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1.
Normalization of a perturbed elliptic oscillator, when executed in Lissajous variables, amounts to averaging over the elliptic anomaly. The reduced Lissajous variables constitute a system of cylindrical coordinates over the orbital spheres of constant energy, but the pole-like singularities are removed by reverting to the subjacent Hopf coordinates. The two-parameter coupling that is a polynomial of degree four admitting the symmetries of the square is studied in detail. It is shown that the normalized elliptic oscillator in that case behaves everywhere in the parameter plane like a rigid body in free rotation about a fixed point, and that it passes through butterfly bifurcations wherever its phase flow admits non isolated equilibria.  相似文献   

2.
Action-angle variables for the Levi-Civita regularized planar Kepler problem were introduced independently first by Chenciner and then by Deprit and Williams. The latter used explicitly the so-called Lissajous variables. When applied to the transformed Keplerian Hamiltonian, the Lissajous transformation encounters the difficulty of being defined in terms of the constant frequency parameter, whereas the Kepler problem transformed into a harmonic oscillator involves the frequency as a function of an energy-related canonical variable. A simple canonical transformation is proposed as a remedy for this inconvenience. The problem is circumvented by adding to the physical time a correcting term, which occurs to be a generalized Kepler’s equation. Unlike previous versions, the transformation is symplectic in the extended phase space and allows the treatment of time-dependent perturbations. The relation of the extended Lissajous–Levi-Civita variables to the classical Delaunay angles and actions is given, and it turns out to be a straightforward generalization of the results published by Deprit and Williams.  相似文献   

3.
A new canonical transformation is proposed to handle elliptic oscillators, that is, Hamiltonian systems made of two harmonic oscillators in a 1-1 resonance. Lissajous elements pertain to the ellipse drawn with a light pen whose coordinates oscillate at the same frequency, hence their name. They consist of two pairs of angle-action variables of which the actions and one angle refer to basic integrals admitted by an elliptic oscillator, namely, its energy, its angular momentum and its Runge-Lenz vector. The Lissajous transformation is defined in two ways: explicitly in terms of Cartesian variables, and implicitly by resolution of a partial differential equation separable in polar variables. Relations between the Lissajous variables, the common harmonic variables, and other sets of variables are discussed in detail.  相似文献   

4.
A coordinate system is defined on the phase space of a perturbed Keplerian system after the mean anomaly has been averaged out, for the purpose of explaining how eliminating the longitude of the ascending node reduces the orbital space to a two-dimensional sphere in case the system admits an axial symmetry. Concomitantly, on the submanifold of direct osculating ellipses, the CDM variables are replaced by functions which form the basis of a Poisson algebra isomorphic to the Lie algebra so(3) of the rotation group SO(3); furthermore, in these variables, the doubly reduced phase flow appears like a rotation of the reduced phase space.  相似文献   

5.
An effective Microcanonical Thermodynamics of self gravitating systems(SGS) is proposed, analyzing the well known obstacles thought to prevent the formulation of a rigorous Statistical Mechanics (SM), as those due to the formal unboundedness of available phase space and to the unscreened, long range, nature of the interaction. The latter feature entails the well known inequivalence of statistical ensembles, puts clearly into question the meaning, for these systems, of the Thermodynamic Limit, and rules out the use of canonical and grand-canonical ensembles. As to the first obstacle, we argue nevertheless that a hierarchy of timescales exist such that, at any finite time, the volume of the effectively available region of phase space is indeed finite, and that the dynamics satisfies a strong chaos criterion, leading to a fast, increasingly uniform, spreading of orbits over an effectively invariant subset of the constant (N,V,E) surface; thus leading to the definition of a secularly evolving, generalized microcanonical ensemble, which allows to define an (almost extensive) effective entropy and to derive self-consistent definitions for other thermodynamic variables, giving thus an orthode for SGS. Moreover, a Second Law-like criterion allows to single out the hierarchy of secular equilibria describing, for any finite time, the macroscopic behaviour of SGS. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

