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1.
Efi Meletlidou Galina Stagika Simos Ichtiaroglou 《Celestial Mechanics and Dynamical Astronomy》2005,91(3-4):323-335
The structure of the resonance zone in nearly integrable Hamiltonian systems is studied by a more general method than the
pendulum approximation. This method applies to the case of a non-degenerate integrable part in the Hamiltonian. This problem
may be overcome in a class of galactic-type polynomial potentials, in the case where the higher-order term is by itself integrable.
An illustrative example is worked out. 相似文献
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Deprit and Miller have conjectured that normalization of integrable Hamiltonians may produce normal forms exhibiting degenerate equilibria to very high order. Several examples in the class of coupled elliptic oscillators are known. In order to test the utility of normalization as a detector of integrability we normalize, to high order, a perturbed Keplerian system known to have several integrable limits; the generalized van der Waals Hamiltonian for a hydrogen atom. While the separable limits give rise to high order degeneracy we find a non-separable, integrable limit for which the normal form does not exhibit degeneracy. We conclude that normalization may, in certain cases, indicate integrability but is not guaranteed to uncover all integrable limits. 相似文献
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There are two parts to this paper. In the first we calculate the hydrodynamic response of the solar atmosphere to the injection of an intense beam of electrons in a numerical simulation of a solar flare. In the second we predict the spectroscopic consequences of the hydrodynamic behaviour calculated in the first part. The hydrodynamics is predicted by solving the equations of conservation of mass, momentum, and energy. The latter is expressed as two temperature equations; one for the electrons and the other for the neutral atoms and positive ions of hydrogen. The equations are solved in one dimension and the geometric form is of a semi-circular loop having its ends in the photosphere. The results show how the loop is filled at supersonic speed with plasma at temperatures characteristic of flares. At the same time a compression wave is predicted to propagate down towards the photosphere. After the heating pulse stops, the plasma that has risen into the loop, starts to decay and return to the condition it was in before the pulse started. In predicting the spectrum that would be emitted by such a plasma calcium was chosen for illustration. The first and main part of this calculation was setting up and solving the time-dependent equations of ionization/recombination. In order to provide a standard for comparison the same ionization and recombination rate coefficients are used to predict the steady-state distribution of populations of ionization stages. This is then compared with the distribution found from the time-dependent solution and shows that there is a negligibly small time lag predicted by the time-dependent result. However the more significant comparisons to make are between the temperatures of the peak abundances of the various ions under the assumptions of steady-state and time-dependent ionization. For the particular circumstances chosen here the temperature differences are predicted to be in the neighbourhood of 10% or less and in view of the overall accuracy of the atomic data are not significant. It would appear therefore that the much simpler assumption of steady-state ionization balance leads to results of acceptable accuracy for the particular case considered. 相似文献
6.
The twice-averaged Hill problem with the oblateness of the central planet is considered in the case where its equatorial plane coincides with the plane of its orbital motion relative to the perturbing body. A qualitative study of this so-called coplanar integrable case was begun by Y. Kozai in 1963 and continued by M.L. Lidov and M.V. Yarskaya in 1974. However, no rigorous analytical solution of the problem can be obtained due to the complexity of the integrals. In this paper we obtain some quantitative evolution characteristics and propose an approximate constructive-analytical solution of the evolution system in the form of explicit time dependences of satellite orbit elements. The methodical accuracy has been estimated for several orbits of artificial lunar satellites by comparison with the numerical solution of the evolution system. 相似文献
7.
The time-transformed leapfrog scheme of Mikkola Aarseth was specifically designed for a second-order differential equation with two individually separable forms of positions and velocities.It can have good numerical accuracy for extremely close two-body encounters in gravitating few-body systems with large mass ratios,but the non-time-transformed one does not work well.Following this idea,we develop a new explicit symplectic integrator with an adaptive time step that can be applied to a time-dependent Hamiltonian.Our method relies on a time step function having two distinct but equivalent forms and on the inclusion of two pairs of new canonical conjugate variables in the extended phase space.In addition,the Hamiltonian must be modified to be a new time-transformed Hamiltonian with three integrable parts.When this method is applied to the elliptic restricted three-body problem,its numerical precision is explicitly higher by several orders of magnitude than the nonadaptive one's,and its numerical stability is also better.In particular,it can eliminate the overestimation of Lyapunov exponents and suppress the spurious rapid growth of fast Lyapunov indicators for high-eccentricity orbits of a massless third body.The present technique will be useful for conservative systems including N-body problems in the Jacobian coordinates in the the field of solar system dynamics,and nonconservative systems such as a time-dependent barred galaxy model in a rotating coordinate system. 相似文献
8.
