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1.
Periodic Solutions of the Elliptic Isosceles Restricted Three-body Problem with Collision 总被引:1,自引:0,他引:1
Lúcia de Fatima Brandão Claudio Vidal 《Journal of Dynamics and Differential Equations》2008,20(2):377-423
The elliptic isosceles restricted three-body problem with collision, is a restricted three-body problem where the primaries
move having consecutive elliptic collisions and the infinitesimal mass is moving in the plane perpendicular to the primaries
motion that passes through the center of mass of the primary system. Our purpose in this paper is to prove the existence of
many families of periodic solutions using Continuation’s method, where the perturbing parameter is related with the energy
of the primaries. This work is merely analytic and uses symmetry conditions and appropriate coordinates.
Partially supported by Dirección de Investigación UBB, 064608 3/RS. 相似文献
2.
Eder Mateus Andrea Venturelli Claudio Vidal 《Archive for Rational Mechanics and Analysis》2013,210(1):165-176
We consider the spatial isosceles Newtonian three-body problem, with one particle on a fixed plane, and the other two particles (with equal masses) located symmetrically with respect to this plane. Using variational methods, we find a one-parameter family of collision solutions for this system. All these solutions are periodic in a rotating frame. 相似文献
3.
J. F. Palacián C. Vidal J. Vidarte P. Yanguas 《Journal of Dynamics and Differential Equations》2018,30(4):1757-1774
The existence and stability of periodic solutions for different types of perturbations associated to the Charged Restricted Circular Three Body Problem (shortly, CHRCTBP) is tackled using reduction and averaging theories as well as the technique of continuation of Poincaré for the study of symmetric periodic solutions. The determination of KAM 2-tori encasing some of the linearly stable periodic solutions is proved. Finally, we analyze the occurrence of Hamiltonian-Hopf bifurcations associated to some equilibrium points of the CHRCTBP. 相似文献
4.
Periodicity of motion around the collinear libration point associated with the Elliptic Restricted Three-Body Problem is studied. A survey of periodic solutions in the Circular Restricted Three-Body Problem is presented considering both Sun–Earth and Earth–Moon systems. Halo, Lyapunov and Vertical families around L1, L2 and L3 points are investigated, and their orbital period ranges through the entire family are reported. Resonant motions within the orbit families in the circular problem are identified and selected as suitable initial guess to find periodic orbits in the elliptic problem, which are targeted using a differential correction algorithm. Periodic solutions found are cataloged depending on the number of revolutions around libration points. Geometry, dynamical behavior and stability properties of single-revolution orbits are shown, as well as double-, triple- and quadruple-revolution solutions. 相似文献
5.
Luca Biasco Luigi Chierchia Enrico Valdinoci 《Archive for Rational Mechanics and Analysis》2003,170(2):91-135
The spatial planetary three-body problem (i.e., one star and two planets, modelled by three massive points, interacting through gravity in a three dimensional space) is considered. It is proved that, near the limiting stable solutions given by the two planets revolving around the star on Keplerian ellipses with small eccentricity and small non-zero mutual inclination, the system affords two-dimensional, elliptic, quasi-periodic solutions, provided the masses of the planets are small enough compared to the mass of the star and provided the osculating Keplerian major semi-axes belong to a two-dimensional set of density close to one. 相似文献
6.
Hsin-Yuan Huang 《Archive for Rational Mechanics and Analysis》2011,202(1):247-267
In this paper, we study the minimizing triple collision orbits in the planar Newtonian three-body problem with arbitrary masses.
We show that for a given non-collinear initial configuration, the minimizing triple collision orbit is collision-free until
a simultaneous collision, and its limiting configuration is the Lagrangian configuration with the same orientation as the
initial configuration. For the collinear initial configuration, under a certain technical assumption, there exist two minimizing
orbits. The limiting configurations of these orbits are the two opposite Lagrangian configurations. 相似文献
7.
The equations of the restricted three-body problem describe the motion of a massless particle under the influence of two primaries of masses 1 −μ and μ, 0≤μ≤ 1/2, that circle each other with period equal to 2π. When μ=0, the problem admits orbits for the massless particle that are ellipses of eccentricity e with the primary of mass 1 located at one of the focii. If the period is a rational multiple of 2π, denoted 2π p/q, some of these orbits perturb to periodic motions for μ > 0. For typical values of e and p/q, two resonant periodic motions are obtained for μ > 0. We show that the characteristic multipliers of both these motions are given by expressions of the form
in the limit μ→ 0. The coefficient C(e,p,q) is analytic in e at e=0 and C(e,p,q)=O(e|p-q|). The coefficients in front of e|p-q|, obtained when C(e,p,q) is expanded in powers of e for the two resonant periodic motions, sum to zero. Typically, if one of the two resonant periodic motions is of elliptic type the other is of hyperbolic type. We give similar results for retrograde periodic motions and discuss periodic motions that nearly collide with the primary of mass 1 −μ. 相似文献
8.