6.
The present paper addresses the existence of J 2 invariant relative orbits with arbitrary relative magnitude over the infinite time using the Routh reduction and Poincaré techniques in the J 2 Hamiltonian problem. The current research also proposes a novel numerical searching approach for J 2 invariant relative orbits from the dynamical system point of view. A new type of Poincaré mapping is defined from different central manifolds of the pseudo-circular orbits (parameterized by the Jacobi energy E, the polar component of momentum H z and the measure of distance Δr between the fixed point and its central manifolds) to the nodal periods T d and the drifts of longitude of the ascending node during one period (ΔΩ), which differs from Koon et al.’s (AIAA 2001) definition on central manifolds parameterized by the same fixed point. The Poincaré mapping is surjective because it compresses the three-dimensional variables into two-dimensional images, and the mapping degenerates into a bijective mapping in consideration of the fixed points. An iteration algorithm to the degenerated bijective mapping is proposed from the continuation procedure to perform the ergodic representation of E- and H z -contour maps on the space of T d –ΔΩ. For the surjective mapping with Δr ≠ 0, different pseudo-circular or elliptical orbits may share the same images. Hence, the inverse surjective mapping may achieve non-unique variables from a single image, which makes the generation of J 2 invariant relative orbits possible. The pseudo-circular or elliptical orbits generated from the surjective mapping will be defined in different meridian planes. Hence, the critical contribution of the present paper is the assignment of J 2 invariant relative orbits to different invariant parameters E and H z depending on the E- and H z -contour map, which will hold J 2 invariant relative orbits for extended durations. To investigate the high-order nonlinearity neglected by previous studies, a formation configuration with a large magnitude of 500 km is successfully generated from the theory developed in the present work, which is beyond the scope of the linear conditions of J 2 invariant relative orbits. Therefore, the existence of J 2 invariant relative orbit with an arbitrary relative magnitude over the infinite time is achieved from the dynamical system point of view.  相似文献   

7.
The phase space of a light quantum in a given volume is subdivided into “cells” of magnitudeh 3. The number of possible distributions of the light quanta of a macroscopically defined radiation over these cells gives the entropy and with it all thermodynamic properties of the radiation.  相似文献   

8.
It has recently been shown that Jupiter Trojans may exhibit chaotic behavior, a fact that has put in question their presumed long term stability. Previous numerical results suggest a slow dispersion of the Trojan swarms, but the extent of the ‘effective’ stability region in orbital elements space is still an open problem. In this paper, we tackle this problem by means of extensive numerical integrations. First, a set of 3,200 fictitious objects and 667 numbered Trojans is integrated for 4 Myrs and their Lyapunov time, TL, is estimated. The ones following chaotic orbits are then integrated for 1 Gyr, or until they escape from the Trojan region. The results of these experiments are presented in the form of maps of TLand the escape time, TE, in the space of proper elements. An effective stability region for 1 Gyr is defined on these maps, in which chaotic orbits also exist. The distribution of the numbered Trojans follows closely the TE=1 Gyr level curve, with 86% of the bodies lying inside and 14% outside the stability region. This result is confirmed by a 4.5 Gyr integration of the 246 chaotic numbered Trojans, which showed that 17% of the numbered Trojans are unstable over the age of the solar system. We show that the size distributions of the stable and unstable populations are nearly identical. Thus, the existence of unstable bodies should not be the result of a size-dependent transport mechanism but, rather, the result of chaotic diffusion. Finally, in the large chaotic region that surrounds the stability zone, a statistical correlation between TLandTE is found.  相似文献   

9.
Some analytical relations for the phase space functions of a self-consistent spherical stellar system are derived. The integral constraints on the distribution function by imposing a given (r) density distribution andN(E) fractional energy distribution are determined. For the case of radially-anisotropic velocity distribution in theE0 limit the constraint by an exponentialN(E) implies thatf(E, J 2) tends to zero in the order (–E)3/2. This lends analytical support to the use of the Stiavelli and Bertin (1985) distribution function for modeling elliptical galaxies. Maximum phase space density constraint confirms the necessity of high collapse factors to produce such a distribution function. Limits on the steepness of an exponentialN(E) for the case when (r) resembles the emissivity law of ellipticals are also derived.  相似文献   

