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1.
In this paper we study the relative equilibria and their stability for a system of three point particles moving under the action of a Lennard-Jones potential. A central configuration is a special position of the particles where the position and acceleration vectors of each particle are proportional, and the constant of proportionality is the same for all particles. Since the Lennard-Jones potential depends only on the mutual distances among the particles, it is invariant under rotations. In a rotating frame the orbits coming from central configurations become equilibrium points, the relative equilibria. Due to the form of the potential, the relative equilibria depend on the size of the system, that is, depend strongly of the momentum of inertia I. In this work we characterize the relative equilibria, we find the bifurcation values of I for which the number of relative equilibria is changing, we also analyze the stability of the relative equilibria. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

2.
The aim of the planar inverse problem of dynamics is: given a monoparametric family of curves f(x, y) = c, find the potential V (x, y) under whose action a material point of unit mass can describe the curves of the family. In this study we look for V in the class of the anisotropic potentials V(x, y) = v(a2x2 + y2), (a=constant). These potentials have been used lately in the search of connections between classical, quantum, and relativistic mechanics. We establish a general condition which must be satisfied by all the families produced by an anisotropic potential. We treat special cases regarding the families (e. g. families traced isoenergetically) and we present certain pertinent examples of compatible pairs of families of curves and anisotropic potentials. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

3.
To illustrate his theory of coronal heating, Parker initially considers the problem of disturbing a homogeneous vertical magnetic field that is line-tied across two infinite horizontal surfaces. It is argued that, in the absence of resistive effects, any perturbed equilibrium must be independent of z. As a result random footpoint perturbations give rise to magnetic singularities, which generate strong Ohmic heating in the case of resistive plasmas. More recently these ideas have been formalized in terms of a magneto-static theorem but no formal proof has been provided. In this paper we investigate the Parker hypothesis by formulating the problem in terms of the fluid displacement. We find that, contrary to Parker's assertion, well-defined solutions for arbitrary compressibility can be constructed which possess non-trivial z-dependence. In particular, an analytic treatment shows that small-amplitude Fourier disturbances violate the symmetry ∂z = 0 for both compact and non-compact regions of the (x, y) plane. Magnetic relaxation experiments at various levels of gas pressure confirm the existence and stability of the Fourier mode solutions. More general footpoint displacements that include appreciable shear and twist are also shown to relax to well-defined non-singular equilibria. The implications for Parker's theory of coronal heating are discussed.  相似文献   

4.
P. Hoyng 《Solar physics》1991,133(1):43-50
The resonant scattering spectrometers of the IRIS ground-based network for measuring whole-disc solar velocity oscillations make use of a piezoelastic modulator. The velocity noise generated by this optical component is analysed with particular emphasis on the required stability of the amplitude of oscillation, a. The product of the absolute stability ¦ a – a m ¦/a m and the relative stability a r.m.s./a m may not be larger than 10 –4 to 10 –5 (depending on specific wishes), where a m is the optimum amplitude. The velocity noise due to photon statistics is slightly enhanced, but other instrumental sources of velocity noise remain unaffected.  相似文献   

5.
In this paper we study the evolution of the dark energy parameter within the scope of a spatially homogeneous and isotropic FRW universe filled with barotropic fluid and dark energy. The scale factor is considered as a power law function of time which yields a constant deceleration parameter. We consider the case when the dark energy is minimally coupled to the perfect fluid as well as direct interaction with it. The cosmic jerk parameter in our derived models is consistent with the recent data of astrophysical observations. It is concluded that in non-interacting case, all the three open, close and flat universes cross the phantom region whereas in interacting case only open and flat universes cross the phantom region. We find that during the evolution of the universe, the equation of state (EoS) for dark energy ω D changes from ω D >−1 to ω D <−1, which is consistent with recent observations.  相似文献   

6.
We prove that, in general, a given two-dimensional inhomogeneous potential V(x,y) does not allow for the creation of homogeneous families of orbits. Yet, depending on the case at hand, if the given potential satisfies certain conditions, this potential is compatible either with one (or two) monoparametric homogeneous families of orbits or at most with five such familes. The orbits are then found on the grounds of the given potential.  相似文献   

