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1.
Xinhao Liao 《Celestial Mechanics and Dynamical Astronomy》1996,66(3):243-253
In this paper, following the idea of constructing the mixed symplectic integrator (MSI) for a separable Hamiltonian system, we give a low order mixed symplectic integrator for an inseparable, but nearly integrable, Hamiltonian system, Although the difference schemes of the integrators are implicit, they not only have a small truncation error but, due to near integrability, also a faster convergence rate of iterative solution than ordinary implicit integrators, Moreover, these second order integrators are time-reversible. 相似文献
2.
The geometric approach to mechanics based on the Jacobi metric allows to easily construct natural mechanical systems which are integrable (actually separable) at a fixed value of the energy. The aim of the present paper is to investigate the dynamics of a simple prototype system outside the zero-energy hypersurface. We find that the general situation is that in which integrability is not preserved at arbitrary values of the energy. The structure of the Hamiltonian in the separating coordinates at zero energy allows a perturbation treatment of this system at energies slightly different from zero, by which we obtain an analytical proof of non-integrability.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
3.
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
4.
Alessandro Fortunati Stephen Wiggins 《Celestial Mechanics and Dynamical Astronomy》2016,125(2):247-262
The paper deals with the problem of the existence of a normal form for a nearly-integrable real-analytic Hamiltonian with aperiodically time-dependent perturbation decaying (slowly) in time. In particular, in the case of an isochronous integrable part, the system can be cast in an exact normal form, regardless of the properties of the frequency vector. The general case is treated by a suitable adaptation of the finite order normalization techniques usually used for Nekhoroshev arguments. The key point is that the so called “geometric part” is not necessary in this case. As a consequence, no hypotheses on the integrable part are required, apart from analyticity. The work, based on two different perturbative approaches developed by Giorgilli et al., is a generalisation of the techniques used by the same authors to treat more specific aperiodically time-dependent problems. 相似文献
5.
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
6.
Since the discovery of companions to B1257+12, it has been known that planets can exist around pulsars. Such planets may be formed in discs analogous to those around young stars, so we have searched for dust grain emission towards a sample of nine nearby millisecond pulsars. No emission is detected down to typical 2 σ limits of 5 mJy, at a wavelength of 850 μm. Using a model in which grains are heated by the pulsar spin-down luminosity, these dust flux limits correspond to disc masses of typically 10 Earth masses. The low dust limits show that nearby pulsar planets must already exist, rather than be in the process of forming, but only B1257+12 is known to have such planets. Planetary systems appear to occur around only a few per cent of pulsars and main-sequence stars, and are thus a rare phenomenon irrespective of circumstellar environment. 相似文献
7.
Despina Voyatzi Simos Ichtiaroglou 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):331-342
We study the problem of the motion of a unit mass on the unit sphere
and examine the relation between integrability and certain monoparametric families of orbits. In particular we show that
if the potential is compatible with a family of meridians, it is integrable with an integral linear in the velocities, while
a family of parallels guarantees integrability with an integral quadratic in the velocities. 相似文献
8.
Francesco Fassò 《Celestial Mechanics and Dynamical Astronomy》1995,62(1):43-69
This paper deals with Hamiltonian perturbation theory for systems which, like Euler-Poinsot (the rigid body with a fixed point and no torques), are degenerate and do not possess a global system of action-angle coordinates. It turns out that the usual methods of perturbation theory, which are essentially local being based on the construction of normal forms within the domain of a local coordinate system, are not immediately usable to study perturbations of these systems, since degeneracy makes impossible to control that the system does not fall into a singularity of the coordinates. To overcome this difficulty, we develop a global formulation of Hamiltonian perturbation theory, in which the normal forms are globally defined on the phase space manifold. The key for this study lies in the geometry of the fibration by the invariant tori of an integrable degenerate Hamiltonian system, which is described by some generalizations of the Liouville-Arnol'd theorem and is reviewed in the paper. As an application, we provide a global formulation of Nekhoroshev's theorem on the stability for exponentially long times. 相似文献
9.