Giuseppe Pucacco 《Celestial Mechanics and Dynamical Astronomy》2004,90(1-2):109-123
We study strongly and weakly integrable 2-dimensional Hamiltonian systems with velocity dependent potentials. We determine
the set of conditions which must be satisfied in order to allow the existence of an independent second invariant polynomial
in the momenta. We then investigate the linear case for which a complete solution of the problem can be obtained. We recover
the classical set of linear strongly integrable systems and provide several new examples of weakly integrable systems whose
equations of motion can be explicitly solved at a fixed value of the energy. 相似文献
9.
M. A. Vashkov’yak 《Solar System Research》2017,51(4):315-326
The well-known twice-averaged Hill problem is considered by taking into account the oblateness of the central body. This problem has several integrable cases that have been studied qualitatively by many scientists, beginning with M.L. Lidov and Y. Kozai. However, no rigorous analytical solution can be obtained in these cases due to the complexity of the integrals. This paper is devoted to studying the case where the equatorial plane of the central body coincides with the plane of its orbital motion relative to the perturbing body, while the satellite itself moves in a polar orbit. A more detailed qualitative study is performed, and an approximate constructive-analytical solution of the evolution system in the form of explicit time dependences of the eccentricity and pericenter argument of the satellite orbit is proposed. The methodical accuracy for the polar orbits of lunar satellites has been estimated by comparison with the numerical solution of the system. 相似文献
10.
John A. Adam 《Astrophysics and Space Science》1986,127(2):309-320
This paper is part II of a limited review of the applications of the spectral theory of linear operators in an astrophysical context. A major part of the paper is devoted to describing the results obtained by Dyson and Schutz (1979) for differentially rotating perfect fluid stars. The functional-analytic techniques used and the results so obtained are compared with those in Paper I. As in the case of ideal magnetohydrodynamics, the mathematical structure of the rotating star problem is very rich indeed. Many questions remain unanswered in both areas. 相似文献
11.
Giancarlo Benettin Luigi Galgani Antonio Giorgilli 《Celestial Mechanics and Dynamical Astronomy》1985,37(1):1-25
In the present paper we give a proof of Nekhoroshev's theorem, which is concerned with an exponential estimate for the stability times in nearly integrable Hamiltonian systems. At variance with the already published proof, which refers to the case of an unperturbed Hamiltonian having the generic property of steepness, we consider here the particular case of a convex unperturbed Hamiltonian. The corresponding simplification in the proof might be convenient for an introduction to the subject. 相似文献
12.
D. Bancelin D. Hestroffer W. Thuillot 《Celestial Mechanics and Dynamical Astronomy》2012,112(2):221-234
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. 相似文献
13.
Emmanuel Davoust 《Celestial Mechanics and Dynamical Astronomy》1983,31(3):303-315
We use the analytical method of Lindstedt to make an inventory of the families of periodic orbits in a two-dimensional galactic potential first introduced by Contopoulos (1960). We examine the general case of orbital resonance and its neighborhood; two special cases, the 1∶1 and 2∶1 resonances are dealt with separately. The present paper provides a synthesis and an extension of earlier works on this potential in the neighborhood of the integrable case (ε?1). 相似文献
14.
Angelo B. Mingarelli Chiara M. F. Mingarelli 《Celestial Mechanics and Dynamical Astronomy》2005,91(3-4):391-401
In an effort to understand the nature of almost periodic orbits in the n-body problem (for all time t) we look first to the more basic question of the oscillatory nature of solutions of this problem (on a half-line, usually
taken as R
+). Intimately related to this is the notion of a conjugate point(due to A. Wintner) of a solution. Specifically, by rewriting the mass unrestricted general problem of n-bodies in a symmetric form we prove that in the gravitational Newtonian n-body problem with collisionless motions there exists arbitrarily large conjugate points in the case of arbitrary (positive)
masses whenever the cube of the reciprocal of at least one of the mutual distances is not integrable at infinity. The implication
of this result is that there are possibly many Wintner oscillatorysolutions in these cases (some of which may or may not be almost periodic). As a consequence, we obtain sufficient conditions
for all continuable solutions (to infinity) to be either unbounded or to allow for near misses (at infinity). The results
also apply to potentials other than Newtonian ones. Our techniques are drawn from results in systems oscillation theory and
are applicable to more general situations.
Dedicated to the memory of Robert M. (Bob) Kauffman, formerly Professor of the
University of Alabama in Birmingham 相似文献
15.