7.
The motion of a point mass in the J 2 problem is generalized to that of a rigid body in a J 2 gravity field. The linear and nonlinear stability of the classical type of relative equilibria of the rigid body, which have been obtained in our previous paper, are studied in the framework of geometric mechanics with the second-order gravitational potential. Non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are obtained through a Poisson reduction process by means of the symmetry of the problem. The linear system matrix at the relative equilibria is given through the multiplication of the Poisson tensor and Hessian matrix of the variational Lagrangian. Based on the characteristic equation of the linear system matrix, the conditions of linear stability of the relative equilibria are obtained. The conditions of nonlinear stability of the relative equilibria are derived with the energy-Casimir method through the projected Hessian matrix of the variational Lagrangian. With the stability conditions obtained, both the linear and nonlinear stability of the relative equilibria are investigated in details in a wide range of the parameters of the gravity field and the rigid body. We find that both the zonal harmonic J 2 and the characteristic dimension of the rigid body have significant effects on the linear and nonlinear stability. Similar to the classical attitude stability in a central gravity field, the linear stability region is also consisted of two regions that are analogues of the Lagrange region and the DeBra-Delp region respectively. The nonlinear stability region is the subset of the linear stability region in the first quadrant that is the analogue of the Lagrange region. Our results are very useful for the studies on the motion of natural satellites in our solar system.  相似文献   

8.
We study the equilibrium points and the zero-velocity curves of Chermnykh’s problem when the angular velocity ω varies continuously and the value of the mass parameter is fixed. The planar symmetric simple-periodic orbits are determined numerically and they are presented for three values of the parameter ω. The stability of the periodic orbits of all the families is computed. Particularly, we explore the network of the families when the angular velocity has the critical value ω = 2√2 at which the triangular equilibria disappear by coalescing with the collinear equilibrium point L1. The analytic determination of the initial conditions of the family which emanate from the Lagrangian libration point L1 in this case, is given. Non-periodic orbits, as points on a surface of section, providing an outlook of the stability regions, chaotic and escape motions as well as multiple-periodic orbits, are also computed. Non-linear stability zones of the triangular Lagrangian points are computed numerically for the Earth–Moon and Sun–Jupiter mass distribution when the angular velocity varies.  相似文献   

9.
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   

10.
The motion of a point mass in the J 2 problem has been generalized to that of a rigid body in a J 2 gravity field for new high-precision applications in the celestial mechanics and astrodynamics. Unlike the original J 2 problem, the gravitational orbit-rotation coupling of the rigid body is considered in the generalized problem. The existence and properties of both the classical and non-classical relative equilibria of the rigid body are investigated in more details in the present paper based on our previous results. We nondimensionalize the system by the characteristic time and length to make the study more general. Through the study, it is found that the classical relative equilibria can always exist in the real physical situation. Numerical results suggest that the non-classical relative equilibria only can exist in the case of a negative J 2, i.e., the central body is elongated; they cannot exist in the case of a positive J 2 when the central body is oblate. In the case of a negative J 2, the effect of the orbit-rotation coupling of the rigid body on the existence of the non-classical relative equilibria can be positive or negative, which depends on the values of J 2 and the angular velocity Ω e . The bifurcation from the classical relative equilibria, at which the non-classical relative equilibria appear, has been shown with different parameters of the system. Our results here have given more details of the relative equilibria than our previous paper, in which the existence conditions of the relative equilibria are derived and primarily studied. Our results have also extended the previous results on the relative equilibria of a rigid body in a central gravity field by taking into account the oblateness of the central body.  相似文献   

