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1.
We have extend Stormer’s problem considering four magnetic dipoles in motion trying to justify the phenomena of extreme “orderlines”
such as the ones observed in the rings of Saturn; the aim is to account the strength of the Lorentz forces estimating that
the Lorentz field, co-acting with the gravity field of the planet, will limit the motion of all charged particles and small
size grains with surface charges inside a layer of about 200 m thickness as that which is observed in the rings of Saturn.
For this purpose our interest feast in the motion of charged particles with neglected mass where only electromagnetic forces
accounted in comparison to the weakness of the Newtonian fields. This study is particularly difficult because in the regions
we investigate these motions there is enormous three dimensional instability. Following the Poincare’s hypothesis that periodic
solutions are ‘dense’ in the set of all solutions in Hamiltonian systems we try to calculate many families of periodic solutions
and to study their stability. In this work we prove that in this environment charged particles can trace planar symmetric
periodic orbits. We discuss these orbits in details and we give their symplectic relations using the Hamiltonian formulation
which is related to the symplectic matrix. We apply numerical procedures to find families of these orbits and to study their
stability. Moreover we give the bifurcations of these families with families of planar asymmetric periodic orbits and families
of three dimensional symmetric periodic orbits. 相似文献
2.
For the n-centre problem of one particle moving in the potential of attracting centres of small mass fixed in an arbitrary smooth potential
and magnetic field, we prove the existence of periodic and chaotic trajectories shadowing sequences of collision orbits. In
particular, we obtain large subshifts of solutions of this type for the circular restricted 3-body problem of celestial mechanics.
Poincaré had conjectured existence of the periodic ones and given them the name ‘second species solutions’.
This revised version was published online in July 2006 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.
Edward Belbruno Jaume Llibre Mercè Ollé 《Celestial Mechanics and Dynamical Astronomy》1994,60(1):99-129
In this paper we deal with the circular Sitnikov problem as a subsystem of the three-dimensional circular restricted three-body problem. It has a first analytical part where by using elliptic functions we give the analytical expressions for the solutions of the circular Sitnikov problem and for the period function of its family of periodic orbits. We also analyze the qualitative and quantitative behavior of the period function. In the second numerical part, we study the linear stability of the family of periodic orbits of the Sitnikov problem, and of the families of periodic orbits of the three-dimensional circular restricted three-body problem which bifurcate from them; and we follow these bifurcated families until they end in families of periodic orbits of the planar circular restricted three-body problem. We compare our results with the previous ones of other authors on this problem. Finally, the characteristic curves of some bifurcated families obtained for the mass parameter close to 1/2 are also described. 相似文献
5.
Mercè Ollé Joan R. Pacha Jordi Villanueva 《Celestial Mechanics and Dynamical Astronomy》2004,90(1-2):87-107
The paper deals with different kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that
takes place close to the Lyapunov vertical family of periodic orbits of the triangular equilibrium point L4 in the 3D restricted three-body problem (RTBP) for the mass parameter, μ greater than (and close to) μR (Routh’s mass parameter). Consequences of such bifurcation, concerning the confinement of the motion close to the hyperbolic
orbits and the 3D nearby tori are also described. 相似文献
6.
Giacomo Tommei 《Celestial Mechanics and Dynamical Astronomy》2006,94(2):173-195
The purpose of this paper is to find a set of canonical elements to use within the framework of Öpik theory of close encounters of a small body with a planet (Öpik, Interplanetary Encounters, 1976). Since the small body travels along a planetocentric hyperbola during the close approach and Öpik formulas are valid, without approximations, only at collision, we derive a set of canonical elements for hyperbolic collision orbits (eccentricity e → 1+, semi-major axis a fixed) and then we introduce the unperturbed velocity of the small body and the distance covered along the asymptote as a new canonically conjugate pair of orbital elements. An interesting result would be to get a canonical set containing the coordinates in the Target Plane (TP), useful for the analysis of the future encounters: in the last part we prove that this is not possible. 相似文献
7.
P. S. Soulis K. E. Papadakis T. Bountis 《Celestial Mechanics and Dynamical Astronomy》2008,100(4):251-266
We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits
in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the
fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted
three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” ż0 varies within a finite interval (while z
0 tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve
corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D
branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these
orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the
z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion
near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of
ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity. 相似文献
8.