9.
Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I
Yu. N. Chelnokov 《Mechanics of Solids》2017,52(6):613-639
We develop a quaternion method for regularizing the differential equations of the perturbed spatial restricted three-body problem by using the Kustaanheimo–Stiefel variables, which is methodologically closely related to the quaternion method for regularizing the differential equations of perturbed spatial two-body problem, which was proposed by the author of the present paper.A survey of papers related to the regularization of the differential equations of the two- and threebody problems is given. The original Newtonian equations of perturbed spatial restricted three-body problem are considered, and the problem of their regularization is posed; the energy relations and the differential equations describing the variations in the energies of the system in the perturbed spatial restricted three-body problem are given, as well as the first integrals of the differential equations of the unperturbed spatial restricted circular three-body problem (Jacobi integrals); the equations of perturbed spatial restricted three-body problem written in terms of rotating coordinate systems whose angular motion is described by the rotation quaternions (Euler (Rodrigues–Hamilton) parameters) are considered; and the differential equations for angular momenta in the restricted three-body problem are given.Local regular quaternion differential equations of perturbed spatial restricted three-body problem in the Kustaanheimo–Stiefel variables, i.e., equations regular in a neighborhood of the first and second body of finite mass, are obtained. The equations are systems of nonlinear nonstationary eleventhorder differential equations. These equations employ, as additional dependent variables, the energy characteristics of motion of the body under study (a body of a negligibly small mass) and the time whose derivative with respect to a new independent variable is equal to the distance from the body of negligibly small mass to the first or second body of finite mass.The equations obtained in the paper permit developing regular methods for determining solutions, in analytical or numerical form, of problems difficult for classicalmethods, such as the motion of a body of negligibly small mass in a neighborhood of the other two bodies of finite masses. 相似文献
10.
Xijun Hu Yiming Long Shanzhong Sun 《Archive for Rational Mechanics and Analysis》2014,213(3):993-1045
It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter \({\beta=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2 \in [0, 9]}\) and the eccentricity \({e \in [0, 1)}\) . We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle [0, 9] × [0, 1), aside from perturbation methods for e > 0 small enough, blow-up techniques for e sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full (β, e) range [0, 9] × [0, 1) via the ω-index theory of symplectic paths for ω belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the ω-index decreasing property of the solutions in β for fixed \({e\in [0, 1)}\) , we prove the existence of three curves located from left to right in the rectangle [0, 9] × [0, 1), among which two are ?1 degeneracy curves and the third one is the right envelope curve of the ω-degeneracy curves, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter (β, e) passes through each of these three curves. Interesting symmetries of these curves are also observed. The linear stability of the singular case when the eccentricity e approaches 1 is also analyzed in detail. 相似文献
11.
Christopher McCord Kenneth R. Meyer 《Journal of Dynamics and Differential Equations》2000,12(2):247-271
In Dynamical Systems, Birkhoff gave a clear formulation of a cross section, suggested a possible generalization to cross sections with boundary, and raised the question of whether or not such cross sections exist in the three-body problem. In this work, we explicitly develop Birkhoff's notion of a generalized cross section, formulate homological necessary conditions for the existence of a cross section or generalized cross section, and show that these conditions are not satisfied in the three-body problem. 相似文献
12.
Consider the three-body problem, in the regime where one body revolves far away around the other two, in space, the masses of the bodies being arbitrary but fixed; in this regime, there are no resonances in mean motions. The so-called secular dynamics governs the slow evolution of the Keplerian ellipses. We show that it contains a horseshoe and all the chaotic dynamics which goes along with it, corresponding to motions along which the eccentricity of the inner ellipse undergoes large, random excursions. The proof goes through the surprisingly explicit computation of the homoclinic solution of the first order secular system, its complex singularities and the Melnikov potential. 相似文献
13.
Alberto Castro Ortega Ernesto A. Lacomba 《Journal of Dynamics and Differential Equations》2012,24(1):85-100
Consider three charged masses moving along the line. For this model we study the solutions near total collision using blow
up techniques obtaining that for given masses and charges the vector field on the collision manifold has a non-hyperbolic
equilibrium point. To study this situation the vector field is written in normal form and the center manifold theory is used
obtaining that all nonzero solutions near the origin escape to infinity. 相似文献
14.