S. Ferraz-Mello J. C. Klafke T. A. Michtchenko D. Nesvorný 《Celestial Mechanics and Dynamical Astronomy》1996,64(1-2):93-105
The utilization of chaotic dynamics approaches allowed the identification of many modes of motion in resonant asteroidal dynamics. As these dynamical systems are not integrable, the motion modes are not separated and one orbit may transit from one mode to another. In some cases, as in the \31 resonance, these transitions may lead, in a relatively short time scale, to eccentricities so high that the asteroid may approach the Sun and be destroyed. In the \21 and \32 resonances these transitions are much slower and only indirect estimations of the time which is needed for a generic asteroid to leave the resonance are possible. It may reach hundreds of million years in the more robust regions of the \21 resonance and a time of the order of billions of years in those of the \32 resonance. These values are consistent with the observed depletion of the \21 resonance (only a few asteroids known while almost 60 asteroids are known in the \32 resonance). 相似文献
10.
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
11.
We consider numerical integration of nearly integrable Hamiltonian systems. The emphasis is on perturbed Keplerian motion, such as certain cases of the problem of two fixed centres and the restricted three-body problem. We show that the presently known methods have useful generalizations which are explicit and have a variable physical timestep which adjusts to both the central and perturbing potentials. These methods make it possible to compute accurately fairly close encounters. In some cases we suggest the use of composite (instead of symplectic) alternatives which typically seem to have equally good energy conservation properties.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
12.
Shannon L. Coffey André Deprit Etienne Deprit 《Celestial Mechanics and Dynamical Astronomy》1994,59(1):37-72
We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ
2 throughJ
9. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J
2 only). In particular, we examine the manner in which the odd zonalJ
3 breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J
4+J
2
2
=0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions. 相似文献
13.
M. Dominik † 《Monthly notices of the Royal Astronomical Society》2009,393(3):816-821
Some of the difficulties in determining the underlying physical properties that are relevant for observed anomalies in microlensing light curves, such as the mass and separation of extrasolar planets orbiting the lens star, or the relative source–lens parallax, are already anchored in factors that limit the amount of information available from ordinary microlensing events and in the way these are being parametrized. Moreover, a real-time detection of deviations from an ordinary light curve while these are still in progress can only be done against a known model of the latter, and such is also required for properly prioritizing ongoing events for monitoring in order to maximize scientific returns. Despite the fact that ordinary microlensing light curves are described by an analytic function that only involves a handful of parameters, modelling these is far less trivial than one might be tempted to think. A well-known degeneracy for small impacts, and another one for the initial rise of an event, makes an interprediction of different phases impossible, while in order to determine a complete set of model parameters, the fundamental characteristics of all these phases need to be properly assessed. While it is found that the wing of the light curve provides valuable information about the time-scale that absorbs the physical properties, the peak flux of the event can be meaningfully predicted only after about a third of the total magnification has been reached. Parametrizations based on observable features not only ease modelling by bringing the covariance matrix close to diagonal form, but also allow good predictions of the measured flux without the need to determine all parameters accurately. Campaigns intending to infer planet populations from observed microlensing events need to invest some fraction of the available time into acquiring data that allow to properly determine the magnification function. 相似文献
14.
Antonio Giorgilli Ugo Locatelli Marco Sansottera 《Celestial Mechanics and Dynamical Astronomy》2014,119(3-4):397-424
We give a constructive proof of the existence of elliptic lower dimensional tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov normalization algorithm to the case of planetary systems, for which elliptic tori may be used as replacements of elliptic Keplerian orbits in Lagrange-Laplace theory. With this paper we support with rigorous convergence estimates the semi-analytic work in our previous article (Sansottera et al., Celest Mech Dyn Astron 111:337–361, 2011), where an explicit calculation of an invariant torus for a planar model of the Sun-Jupiter-Saturn-Uranus system has been made. With respect to previous works on the same subject we exploit the characteristic of Lie series giving a precise control of all terms generated by our algorithm. This allows us to slightly relax the non-resonance conditions on the frequencies. 相似文献
15.
Thierry Combot 《Celestial Mechanics and Dynamical Astronomy》2012,114(4):319-340
We prove an integrability criterion and a partial integrability criterion for homogeneous potentials of degree ?1 which are invariant by rotation. We then apply it to the proof of the meromorphic non-integrability of the n-body problem with Newtonian interaction in the plane on a surface of equation (H, C) = (H 0, C 0) with (H 0, C 0) ?? (0, 0) where C is the total angular momentum and H the Hamiltonian, in the case where the n masses are equal. Several other cases in the 3-body problem are also proved to be non integrable in the same way, and some examples displaying partial integrability are provided. 相似文献
16.
