (C.44) in Proposition C.3, Section C.2.2, Appendix C is not correct. The Z∗ j ’s (called in [BCV] “Poincare integrals”) are not global integrals. Therefore, Section C.2.2 requires amendment. What now follows, replaces Section C.2.2 in [BCV], correcting the above mentioned flaw. Notations are as in [BCV]. Recall that in [BCV] we consider the 10-dimensional invariant symplectic manifold, Mver, defined by taking the non-vanishing total angular momentum to be vertical, i.e., C:=C(1) + C(2) = ck3 where C denotes the angular momentum of the ith planet with respect to the origin of an inertial heliocentric frame {k1, k2, k3} and c ∈ R\{0}; compare (C.30) and (C.31). Recall, also, that in [BCV] we use the “osculating Poincare variables” ( ∗, η∗, p∗, λ∗, ξ∗, q∗) as described in Section C.2.1 of [BCV], where ∗ = ( ∗1, ∗2), η∗ = (η∗ 1, η∗ 2), etc., and the index i = 1, 2 refers to the two-body system ith planet-star. Finally, recall that −ζ ∗ i = θi is the longitude of the ith node on the plane {k1, k2}, while −Z∗ i = i = C · k3. Let us now describe, analytically, Mver. From the definition of the Delaunay variables (Li, i,Gi, gi, i, θi) (compare to Section C.1.1) it follows that the relations defining Mver, i.e., C · k1 = 0 = C · k2, are equivalent to ζ ∗ 2 − ζ ∗ 1 = π and G1 − 1 = G2 − 2. Therefore, since Gi = ∗i − H ∗ i , with H ∗ i :=(η∗ i 2 + ξ∗ i )/2 (compare to (C.5), (C.15) and (C.17)), we find

The post-Newtonian effects in the critical inclination problem in artificial satellite theory

The critical inclination problem in artificial satellite theory and the post-Newtonian (PN) effects on it are discussed. The Hamiltonian is constructed and is expressed in terms of the Delaunay variables containing the zonal harmonics up to J 4 and the PN contribution up to order J 2/ c l 2. The Hamiltonian is ordered out such that the PN terms are of order J 2. Two successive canonical transformations based on Lie series and transform are carried out to remove short and long period terms from the Hamiltonian, respectively. Terms are retained up to the third order in the secular perturbations and to the second order in the periodic perturbations, where J 2 is considered of first order.

Canonical solution of the critical inclination problem taking into account PN-corrections

The critical inclination problem in artificial satellite theory and the post-Newtonian (PN) effects on it are discussed. The Hamiltonian is constructed and is expressed in terms of the Delaunay variables containing the zonal harmonics up to J 4 and the PN contributions up to order J 2/ c l 2. The Hamiltonian is ordered out such that the PN terms are of order J 2. Two successive canonical transformations based on Lie series and transform are carried out to remove short- and long-period terms from the Hamiltonian, respectively. Terms are retained up to third order in the secular and to the second order in the periodic terms, where J 2 is considered of first order.

Non-Continuation of Integrals of the Rotating Two-body Problem in Hill’s Lunar Problem

We consider Hill's lunar problem as a perturbation of the integrable two-body problem. For this we avoid the usual normalization in which the angular velocity ω of the rotating frame of reference is put equal to unity and consider ω as the perturbation parameter. We first express the Hamiltonian H of Hill's lunar problem in the Delaunay variables. More precisely we deduce the expressions of H along the orbits of the two-body problem. Afterwards with the help of the conserved quantities of the planar two-body problem (energy, angular momentum and Laplace–Runge–Lenz vector) we prove that Hill's lunar problem does not possess a second integral of motion, independent of H, in the sense that there exist no analytic continuation of integrals, which are linear functions of ω in the rotating two-body problem. In connection with the proof of this main result we give a further restrictive statement to the nonintegrability of Hill's lunar problem.

The equations of motion of the joint effects of the relativistic and solar radiation pressure on an earth artificial satellite

This work deals with the study of the post-Newtonian (PN) geopotential and direct solar radiation pressure (SRP) effects on the motion of an Earth artificial satellite. The potential of the Earth, including the PN terms, is developed in terms of the Delaunay variables up to 0(4) assuming J 2 of first order. A potential like function representing the SRP is formed to enable this effect to be treated in the Hamiltonian framework. The Hamiltonian is then formed in terms of the Delaunay variables representing the problem, augmented to remove the explicit time dependence through the motion of the sun.

Geometric Derivation of the Delaunay Variables and Geometric Phases

We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus T3 on the phase space of the Kepler problem, computing its associated momentum map and using the geometry associated with this structure. A central feature in this derivation is the identification of the mean anomaly as the angle variable for a symplectic S1 action on the union of the non-degenerate elliptic Kepler orbits. This approach is geometrically more natural than traditional ones such as directly solving Hamilton–Jacobi equations, or employing the Lagrange bracket. As an application of the new derivation, we give a singularity free treatment of the averaged J2-dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J2-Hamiltonian is a collective Hamiltonian of the T3 momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J2 effect as geometric phases.