10.
Relations between integrable systems in plane and curved spaces   总被引：1，自引：0，他引：1  
We consider trajectory isomorphisms between various integrable systems on an n-dimensional sphere S n and a Euclidean space . Some of the systems are classical integrable problems of Celestial Mechanics in plane and curved spaces. All the systems under consideration have an additional first integral quadratic in momentum and can be integrated analytically by using the separation of variables. We show that some integrable problems in constant curvature spaces are not essentially new from the viewpoint of the theory of integration, and they can be analyzed using known results of classical Celestial Mechanics.  相似文献   

11.
We present a global view of the resonant structure of the phase space of a planetary system with two planets, moving in the same plane, as obtained from the set of the families of periodic orbits. An important tool to understand the topology of the phase space is to determine the position and the stability character of the families of periodic orbits. The region of the phase space close to a stable periodic orbit corresponds to stable, quasi periodic librations. In these regions it is possible for an extrasolar planetary system to exist, or to be trapped following a migration process due to dissipative forces. The mean motion resonances are associated with periodic orbits in a rotating frame, which means that the relative configuration is repeated in space. We start the study with the family of symmetric periodic orbits with nearly circular orbits of the two planets. Along this family the ratio of the periods of the two planets varies, and passes through rational values, which correspond to resonances. At these resonant points we have bifurcations of families of resonant elliptic periodic orbits. There are three topologically different resonances: (1) the resonances (n + 1):n, (2:1, 3:2, ...), (2) the resonances (2n + 1):(2n-1), (3:1, 5:3, ...) and (3) all other resonances. The topology at each one of the above three types of resonances is studied, for different values of the sum and of the ratio of the planetary masses. Both symmetric and asymmetric resonant elliptic periodic orbits exist. In general, the symmetric elliptic families bifurcate from the circular family, and the asymmetric elliptic families bifurcate from the symmetric elliptic families. The results are compared with the position of some observed extrasolar planetary systems. In some cases (e.g., Gliese 876) the observed system lies, with a very good accuracy, on the stable part of a family of resonant periodic orbits.  相似文献   

12.
In a microwave background polarization map that covers only part of the sky, it is impossible to separate the E and B components perfectly. This difficulty in general makes it more difficult to detect the B component in a data set. Any polarization map can be separated in a unique way into “pure E”, “pure B” and “ambiguous” components. Power that resides in the pure E(B) component is guaranteed to be produced by E(B) modes, but there is no way to tell whether the ambiguous component comes from E or B modes. A polarization map can be separated into the three components either by finding an orthonormal basis for each component, or directly in real space by using Green functions or other methods.  相似文献   

13.
It is usually believed that we know everything to be known for any separable Hamiltonian system, i.e. an integrable system in which we can separate the variables in some coordinate system (e.g. see Lichtenberg and Lieberman 1992, Regular and Chaotic Dynamics, Springer). However this is not always true, since through the separation the solutions may be found only up to quadratures, a form that might not be particularly useful. A good example is the two-fixed-centers problem. Although its integrability was discovered by Euler in the 18th century, the problem was far from being considered as completely understood. This apparent contradiction stems from the fact that the solutions of the equations of motion in the confocal ellipsoidal coordinates, in which the variables separate, are written in terms of elliptic integrals, so that their properties are not obvious at first sight. In this paper we classify the trajectories according to an exhaustive scheme, comprising both periodic and quasi-periodic ones. We identify the collision orbits (both direct and asymptotic) and find that collision orbits are of complete measure in a 3-D submanifold of the phase space while asymptotically collision orbits are of complete measure in the 4-D phase space. We use a transformation, which regularizes the close approaches and, therefore, enables the numerical integration of collision trajectories (both direct and asymptotic). Finally we give the ratio of oscillation period along the two axes (the ‘rotation number’) as a function of the two integrals of motion. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