The development of an ultra high energy air shower has an intrinsic energy fluctuation due both to the first interaction point and to the cascade development. Here we show that for a given primary energy this fluctuation has a lognormal distribution and thus observations will estimate the primary energy with a lognormal error distribution. We analyze the UHECR energy spectrum convolved with the lognormal energy error and demonstrate that the shape of the error distribution will interfere significantly with the ability to observe features in the spectrum. If the standard deviation of the lognormal error distribution is equal or larger than 0.25, both the shape and the normalization of the measured energy spectra will be modified significantly. As a consequence, the GZK cutoff might be sufficiently smeared as not to be seen (without very high statistics). This result is independent of the power law of the cosmological flux. As a conclusion we show that in order to establish the presence or not of the GZK feature, not only more data are needed but also that the shape of the energy error distribution has to be known well. The high energy tail and the sigma of the approximate lognormal distribution of the error in estimating the energy must be at the minimum set by the physics of showers. 相似文献
17.
Timothy A. Davis Peter J. Wheatley 《Monthly notices of the Royal Astronomical Society》2009,396(2):1012-1017
We investigate the evaporation history of known transiting exoplanets in order to consider the origin of observed correlations between mass, surface gravity and orbital period. We show that the survival of the known planets at their current separations is consistent with a simple model of evaporation, but that many of the same planets would not have survived closer to their host stars. These putative closer-in systems represent a lost population that could account for the observed correlations. We conclude that the relation underlying the correlations noted by Mazeh et al. and Southworth et al. is most likely a linear cut-off in the M 2 / R 3 versus a −2 plane, and we show that the distribution of exoplanets in this plane is in close agreement with the evaporation model. 相似文献
18.
《Astroparticle Physics》2007,26(6):375-379
The development of an ultra high energy air shower has an intrinsic energy fluctuation due both to the first interaction point and to the cascade development. Here we show that for a given primary energy this fluctuation has a lognormal distribution and thus observations will estimate the primary energy with a lognormal error distribution. We analyze the UHECR energy spectrum convolved with the lognormal energy error and demonstrate that the shape of the error distribution will interfere significantly with the ability to observe features in the spectrum. If the standard deviation of the lognormal error distribution is equal or larger than 0.25, both the shape and the normalization of the measured energy spectra will be modified significantly. As a consequence, the GZK cutoff might be sufficiently smeared as not to be seen (without very high statistics). This result is independent of the power law of the cosmological flux. As a conclusion we show that in order to establish the presence or not of the GZK feature, not only more data are needed but also that the shape of the energy error distribution has to be known well. The high energy tail and the sigma of the approximate lognormal distribution of the error in estimating the energy must be at the minimum set by the physics of showers. 相似文献
19.
Efi Meletlidou Simos Ichtiaroglou Fabian Josef Winterberg 《Celestial Mechanics and Dynamical Astronomy》2001,80(2):145-156
We prove that Hill's lunar problem does not possess a second analytic integral of motion, independent of the Hamiltonian. In order to obtain this result, we avoid the usual normalization in which the angular velocity of the rotating reference frame is put equal to unit. We construct an artificial Hamiltonian that includes an arbitrary parameter b and show that this Hamiltonian does not possess an analytic integral of motion for in an open interval around zero. Then, by selecting suitable values of , b and using the invariance of the Hamiltonian under scaling in the units of length and time, we show that the Hamiltonian of Hill's problem does not possess an integral of motion, analytically continued from the integrable two–body problem in a rotating frame. 相似文献
20.
Jing-Kun Zhao Gang Zhao Yu-Qin Chen Jian-Rong Shi Yu-Juan Liu Ju-Yong Zhang National Astronomical Observatories Chinese Academy of Sciences Beijing 《中国天文和天体物理学报》2006,6(6):689-696
It is well known that normalization, radial velocity correction and equivalent-width measurement of high-resolution stellar spectra are time-consuming work. In order to improve the efficiency we present an automatic method for these routines. The continuum is determined by fitting the 'high points' in the spectrum. After continuum normalization, the program automatically searches for the position of the Ha line and obtains a rough radial velocity, then computes an accurate radial velocity by cross-correlation between the given spectrum and the solar spectrum. In this method, the equivalent-width is automatically measured using Gaussian fitting. A comparison between our results and those from traditional analysis shows that the typical error for equivalent width is around 3.8% in our method. Developing such automatic routines does not mean to replace the interactive reduction method: it is just for a quick extraction of information from the spectra, especially those obtained in large sky surveys. 相似文献