Jürgen Pöschel 《Celestial Mechanics and Dynamical Astronomy》1982,28(1-2):133-139
Differentiable Hamiltonian systems close to nondegenerate, integrable Hamiltonian systems are shown to be integrable on a Cantor set in the sense that on some Cantor set, (i) the invariant KAM-tori form a smooth foliation, (ii) there exist smooth, independent integrals in involution, and (iii) there exists a complete solution of the Hamilton Jacobi equation. The complement of the Cantor set is shown to be small in measure.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics. 相似文献
16.
Giuseppe Pucacco Dino Boccaletti Cinzia Belmonte 《Celestial Mechanics and Dynamical Astronomy》2008,102(1-3):163-176
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. 相似文献
17.
H. M. Yehia 《Celestial Mechanics and Dynamical Astronomy》1987,41(1-4):275-288
The general problem of motion of a rigid body about a fixed point under the action of stationary non-symmetric potential and gyroscopic forces is considered. The equations of motion in the Euler-Poisson form are derived. An interpretation is given in terms of charged, magnetized gyrostat moving in a superposition of three classical fields. As an example, the problem of motion of a satellite — gyrostat on a circular orbit with respect to its orbital system is reduced to that of its motion in an inertial system under additional magnetic and Lorentz forces.When the body is completely symmetric about one of its axes passing through the fixed point, the above problem is found to be equivalent to another one, in which the body has three equal moments of inertia and the forces are symmetric around a space axis. The last problem is well-studied and the given analogy reveals a number of integrable cases of the original problem. A transformation is found, which gives from each of these cases a class of integrable cases depending on an arbitrary function. The equations of motion are also reduced to a single equation of the second order. 相似文献
18.
M. A. Vashkov’yak 《Solar System Research》2018,52(4):359-370
The well-known problem of motion in a central field integrable in quadratures is considered. The force function of the problem depends only on the particle distance to the chosen coordinate origin. In the general case of an arbitrary central force, a rigorous analytical solution of the problem cannot be obtained due to the complexity of the integrals. In this paper we propose a semi-analytical method of constructing an approximate solution for the case where the distance varies in a limited range that allows the time dependences of the polar coordinates to be obtained using elliptic functions and integrals. As an example, we consider the model problems of the perturbed motion of hypothetical Jovian and lunar equatorial satellites as well as the problem of the motion of a single star in the principal plane of a galaxy. The methodical accuracy has been estimated by a comparison with the numerical solution. 相似文献
19.
We describe analysis methods to estimate parameters of electromagnetic waves based on the multi-component measurements of the DEMETER spacecraft. Using the fact that the wave magnetic field is perpendicular to the wave vector, the wave normal direction can be estimated by different methods. We use these plane-wave estimates to interpret measurements of the observed wave emissions. For instance, we use the recently developed singular value decomposition (SVD) technique. The results of the plane-wave analysis have an advantage that they often allow a straightforward interpretation. These different methods have been successfully tested with the data of previous spacecraft. All these methods are also implemented in the analysis tools designed for the analysis of the DEMETER wave measurements.We show the first results of these analysis techniques for different types of wave emissions observed on board DEMETER. Obliquely propagating right-hand polarized electromagnetic waves at a few hundreds of Hz are usually connected with a multi-ion mode structure below the local proton cyclotron frequency and with a sharp lower cutoff of left-hand polarized waves, as well as with right-hand polarized waves tunelling below the multi-ion cross-over frequency. Electron and proton whistlers are also very frequently observed on DEMETER. An unusual narrow-band emission at 140 Hz (well below the local proton cyclotron frequency) serves us as another case for a detailed analysis. We find that these waves are right-hand polarized and obliquely propagating.Using this example case, we also present analysis methods to estimate continuous distribution of wave energy density as a function of wave vector directions. These techniques of wave distribution function (WDF) analysis need both wave and particle measurements. In the analyzed case, two different methods of WDF analysis give similar results consistent with the results of the plane-wave techniques. To identify the source region we use the backward ray-tracing method. The wave normal direction obtained by the analysis of multi-component data is used for a simulation of wave propagation from the point of measurement. By this procedure, we obtain an inverse trajectory of the wave ray. We can thus follow the ray path back to the anticipated source region which is in our case located a few degrees of latitude to the South from the spacecraft position. 相似文献
20.
The previously derived equation (Agekyan 1974) for the derivative ?f/?n of the field of directions along the normal to a trajectory is approximate, because differentiating along the normal takes the point out of the orbit and changes the third integral of motion. However, on the envelope of the trajectory, i.e., on the contour of an orbit or a fold, ?f/?n undergoes a discontinuity of the second kind. Many authors have used this property to find points of the contours of orbits and folds. Although the integrable equation is approximate, the envelope points are determined accurately. 相似文献