Evolution of the Motion of a Viscoelastic Sphere in a Central Newtonian Field

The evolution of translational-rotational motion of a viscoelastic sphere in a central Newtonian field is studied. By the method of separating motions and averaging in the generalized Andoyer–Delaunay variables, equations are derived that describe the evolution of motion. The analysis of the approximate equations obtained is performed. The condition of existence of a steady-state motion is found, and its stability is investigated using the equations in variations.

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.

Gyldén Model: The Slowly Changing Equivalent Gravitational Parameter

In the Gylden problem, the case of slowly changing equivalent gravitational parameter (e.g.p.) is studied. Assuming the following law for the variation of the e.g.p.: µ (t) = µ0+ eµ (et), e< µ0, we obtained a) a O(e4 T4)-approximation of the solution, on a shortened time scale (0,T), with T of order o(e-1),for the general case (i.e. the support function µ is O(1)-valued and admits an expansion in power series of its argument), and b) a O(e2)-approximation of the solution, on a natural timescale of order O(1/e), for the case of a bounded variation rate (i.e. the support function µ and its derivative are both O(1)-valued). For the study of the first problem we introduced a type of Lissajous variables and used the Lie-Hori normalization scheme; for the second problem we used the Delaunay variables and applied the von Zeipel method for approximate integration. The physical interpretation of the results is in both cases the same: within the corresponding limits of approximation, the variation of e.g.p. (i) has no effect on the size of the osculating ellipse, (ii) it sets the pericenter in slow rotation and (iii) it introduces a secular variation in the longitudes.

Analytical Effects of Gravitational Waves on the Motion of an Artificial Satellite

AbstractThe motion of an artificial satellite in the Earth’s gravitational field is discussed in the post-Newtonian framework including the effect of weak gravitation waves using the perturbation technique of the canonical Lie-transformations. Two successive canonical transformations are used to derive analytical expressions for the short-period, long-period and secular perturbations of orbital elements. The solution is expressed in terms of the Delaunay variables.

An application of the Nekhoroshev theorem to the restricted three-body problem

We studied the stability of the restricted circular three-body problem. We introduced a model Hamiltonian in action-angle Delaunay variables. which is nearly-integrable with the perturbing parameter representing the mass ratio of the primaries. We performed a normal form reduction to remove the perturbation in the initial Hamiltonian to higher orders in the perturbing parameter. Next we applied a result on the Nekhoroshev theorem proved by Poschel [13] to obtain the confinement in phase space of the action variables (related to the elliptic elements of the minor body) for an exponentially long time. As a concrete application. we selected the Sun-Ceres-Jupiter case, obtaining (after the proper normal form reduction) a stability result for a time comparable to the age of the solar system (i.e., 4.9 · 109 years) and for a mass ratio of the primaries less or equal than 10−6.

Passing through resonance: The excitation and dissipation of the lunar free libration in longitude

The acceleration of the mean lunar longitude has a small effect on the periods of most terms in a Fourier expansion of the longitude. There are several planetary perturbation terms that have small amplitudes, but whose periods are close to the resonant period of the lunar libration in longitude. Some of these terms are moving toward resonance, some are moving away from resonance, and the periods of those terms that do not include the Delaunay variables in their arguments are not moving. Because of its acceleration of longitude, the Moon is receding from the Earth, so the magnitude of the restoring torque that the Earth exerts on the rotating Moon is gradually attenuating; thus resonance itself is moving, but at a much slower rate than the periods of the accelerating planetary perturbations. There are five planetary perturbation terms from the ELP-2000 Ephemeris (with amplitudes of 0”.00001 or greater) that have passed through resonance in the past two million years. One of them is of special interest because it appears to be the excitation source of a supposed free libration in longitude that has been detected by the lunar laser ranging experiment. The amplitude of the term is only 0”.00021 but it could be the source of the ~ 1” amplitude free libration term if the viscoelastic properties of the Moon are similar to those of the Earth.

The Lissajous transformation IV. Delaunay and Lissajous variables

Relations are established between the Delaunay variables defined over a phase space E in four dimensions and the Lissajous variables defined over a four-dimensional phase space F when the latter is mapped onto E by a parabolic canonical transformation.

Periodic orbits of the third kind in the restricted three-body problem with oblateness

By use of the secular perturbing potential due to oblateness, the existence of periodic orbits of the third kind is established through Poincare's method of analytic continuation using Delaunay's canonical variables and other three sets of canonical variables which are linear combinations of Delaunay's variables, in the three-dimensional restricted three-body problem when the more massive primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. For two sets of the canonical variables, the singularities are found at the inclinationsi=68 . ° 5833,111 . ° 4167, while for the other two sets of the canonical variables, the singularities are ati=55 . ° 3854, 103 . ° 575.