14.
It was discovered some years ago by Schiff that the equations divE = 4πQ and curlB - (1/c) ∂E/∂t = (4π/c)J for fields in vacuum do not carry over without change from an inertial frame to a frame with rotating axes of space coordinates, even for a region with all velocities of orderv≪c. However, the belief that all four of the field equations are invariant under such conditions is still prevalent and causes misconceptions in physical applications, including astrophysical and geophysical ones. The purpose of the present paper is therefore to call attention to Schiff's discovery, discussing its basis and its extension to fields in material media, and to interpret the additional terms that must be added to the equations in order to obtain valid transformations to rotating axes of coordinates.  相似文献   

15.
This work proposes a Lunar Global Positioning System (LGPS) and a Lunar Global Communication System (LGCS) using two constellations of satellites on Lissajous trajectories around the collinear L 1 and L 2 libration points in the Earth–Moon system. This solution is compared against a Walker constellation around the Moon similar to the one used for the Global Positioning System (GPS) on the Earth to evaluate the main differences between the two cases and the advantages of adopting the Lissajous constellations. The problem is first studied using the Circular Restricted Three Body Problem to find out its main features. The study is then repeated with higher fidelity using a four-body model and higher-order reference trajectories to simulate the Earth-Moon-spacecraft dynamics more accurately. The LGPS performance is evaluated for both on-ground and in-flight users, and a visibility study for the LGCS is used to check that communication between opposite sides of the Moon is possible. The total ΔV required for the transfer trajectories from the Earth to the constellations and the trajectory control is calculated. Finally, the estimated propellant consumption and the total number of satellites for the Walker constellation and the Lissajous constellations is used as a performance index to compare the two proposed solutions.  相似文献   

16.
In this paper, besides general definition of hardness ratio of gamma-ray bursts (GRBs), HR 32, we also presented new definitions of hardness ratios, HR hl and HR ll, then presented the results of correlation studies, examining the association between the hardness ratios and the spectral fitting parameters (E 0 or E peak, α, β) by using the GRB data observed by BATSE. The HR hl is defined as the fluence of channel 4 divided by the sum of the fluences of channels 1 to 3,and the HR ll is defined as the fluence of channel 3 divided by the sum of the fluences of channel 1 and 2. We found that E 0 and E peak are correlated with hardness ratios, and β is only correlated with log (HR hl). The α is not correlated with any hardness ratios in the total sample of GRBs, while it correlated with log (HR ll) and log (HR 32) in the two subsets seperated at log (HR ll) = 0.34 ×α +0.66, respectively. These results show that a harder spectra tends to be steeper and has a higher E peak or E 0; HR hl describes the spectral behaviors in high energy, while both of HR ll and HR 32 reflect the spectral characteristics in low energy; the spectral behaviors in low energy are different for the two subsets. We also presented a brief qualitative analysis discussion to the correlations and suggested that the correlations are caused by both of intrinsic and cosmological effects, but intrinsic effects are dominant. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