11.
We use a formulation of the N-body problem in spaces of constant Gaussian curvature, \({\kappa }\in \mathbb {R}\), as widely used by A. Borisov, F. Diacu and their coworkers. We consider the restricted three-body problem in \(\mathbb {S}^2\) with arbitrary \({\kappa }>0\) (resp. \(\mathbb {H}^2\) with arbitrary \({\kappa }<0\)) in a formulation also valid for the case \({\kappa }=0\). For concreteness when \({\kappa }>0\) we restrict the study to the case of the three bodies at the upper hemisphere, to be denoted as \(\mathbb {S}^2_+\). The main goal is to obtain the totality of relative equilibria as depending on the parameters \({\kappa }\) and the mass ratio \(\mu \). Several general results concerning relative equilibria and its stability properties are proved analytically. The study is completed numerically using continuation from the \({\kappa }=0\) case and from other limit cases. In particular both bifurcations and spectral stability are also studied. The \(\mathbb {H}^2\) case is similar, in some sense, to the planar one, but in the \(\mathbb {S}^2_+\) case many differences have been found. Some surprising phenomena, like the coexistence of many triangular-like solutions for some values \(({\kappa },\mu )\) and many stability changes will be discussed.  相似文献   

12.
We consider the following case of the 3D inverse problem of dynamics: Given a spatial two‐parametric family of curves f (x, y, z) = c1, g (x, y, z) = c2, find possibly existing two‐dimension potentials under whose action the curves of the family are trajectories for a unit mass particle. First we establish the conditions which must be fulfilled by the family so that potentials of the form w (y, z) give rise to the curves of the family, and we present some applications. Then we examine briefly the existence of potentials depending on (x, z), respectively (x, y), which are compatible with the given family (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

13.
The vertical stability character of the families of short and long period solutions around the triangular equilibrium points of the restricted three-body problem is examined. For three values of the mass parameter less than equal to the critical value of Routh (μ R ) i.e. for μ = 0.000953875 (Sun-Jupiter), μ = 0.01215 (Earth-Moon) and μ = μ R = 0.038521, it is found that all such solutions are vertically stable. For μ > (μ R ) vertical stability is studied for a number of ‘limiting’ orbits extended to μ = 0.45. The last limiting orbit computed by Deprit for μ = 0.044 is continued to a family of periodic orbits into which the well known families of long and short period solutions merge. The stability characteristics of this family are also studied.  相似文献   

14.
We study the non-linear stability of the equilibria corresponding to the motion of a particle orbiting around a finite straight segment. The potential is a logarithmic function and may be considered as an approximation to the one generated by elongated celestial bodies. By means of the Arnold's theorem for non-definite quadratic forms we determine the orbital stability of the equilibria, for all values of the parameter k of the problem, resonant cases included.  相似文献   

15.
Scalar field as dark energy accelerating expansion of the Universe   总被引：1，自引：1，他引：0  
The features of a homogeneous scalar field ϕ with classical Lagrangian L = ϕ;i ϕ;i /2 − V(ϕ) and tachyon field Lagrangian L = −V(ϕ)√1 − ϕ;i ϕ;i causing the observable accelerated expansion of the Universe are analyzed. The models with constant equation-of-state parameter w de = p de/ρde < −1/3 are studied. For both cases the fields ϕ(a) and potentials V(a) are reconstructed for the parameters of cosmological model of the Universe derived from the observations. The effect of rolling down of the potential V(ϕ) to minimum is shown. Published in Ukrainian in Kinematika i Fizika Nebesnykh Tel, 2008, Vol. 24, No. 5, pp. 345–359. The article was translated by the authors.  相似文献   

16.
The spectral stability of synchronous circular orbits in a rotating conservative force field is treated using a recently developed Hamiltonian method. A complete set of necessary and sufficient conditions for spectral stability is derived in spherical geometry. The resulting theory provides a general unified framework that encompasses a wide class of relative equilibria, including the circular restricted three-body problem and synchronous satellite motion about an aspherical planet. In the latter case we find an interesting class of stable nonequatorial circular orbits. A new and simplified treatment of the stability of the Lagrange points is given for the restricted three-body problem.  相似文献   

