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1.
The D'Alembert model for the spin/orbit problem in celestial mechanics is considered. Using a Hamiltonian formalism, it is shown that in a small neighborhood of a p:q spin/orbit resonance with (p,q) different from (1,1) and (2,1) the 'effective' D'Alembert Hamiltonian is a completely integrable system with phase space foliated by maximal invariant curves; instead, in a small neighborhood of a p:q spin/orbit resonance with (p,q) equal to (1,1) or (2,1) the 'effective' D'Alembert Hamiltonian has a phase portrait similar to that of the standard pendulum (elliptic and hyperbolic equilibria, separatrices, invariant curves of different homotopy). A fast averaging with respect to the 'mean anomaly' is also performed (by means of Nekhoroshev techniques) showing that, up to exponentially small terms, the resonant D'Alembert Hamiltonian is described by a two-degrees-of-freedom, properly degenerate Hamiltonian having the lowest order terms corresponding to the 'effective' Hamiltonian mentioned above. 相似文献
2.
Christoph Lhotka 《Celestial Mechanics and Dynamical Astronomy》2017,127(4):397-428
In this paper, we consider the elliptic collinear solutions of the classical n-body problem, where the n bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler–Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler–Moulton collinear solution of n-bodies splits into \((n-1)\) independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler 2-body problem at Kepler elliptic orbit, and each of the other \((n-2)\) systems is the essential part of the linearized Hamiltonian system at an elliptic Euler collinear solution of a 3-body problem whose mass parameter is modified. Then the linear stability of such a solution in the n-body problem is reduced to those of the corresponding elliptic Euler collinear solutions of the 3-body problems, which for example then can be further understood using numerical results of Martínez et al. on 3-body Euler solutions in 2004–2006. As an example, we carry out the detailed derivation of the linear stability for an elliptic Euler–Moulton solution of the 4-body problem with two small masses in the middle. 相似文献
3.
When the coordinate system used in perturbation theory presents a geometrical singularity and when the perturbation technique fails to take account of this, the solution developed may present singularities which are no longer easily explained by purely geometrical means. These singularities have been calledvirtual singularities by Deprit and Rom (1970). We propose to demonstrate that virtual singularities can in general be avoided by the use of Lie transforms. In general, it is sufficient to recognize that the original Hamiltonian function presents the d'Alembert characteristic with respect to pairs of action-angle variables and that the averaging operations preserve this characteristic. We then apply this criterion to the artificial satellite theory (for small to moderate eccentricity) showing that all of three possible virtual singularities can be avoided at the same time. A new set of elliptic elements, well suited to the problem at hand, is introduced. 相似文献
4.
Bin Kang Cheng 《Celestial Mechanics and Dynamical Astronomy》1979,19(1):31-41
In this paper the first variational equations of motion about the triangular points in the elliptic restricted problem are investigated by the perturbation theories of Hori and Deprit, which are based on Lie transforms, and by taking the mean equations used by Grebenikov as our upperturbed Hamiltonian system instead of the first variational equations in the circular restricted problem. We are able to remove the explicit dependence of transformed Hamiltonian on the true anomaly by a canonical transformation. The general solution of the equations of motion which are derived from the transformed Hamiltonian including all the constant terms of any order in eccentricity and up to the periodic terms of second order in eccentricity of the primaries is given. 相似文献
5.
R. G. Langebartel 《Astrophysics and Space Science》1981,75(2):437-454
The restricted three-body Hamiltonian is partitioned into a two-center type principal part and its accompanying perturbational part. The mathematical analysis, involving the Jacobian elliptic functions, is adapted for the case of figure-eight orbits winding around the two given mass points. For many such orbits the elliptic function modulusk is small and can serve as a small parameter.Fourier expansions in terms of a parameter related tot are obtained for the intermediate orbit functions which provide representations in terms of elementary functions. 相似文献
6.
M. Šidlichovský 《Celestial Mechanics and Dynamical Astronomy》1993,56(1-2):143-152
A systematic study of the main asteroidal resonances of the third and fourth order is performed using mapping techniques. For each resonance one-parameter family of surfaces of section is presented together with a simple energy graph which helps to understand and predict the changes in the surfaces of section within the family. As the truncated Hamiltonian for the planar, elliptic, restricted three-body problem is used for the mapping, the method is expected to fail for high eccentricities. We compared, therefore, the surfaces of section with trajectories calculated by symplectic integrators of the fourth and six order employing the full Hamiltonian. We found a good agreement for small eccentricities but differences for the higher eccentricities (e 0.3). 相似文献
7.