17.
Employing a sample presented by Kaneko et al. (2006) [Kaneko, Y. et al., 2006. ApJS 166, 298 (Paper I)] and Kocevski et al. (2003) [Kocevski, D. et al., 2003. ApJ 596, 389], we select 42 individual tracking pulses (here we defined tracking as the cases in which the hardness follows the same pattern as the flux or count rate time profile) within 36 gamma-ray bursts (GRBs) containing 527 time-resolved spectra and investigate the spectral hardness, Epeak (where Epeak is the maximum of the νFν spectrum), evolutionary characteristics. The evolution of these pulses follow soft-to-hard-to-soft (the phase of soft-to-hard and hard-to-soft are denoted by rise phase and decay phase, respectively) with time. It is found that the overall characteristics of Epeak of our selected sample are: (1) the Epeak evolution in the rise phase always start on the high state (the values of Epeak are always higher than 50 keV); (2) the spectra of rise phase clearly start at higher energy (the median of Epeak are about 300 keV), whereas the spectra of decay phase end at much lower energy (the median of Epeak are about 200 keV); (3) the spectra of rise phase are harder than that of the decay phase and the duration of rise phase are much shorter than that of decay phase as well. In other words, for a complete pulse the initial Epeak is higher than the final Epeak and the duration of initial phase (rise phase) are much shorter than the final phase (decay phase). This results are in good agreement with the predictions of [Lu, R.J. et al., 2007. ApJ 663, 1110] and current popular view on the production of GRBs. We argue that the spectral evolution of tracking pulses may be relate to both of kinematic and dynamic process even if we currently can not provide further evidences to distinguish which one is dominant. Moreover, our statistical results give some witnesses to constrain the current GRB model.  相似文献   

18.
The existence of the universal quantization law E=n ε E =any energy; n = an integer, ε = the fundamental energy ∼ħ c/R with ħ = the reduced Planck constant, c = the speed of light, R = the curvature radius of the closed cosmological space) is advocated and discussed. A possible connection between ε and the mass of elementary particles is pointed out.  相似文献   

19.
A complete study is made of the resonant motion of two planets revolving around a star, in the model of the general planar three body problem. The resonant motion corresponds to periodic motion of the two planets, in a rotating frame, and the position and stability properties of the periodic orbits determine the topology of the phase space and consequently play an important role in the evolution of the system. Several families of symmetric periodic orbits are computed numerically, for the 2/1 resonance, and for the masses of some observed extrasolar planetary systems. In this way we obtain a global view of all the possible stable configurations of a system of two planets. These define the regions of the phase space where a resonant extrasolar system could be trapped, if it had followed in the past a migration process.The factors that affect the stability of a resonant system are studied. For the same resonance and the same planetary masses, a large value of the eccentricities may stabilize the system, even in the case where the two planetary orbits intersect. The phase of the two planets (position at perihelion or aphelion when the star and the two planets are aligned) plays an important role, and the change of the phase, other things being the same, may destabilize the system. Also, the ratio of the planetary masses, for the same total mass of the two planets, plays an important role and the system, at some resonances and some phases, is destabilized when this ratio changes.The above results are applied to the observed extrasolar planetary systems HD 82943, Gliese 876 and also to some preliminary results of HD 160691. It is shown that the observed configurations are close to stable periodic motion.  相似文献   

20.
Experimental results on the intensity, energy spectrum and time variations in hard X-ray emission from Cyg X-1 based on a balloon observation made on 1971, April 6 from Hyderabad (India) are described. The average energy spectrum of Cyg X-1 in the 22–154 keV interval on 1971 April 6 is best represented by a power law dN/dE=(5.41±1.53)E –(1.92±0.10) photons cm–2s–1 keV–1 which is in very good agreement with the spectrum of Cyg X-1 derived from an earlier observation made by us on 1969 April 16 in the 25–151 keV band and given by dN/dE=(3.54±2.44)E –(1.89±0.22) photons cm–2s–1 keV–1. A thermal bremsstrahlung spectrum fails to give a good fit over the entire energy range for both the observations. Comparison with the observations of other investigators shows that almost all balloon experiments consistently give a spectrum of E –2, while below 20 keV the spectrum varies fromE –1.7 toE –5. There is some indication of a break in the Cyg X-1 spectrum around 20 keV. Spectral analysis of data in different time intervals for the 1971 April 6 flight demonstrates that while the source intensity varies over time scales of a few minutes, there is no appreciable variation in the spectral slope. Analysis of various hard X-ray observations for long term variations shows that over a period of about a week the intensity of Cyg X-1 varies upto a factor of four. The binary model proposed by Dolan is examined and the difficulties in explaining the observed features of Cyg X-1 by this model are pointed out.  相似文献   

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