Alberto Castro Ortega Manuel Falconi 《Journal of Dynamics and Differential Equations》2016,28(2):519-532
In this paper, we study the existence of Schubart-like periodic solutions in a charged collinear three-body problem by applying the notion of turning points and some continuity arguments. We proved the existence of Schubart solutions for the case where the outer particles repel each other. 相似文献
15.
Vivina Barutello Davide L. Ferrario Susanna Terracini 《Archive for Rational Mechanics and Analysis》2008,190(2):189-226
We consider periodic and quasi-periodic solutions of the three-body problem with homogeneous potential from the point of view
of equivariant calculus of variations. First, we show that symmetry groups of the Lagrangian action functional can be reduced
to groups in a finite explicitly given list, after a suitable change of coordinates. Then, we show that local symmetric minimizers
are always collisionless, without any assumption on the group other than the fact that collisions are not forced by the group
itself. Moreover, we describe some properties of the resulting symmetric collisionless minimizers (Lagrange, Euler, Hill-type
orbits and Chenciner–Montgomery figure-eights). 相似文献
16.
Davide L. Ferrario 《Archive for Rational Mechanics and Analysis》2006,179(3):389-412
Periodic and quasi-periodic solutions of the n-body problem can be found as minimizers of the Lagrangian action functional restricted to suitable spaces of symmetric paths.
The main purpose of this paper is to develop a systematic approach to the equivariant minimization for the three-body problem
in three-dimensional space. First we give a finite complete list of symmetry groups fitting to the minimization of the action,
with the property that any other symmetry group can be reduced to be isomorphic to one of these representatives. A second
step is to prove that the resulting (local and global) symmetric action-minimizers are always collisionless (when they are
not already bound to collisions). Furthermore, we prove some results which address the question of whether minimizers are
planar or non-planar; as a consequence of our theory we will give general criteria for a symmetry group to yield planar or
homographic minimizers (either homographic or not, as in the Chenciner-Montgomery eight solution). On the other hand we will
provide a rigorous proof of the existence of some interesting one-parameter families of periodic and quasi-periodic non-planar
orbits. These include the choreographic Marchal's P12 family with equal masses – together with a less-symmetric choreographic family (which anyway probably coincides with the
P12 family). 相似文献
17.
Nan Lu 《Journal of Dynamics and Differential Equations》2013,25(3):715-732
We investigate local dynamics of a 4-dimensional system with small and slowly varying time periodic forcing. By assuming the unperturbed system is autonomous and has a fixed point with eigenvalues $(0,0,i,-i)$ ( 0 , 0 , i , ? i ) , we study homoclinic, subharmonic solutions and Hopf bifurcation in a $O(\epsilon )$ O ( ? ) neighborhood of the fixed point, where $\epsilon $ ? is the perturbation parameter. 相似文献
18.
Chaos Theory and the Problem of Change in Family Systems 总被引:1,自引:0,他引:1
In spite of the fact that nonlinear dynamical models have been used for almost half a century in the area of family process theory, an appreciation of the potential of chaos models is a relatively recent development. The present paper discusses the shift of focus in our understanding of family processes resulting from Prigogine's chaos framework, and outlines a chaos approach to family interaction. It is argued that this approach allows us to more effectively address one of the central outstanding questions in the field, namely, how self regulatory behavior can contribute to structural transformation of the family system. 相似文献
19.
Lúcia Brandão Dias Hildeberto E. Cabral 《Journal of Dynamics and Differential Equations》2018,30(1):81-92
We consider the dynamics of an infinitesimal particle under the gravitational action of P primaries of equal masses. These move in an elliptic homographic solution of the P-body problem and the infinitesimal particle moves along the straight line perpendicular to their plane of motion and passing through the common focus of the ellipses. In this work we consider the parametric stability of the infinitesimal mass located at the focus of the ellipses. We construct the boundary curves of the stability/instability regions in the plane of the parameters \(\mu \) and \(\epsilon \), which are the mass of each primary and the eccentricity of the elliptic orbit, respectively. 相似文献
20.
The restricted planar three-body problem has a rich history, yet many unanswered questions still remain. In the present paper
we prove the existence of a global surface of section near the smaller body in a new range of energies and mass ratios for
which the Hill’s region still has three connected components. The approach relies on recent global methods in symplectic geometry
and contrasts sharply with the perturbative methods used until now. 相似文献