Sebastián Ferrer Juan Félix San-Juan Alberto Abad 《Celestial Mechanics and Dynamical Astronomy》2007,99(1):69-83
In the analytical approach to the main problem in satellite theory, the consideration of the physical parameters imposes a
lower bound for normalized Hamiltonian. We show that there is no elliptic frozen orbits, at critical inclination, when we
consider small values of H, the third component of the angular momentum. The argument used suggests that it might be applied also to more realistic
zonal and tesseral models. Moreover, for almost polar orbits, when H may be taken as another small parameter, a different approach that will simplify the ephemerides generators is proposed. 相似文献
8.
John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1992,53(2):151-183
Four 3 : 1 resonant families of periodic orbits of the planar elliptic restricted three-body problem, in the Sun-Jupiter-asteroid system, have been computed. These families bifurcate from known families of the circular problem, which are also presented. Two of them, I
c
, II
c
bifurcate from the unstable region of the family of periodic orbits of the first kind (circular orbits of the asteroid) and are unstable and the other two, I
e
, II
e
, from the stable resonant 3 : 1 family of periodic orbits of the second kind (elliptic orbits of the asteroid). One of them is stable and the other is unstable. All the families of periodic orbits of the circular and the elliptic problem are compared with the corresponding fixed points of the averaged model used by several authors. The coincidence is good for the fixed points of the circular averaged model and the two families of the fixed points of the elliptic model corresponding to the families I
c
, II
c
, but is poor for the families I
e
, II
e
. A simple correction term to the averaged Hamiltonian of the elliptic model is proposed in this latter case, which makes the coincidence good. This, in fact, is equivalent to the construction of a new dynamical system, very close to the original one, which is simple and whose phase space has all the basic features of the elliptic restricted three-body problem. 相似文献
9.
10.
Bernard De Saedeleer 《Celestial Mechanics and Dynamical Astronomy》2005,91(3-4):239-268
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. 相似文献
11.
R. Broucke 《Astrophysics and Space Science》1980,72(1):33-53
In this article we collect several results related to the classical problem of two-dimensional motion of a particle in the field of a central force proportional to a real power of the distancer. At first we generalize Whittaker's result of the fourteen powers ofr which lead to intergrability with elliptic functions. We enumerate six more general potentials, including Whittaker's fourteen potentials as particular cases (Sections 2 and 3).Next, we study the stability of the circular solutions, which are the singular solutions of the problem, in Whittaker's terminology. The stability index is computed as a function of the exponentn and its properties are explained, especially in terms of bifurcations with other families of ordinary periodic solutions (Sections 4, 5 and 7). In Section 6, the detailed solution of the inverse cube force problem is given in terms of an auxiliary variable which is similar to the eccentric anomaly of the Kepler problem.Finally, it is shown that the stable singular circular solutions of the central force problem generalize to stable singular elliptic solutions of the two-fixed-center problem. The stability and the bifurcations with other families of periodic solutions of the two-fixed-center problem are also described. 相似文献
12.
Alan H. Jupp 《Celestial Mechanics and Dynamical Astronomy》1974,9(1):3-20
It is well known that the equations governing the motion of a freely-rotating rigid body possess an exact analytical solution, involving Jacobi's elliptic functions. Andoyer (1923) and Deprit (1967) have shown that the problem may be very usefully reduced to a one-degree-of-freedom Hamiltonian system. When two of the body's principal moments of inertia are very nearly equal, the Hamiltonian system has the same form as the Ideal Resonance Problem. In earlier publications (Jupp, 1969, 1972, 1973), the author has constructed formal power-series solutions of the latter problem.In this article, the general solution of the Ideal Resonance Problem is employed to formulate a second-order formal series solution of the problem of a freely-rotating rigid body which has two of its principal moments of inertia differing by a small quantity. This solution is firstly expressed in terms of the mean elements, and then in terms of the initial conditions. The latter solution is global in nature being applicable over the whole phase plane. It is demonstrated that the exact solution and the second-order formal series solution, written in terms of the initial conditions, differ by terms of at most third order in the small parameter, over the whole domain of possible motions. This serves as an important check on the general results published in the earlier articles. 相似文献
13.
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 S
1 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 J
2-dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J
2-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 J
2 effect as geometric phases. 相似文献
14.
An explicit symplectic integrator is constructed for the problem of a rotating planetary satellite on a Keplerian orbit. The
spin vector is fixed perpendicularly to the orbital plane. The integrator is constructed according to the Wisdom-Holman approach:
the Hamiltonian is separated in two parts so that one of them is multiplied by a small parameter. The parameter depends on
the satellite’s shape or the eccentricity of its orbit. The leading part of the Hamiltonian for small eccentricity orbits
is similar to the simple pendulum and hence integrable; the perturbation does not depend on angular momentum which implies
a trivial ‘kick’ solution. In spite of the necessity to evaluate elliptic function at each step, the explicit symplectic integrator
proves to be quite efficient.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
15.