17.
This paper concerns the dynamics of a rigid body of finite extent moving under the influence of a central gravitational field. A principal motivation behind this paper is to reveal the hamiltonian structure of the n-body problem for masses of finite extent and to understand the approximation inherent to modeling the system as the motion of point masses. To this end, explicit account is taken of effects arising because of the finite extent of the moving body. In the spirit of Arnold and Smale, exact models of spin-orbit coupling are formulated, with particular attention given to the underlying Lie group framework. Hamiltonian structures associated with such models are carefully constructed and shown to benon-canonical. Special motions, namely relative equilibria, are investigated in detail and the notion of anon-great circle relative equilibrium is introduced. Non-great circle motions cannot arise in the point mass model. In our analysis, a variational characterization of relative equilibria is found to be very useful. Thereduced hamiltonian formulation introduced in this paper suggests a systematic approach to approximation of the underlying dynamics based on series expansion of the reduced hamiltonian. The latter part of the paper is concerned with rigorous derivations of nonlinear stability results for certain families of relative equilibria. Here Arnold's energy-Casimir method and Lagrange multiplier methods prove useful. This work was supported in part by the AFOSR University Research Initiative Program under grant AFOSR-87-0073, by AFOSR grant 89-0376, and by the National Science Foundation's Engineering Research Centers Program: NSFD CDR 8803012. The work of P.S. Krishnaprasad was also supported by the Army Research Office through the Mathematical Sciences Institute of Cornell University.  相似文献   

18.
The linear stability of the inner collinear equilibrium point of the photogravitational elliptic restricted three-body problem is examined and the stability regions are determined in the space of the parameters of mass, eccentricity and radiation pressure. The case of equal radiation factors of the two primaries is considered and the full range of values of the common radiation factor is explored, from the caseq 1 =q 2 =q = 1/8 at which the triangular equilibria disappear by coalescing on the rotating axis of the primaries transferring their stability to the collinear point, down toq = 0 at which value the stability regions in theµ - e plane disappear by shrinking down to zero size. It is found that radiation pressure exerts a significant influence on the stability regions. For certain intervals of radiation values these regions become qualitatively different from the gravitational case as well as the solar system case. They evolve as in the case of the triangular equilibrium point considered in a previous paper. There exist values of the common radiation factor, in the range considered, for which the collinear equilibrium point is stable for the entire range of mass distribution among the primaries and for large eccentricities of their orbits.  相似文献   

19.
Stability of the planar full 2-body problem   总被引：1，自引：0，他引：1  
The stability of the Full Two-Body Problem is studied in the case where both bodies are non-spherical, but are restricted to planar motion. The mutual potential is expanded up to second order in the mass moments, yielding a highly symmetric yet non-trivial dynamical system. For this system we identify all relative equilibria and determine their stability properties, with an emphasis on finding the energetically stable relative equilibria and conditions for Hill stability of the system. The energetically stable relative equilibria always correspond to the classical “gravity gradient” configuration with the long ends of the two bodies pointed at each other, however there always exists a second equilibrium in this configuration at a closer separation that is unstable. For our model system we precisely map out the relations between these different configurations at a given value of angular momentum. This analysis identifies the fundamental physical constraints and limitations that exist on such systems, and has immediate applications to the stability of asteroid systems that are fissioned due to a rapid spin rate. Specifically, we find that all contact binary asteroids which are spun to fission will initially lie in an unstable dynamical state and can always re-impact. If the total system energy is positive, the fissioned system can disrupt directly from this relative equilibrium, while if it is negative the system is bound together.  相似文献   

20.
Maxwell’s ring-type configuration (i.e. an N-body model where the ν = Ν − 1 bodies have equal masses and are located at the vertices of a regular ν-gon while the N-th body with a different mass is located at the center of mass of the system) has attracted special attention during the last 15 years and many aspects of it have been studied by considering Newtonian and post-Newtonian potentials (Mioc and Stavinschi 1998, 1999), homographic solutions (Arribas et al. 2007) and relative equilibrium solutions (Elmabsout 1996), etc. An equally interesting problem, known as the ring problem of (N + 1) bodies, deals with the dynamics of a small body in the combined force field produced by such a configuration. This is the problem we are dealing with in the present paper and our aim is to investigate the variations in the dynamics of the small body in the case that the central primary is also a radiating source and therefore acts on the particle with both gravitation and radiation. Based on the general outlines of Radzievskii’s model, we study the permitted and the existing trapping regions of the particle, its equilibrium locations and their parametric variations as well as the existence of focal points in the zero-velocity diagrams. The distribution of the characteristic curves of families of planar symmetric periodic orbits and their stability for various values of the radiation coefficient of the central body is additionally investigated.  相似文献   

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