Study of the terms of coupling between rotational and translational motions

The equations of the translational-rotational motion of an Earth's artificial satellite, taking into account only the gravitational attraction, can be put in canonical form using the Delaunay and Andoyer variables. Terms due to the oblateness of the Earth will be considered, as well as, different moments of inertia of the satellite, when it is very close to another. Hori's Method is applied to analyse the equations and the solution is presented up to the first order.

Asymptotic solutions of the restricted problem near the equilateral Lagrangian points

The restricted problem in the vicinity of the Lagrangian point L4 is studied by finding a convergent binomial expansion of the disturbing function. Using a Hamiltonian formulation in Delaunay variables and removing the short-period terms a resonance problem (already considered by Giacaglia (1970) in an attempt of enlarging the Ideal Resonance) is obtained. It is shown that this extension is reducible to Garfinkel's ideal resonance in the libration region.

Remark on integrable Hamiltonian systems

An extension ton degrees of freedom of the fact is established that forn=1 the time and the energy constant are canonically conjugate variables. This extension is useful in some cases to get action-angle variables from the general solution of a given integrable Hamiltonian system. As an example the Delaunay variables are proved to be canonical.

Collisions of stars by oscillating orbits of the second kind

Ordinary estimations of the number of star collisions in our galaxy—by simple kinematic considerations—lead to a very small number of such collisions: about one or even less every millions of years. However star collisions can occur through the following indirect way which has a much higher probability. (a) Binary stars are very common in our galaxy, about 30–50% of the stars. (b) If two binary stars meet a triple system can be formed by an ordinary exchange type motion. (c) A triple system is generally decomposed into the “inner orbit” (i.e. the relative orbit of the two nearest stars) and the “outer orbit” (i.e. the relative orbit of the third star with respect to the center of mass of the two nearest stars). The major axes of these two orbits have generally small perturbations and it is the same for the eccentricity of the outer orbit. On the contrary, if the relative inclination of the two orbits is large, the perturbations of the eccentricity of the inner orbit are important and can even in some cases lead to an eccentricity equal to one, that is to a collision of the two stars of the inner orbit. Such orbits can be called “oscillating orbits of the second kind”, indeed the first oscillating orbits—conceived by Khilmi and described for the first time in an example by Sitnikov—have unbounded mutual distances r ij , but the system always come back to small sizes, it has an infinite number of very large expansions followed by strong contractions and, in the three-body case, an upper bound of lim inf ( r 1.2 + r 1.3 + r 2.3) can be given in terms of the three masses and the integrals of motion. For the oscillating orbits of the second kind the mutual distances r ij are bounded, but the velocities are unbounded (i.e. lim inf r ij = 0 for at least one r ij ) and the system goes to a collision if the bodies have non-zero radius even small. The analytical study of the oscillating orbits of the second kind is a part of the general analytical study of the three-body problem, a part which must be valid for large eccentricities and large inclinations. The use of Delaunay's variables and of a Von Zeipel transformation lead to a first order integrable approximation, valid for any eccentricities and any inclinations, and giving the following results: (a) The oscillating orbits of the second kind occur when the angular momentum of the outer orbit has a modulus sufficiently close to the modulus of the total angular momentum of the three-body system. Hence these orbits occur for inclinations in the vicinity of 90°. (b) The oscillating orbits represent a set of positive measure of phase space and the first order study allows to give a rough estimation of the probability of collisions—even for stars of infinitely small radius. This probability, for given initial major axes and eccentricities and for isotropic arbitrary initial orientations, is generally of the order of m 3 R M ( m 3 being the mass of the outer star, M the total mass and R the ratio of the period of the inner orbit to that of outer orbit). One question remains to be solved: how many collisions of stars are due to that phenomenon? That question is difficult because the probability of formation of a triple system by a random meeting of two binaries is very uneasy to estimate. However it seems that, compared to the usual evaluations based on pure kinematic considerations without gravitational effects, the number of collisions must be multiplied by a factor between one thousand and one million.

New Formulation of De Sitter’s Theory of Motion for Jupiter I-IV. I. Equations of Motion and the Disturbing Function

AbstractA brief discussion is given of the basic features of de Sitter’s theory. The main advantage of his theory is that it contains no small divisors, thanks to the use of elliptic rather than circular intermediate orbits in the first approximation. A 50-year extension of the satellite observations available to de Sitter makes it desirable to rederive the elements of his intermediate orbits, whose perijoves have a common retrograde motion. Furthermore, the theory suffers from a convergence problem, which can be avoided by reformulating the theory in terms of canonical variables, a task that is begun here. We adopt a formulation in Poincaré’s canonical relative coordinates rather than, as customary, in ordinary relative coordinates or in the Jacobian canonical coordinates. By means of the generalized Newcomb operators devised by Izsak, the disturbing function is expanded in a form that is very convenient for use with the modified Delaunay variables, G, L – G, H – G,l+ ω + Ω,l, and ω and their associated Poincaré variables.