Transition from elliptic to hyperbolic orbits in the two-body problem with slowly decreasing mass is investigated by means of asymptotic approximations.Analytical results by Verhulst and Eckhaus are extended to construct approximate solutions for the true anomaly and the eccentricity of the osculating orbit if the initial conditions are nearly-parabolic. It becomes clear that the eccentricity will monotonously increase with time for all mass functions satisfying a Jeans-Eddington relation and even for a larger set of functions. To illustrate these results quantitatively we calculate the eccentricity as a function of time for Jeans-Eddington functionsn=0(1) 5 and 18 nearly-parabolic initial conditions to find that 93 out of 108 elliptic orbits become hyperbolic. 相似文献
16.
K. T. Alfriend R. Dasenbrock H. Pickard A. Deprit 《Celestial Mechanics and Dynamical Astronomy》1977,16(4):441-458
The Vinti problem, motion about an oblate spheroid, is formulated using the extended phase space method. The new independent variable, similar to the true anomaly, decouples the radius and latitude equations into two perturbed harmonic oscillators whose solutions toO(J
2
4
) are obtained using Lindstedt's method. From these solutions and the solution to the Hamilton-Jacobi equation suitable angle variables, their canonical conjugates and the new Hamiltonian are obtained. The new Hamiltonian, accurate toO(J
2
4
) is function of only the momenta. 相似文献
17.
We deal with the study of the spatial restricted three-body problem in the case where the small particle is far from the primaries, that is, the so-called comet case. We consider the circular problem, apply double averaging and compute the relative equilibria of the reduced system. It appears that, in the circular problem, we find not only part of the equilibria existing in the elliptic case, but also new ones. These critical points are in correspondence with periodic and quasiperiodic orbits and invariant tori of the non-averaged Hamiltonian. We explain carefully the transition between the circular and the elliptic problems. Moreover, from the relative equilibria of elliptic type, we obtain invariant 3-tori of the original system. 相似文献
18.
Victor A. Brumberg Eugene V. Brumberg: John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1999,75(4):301-301
The aim of this book is to present techniques for the study of motion of solar system objects in highly eccentric orbits. Instead of using the usual anomalies (mean, true, eccentric), the authors define and use a new kind of anomaly, the elliptic anomaly.In this way, it is possible, in a theory using perturbation series expansions, to make the ratio: (accuracy)/(number of needed terms), higher than in the classical techniques. The book consists of six chapters. The first chapter deals with the elliptic anomaly in the two-body problem and the second chapter presents the general technique to construct first-order perturbation theory in elliptic function expansions. The next three chapters deal with applications of the new technique to artificial satellites and asteroids, in highly eccentric orbits. The last chapter describes the basic algorithms of the theory.The tools developed in the book demand the use of computer algebra, which is implemented by means of Mathematica 3.0.The book is well written and the new technique is clearly presented and related to the existing techniques, making it useful to all those who use analytical or semi-analytical methods for the study of highly eccentric motion.
Celestial Mechanics at High Eccentricities, Gordon and Breach Publishers, US$95, GBP 59, EUR 79, ISBN 90-5699-212-0 相似文献
19.
When the elimination of the parallax and the elimination of the perigee is applied to the zonal problem of the artificial satellite, a one-degree of freedom Hamiltonian is obtained. The classical way to integrate this Hamiltonian is by applying the Delaunay normalization, however, changing the time to the perturbed true anomaly and the variable to the inverse of the distance, the Hamilton equations become a perturbed harmonic oscillator. In this paper we apply the Krylov—Bogoliubov—Mitropolsky (KBM) method to integrate the perturbed harmonic oscillator as an alternative method to the Delaunay normalization. This method has no problem with small eccentricities and inclinations, and shows good numerical results in the evaluation of ephemeris of satellites.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
20.
We present a symplectic mapping model to study the evolution of a small body at the 3/4 exterior resonance with Neptune, for planar and for three dimensional motion. The mapping is based on the averaged Hamiltonian close to this resonance and is constructed in such a way that the topology of its phase space is similar to that of the Poincaré map of the elliptic restricted three-body problem. Using this model we study the evolution of a small object near the 3/4 resonance. Both chaotic and regular motions are found, and it is shown that the initial phase of the object plays an important role on the appearance of chaos. In the planar case, objects that are phase-protected from close encounters with Neptune have regular orbits even at eccentricities up to 0.44. On the other hand objects that are not phase protected show chaotic behaviour even at low eccentricities. The introduction of the inclination to our model affects the stable areas around the 3/4 mean motion resonance, which now become thinner and thinner and finally at is=10° the whole resonant region becomes chaotic. This may justify the absence of a large population of objects at this resonance. 相似文献