A dynamical model composed of a disk galaxy with an elliptic companion, moving in a circular orbit, is used in order to study
the stellar orbits in a binary galaxy. Using the Poincare surface of section we study the evolution of the stochastic regions
in the primary galaxy considering the mass of the companion or the value of the Jacobi’s integral as a parameter. Our numerical
calculations suggest that the regions of stochasticity increase, as the mass of the companion or the value of the Jacobi’s
integral increase. An interesting observation is that only direct orbits become stochastic. 相似文献
9.
For the circular restricted three-body problem of celestial mechanics with small secondary mass, we prove the existence of uniformly hyperbolic invariant sets of non-planar periodic and chaotic almost collision orbits. Poincaré conjectured existence of periodic ones and gave them the name “second species solutions”. We obtain large subshifts of finite type containing solutions of this type. 相似文献
10.
Sergey Bolotin 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):343-371
We consider the plane restricted elliptic 3 body problem with small mass ratio and small eccentricity and prove the existence
of many periodic orbits shadowing chains of collision orbits of the Kepler problem. Such periodic orbits were first studied
by Poincaré for the non-restricted 3 body problem. Poincaré called them second species solutions. 相似文献
11.
Michel Hénon 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):87-100
We present five families of periodic solutions of Hill’s problem which are asymmetric with respect to the horizontal ξ axis. In one of these families, the orbits are symmetric with respect to the vertical η axis; in the four others, the orbits are without any symmetry. Each family consists of two branches, which are mirror images
of each other with respect to the ξ axis. These two branches are joined at a maximum of Γ, where the family of asymmetric periodic solutions intersects a family
of symmetric (with respect to the ξ axis) periodic solutions. Both branches can be continued into second species families for Γ → − ∞. 相似文献
12.
Ernesto A. Lacomba Ernesto Pérez-Chavela 《Celestial Mechanics and Dynamical Astronomy》1993,57(3):411-437
In this work we study escape and capture orbits in the planar rhomboidal 4-body problem in a level of constant negative energy. There are only two different values of the masses here. By using numerical analysis, we show certain transversal intersections of the invariant manifolds of parabolic orbits. We then introduce Symbolic Dynamics when the mass ratio is small, and when it is close to one. In the first case the escapes or captures predominate in the direction of one of the diagonals of the rhombus, while in the second case we find solutions escaping or being captured in the direction of both possible diagonals. 相似文献
13.
G. Tancredi 《Celestial Mechanics and Dynamical Astronomy》1998,70(3):181-200
The dynamics of two families of minor inner solar system bodies that suffer frequent close encounters with the planets is
analyzed. These families are: Jupiter family comets (JF comets) and Near Earth Asteroids (NEAs).
The motion of these objects has been considered to be chaotic in a short time scale,and the close encounters are supposed
to be the cause of the fast chaos. For a better understanding of the chaotic behavior we have computed Lyapunov Characteristic
Exponents (LCEs) for all the observed members of both populations. LCEs are a quantitative measure of the exponential divergence
of initially close orbits. We have observed that most members of the two families show a concentration of Lyapunov times (inverse
of LCE) around 50–100yr. The concentration is more pronounced for JF comets than for NEAs, among which a lesser spread is
observed for those that actually cross the Earth's orbit (mean perihelion distance q < 1.05 AU). It is also observed that
a general correspondence exists between Lyapunov times and the time between consecutive encounters.
A simple model is introduced to describe the basic characteristics of the dynamical evolution. This model considers an impulsive
approach, where the particles evolve unperturbedly between encounters and suffer ‘kicks’ in semimajor axis at the encounters.
It also reproduces successfully the short Lyapunov times observed in the numerical integrations and is able to estimate the
dynamical lifetimes of comets during a stay in the Jupiter family in correspondence with previous estimates.
It has been demonstrated with the model that the encounters with the largest effect on the exponential growth of the distance
between initially nearby orbits are neither the infrequent deep encounters, nor the frequent and far ones; instead, the intermediate
approaches have the most relevant contribution to the error growth. Such encounters are at a distance a few times the radius
of the Hill's sphere of the planet (e.g. 3).
An even simpler model allows us to get analytical estimates of the Lyapunov times in good agreement with the values coming
from the model above and the numerical integrations.
The predictability of the medium‐term evolution and the hazard posed to the Earth by those objects are analysed in the Discussion
section.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
14.
Families of Periodic Orbits Emanating From Homoclinic Orbits in the Restricted Problem of Three Bodies 总被引:2,自引:1,他引:1
We describe and comment the results of a numerical exploration on the evolution of the families of periodic orbits associated
with homoclinic orbits emanating from the equilateral equilibria of the restricted three body problem for values of the mass
ratio larger than μ
1. This exploration is, in some sense, a continuation of the work reported in Henrard [Celes. Mech. Dyn. Astr. 2002, 83, 291]. Indeed it shows how, for values of μ. larger than μ
1, the Trojan web described there is transformed into families of periodic orbits associated with homoclinic orbits. Also we describe how families
of periodic orbits associated with homoclinic orbits can attach (or detach) themselves to (or from) the best known families
of symmetric periodic orbits.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
15.
The concept of closest approach is analyzed in Hill’s problem, resulting in a partitioning of the position space. The different
behavior between the direct and retrograde motion is explained analytically, resulting in a simple estimate of the variation
of Hill’s periodic and quasi-circular orbits as a function of the Jacobi constant. The local behavior of the orbits on the
zero velocity surfaces and an analytical definition of local escape and capture in Hill’s problem are also given. 相似文献
16.
G. A. Tsirogiannis E. A. Perdios V. V. Markellos 《Celestial Mechanics and Dynamical Astronomy》2009,103(1):49-78
We present an improved grid search method for the global computation of periodic orbits in model problems of Dynamics, and
the classification of these orbits into families. The method concerns symmetric periodic orbits in problems of two degrees
of freedom with a conserved quantity, and is applied here to problems of Celestial Mechanics. It consists of two main phases;
a global sampling technique in a two-dimensional space of initial conditions and a data processing procedure for the classification
(clustering) of the periodic orbits into families characterized by continuous evolution of the orbital parameters of member
orbits. The method is tested by using it to recompute known results. It is then applied with advantage to the determination
of the branch families of the family f of retrograde satellites in Hill’s Lunar problem, and to the determination of irregular families of periodic orbits in a
perturbed Hill problem, a species of families which are difficult to find by continuation methods.
相似文献
17.
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
18.
The effect of the eccentricity of a planet’s orbit on the stability of the orbits of its satellites is studied. The model
used is the elliptic Hill case of the planar restricted three-body problem. The linear stability of all the known families
of periodic orbits of the problem is computed. No stable orbits are found, the majority of them possessing one or two pairs
of real eigenvalues of the monodromy matrix, while a part of a family with complex instability is found. Two families of periodic
orbits, bifurcating from the Lagrangian points L1, L2 of the corresponding circular case are found analytically. These orbits are very unstable and the determination of their
stability coefficients is not accurate, so we compute the largest Liapunov exponent in their vicinity. In all cases these
exponents are positive, indicating the existence of chaotic motions 相似文献
19.
We introduce a three-dimensional version of Hill’s problem with oblate secondary, determine its equilibrium points and their
stability and explore numerically its network of families of simple periodic orbits in the plane, paying special attention
to the evolution of this network for increasing oblateness of the secondary. We obtain some interesting results that differentiate
this from the classical problem. Among these is the eventual disappearance of the basic family g′ of the classical Hill problem and the existence of out-of-plane equilibrium points and a family of simple-periodic plane
orbits non-symmetric with respect to the x-axis. 相似文献
20.
The second species periodic solutions of the restricted three body problem are investigated in the limiting case of μ=0. These orbits, called consecutive collision orbits by Hénon and generating orbits by Perko, form an infinite number of continuous one-parameter families and are the true limit, for μ→0, of second species periodic solutions for μ>0. By combining a periodicity condition with an analytic relation, for criticality, isolated members of several families are obtained which possess the unique property that the stability indexk jumps from ±∞ to ?∞ at that particular orbit. These orbits are of great interest since, for small μ>0, ‘neighboring’ orbits will then have a finite (but small) region of stability. 相似文献