Investigating the planar circular restricted three-body problem with strong gravitational field

The case of the planar circular restricted three-body problem where one of the two primaries has a stronger gravitational field with respect to the classical Newtonian field is investigated. We consider the case where two primaries have the same mass, so as the the only difference between them to be the strength of the gravitational field which is controlled by the power p of the potential. A thorough numerical analysis takes place in several types of two dimensional planes in which we classify initial conditions of orbits into three main categories: (1) bounded, (2) escaping and (3) collision. Our results reveal that the power of the gravitational potential has a huge impact on the nature of orbits. Interpreting the collision motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We successfully located the escape as well as the collision basins and we managed to correlate them with the corresponding escape and collision time of the orbits. We hope our contribution to be useful for a further understanding of the escape and collision properties of motion in this interesting version of the restricted three-body problem.     

Dynamical properties of the restricted four-body problem with radiation pressure

The paper deals with the photo-gravitational restricted four-body problem. The dynamical behavior of the small particle which is subjected to both the gravitational attraction and the radiation pressure of the three primaries is studied and some features of this model, like the zero-velocity curves and surfaces, as well as the equilibrium positions and their stability, are investigated.     

STUDIES OF MELNIKOV METHOD AND TRANSVERSAL HOMOCLINIC ORBITS IN THE CIRCULAR PLANAR RESTRICTED THREE－BODY PROBLEM

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The region of possible motion of the planar circular restricted three-body problem under the influence of a ring

This paper discusses the region of possible motion of planar cir-cular restricted three-body problem that one primarg in them has aring and it gives the equation of motion of the third body.Some re-sults are as follows:(1)The location of equilibrium points depends on the parameter μof the system;inner radius a and outer radius b of the ring;the dis-tance l between two primaries;and the ratio θ of the mass of the ringwith the sum of masses of the primary which has the ring and the ringitself.when α,b,l,θare constant.the number of the equilibriumpoints varies with μ,it is five at most and three at least.Besideseach triangular point and two primaries form an isosceles triangle.(2)When α.b.l.θ are constant,we give a range of μ.If μ isin this range.the structure of the region of possible motion of thethird body is the same as that of general planar circular restrictedthree-body problem.     

Relative equilibrium stability of a mechanical system with deformable elements in a circular orbit

The relative equilibrium stability is analyzed for a mechanical system in an orbit. The system consists of two rigid bodies connected by a thin inextensible elastic rod. The problem of stability of steady motions is reduced to the minimization problem for the system’s potential energy consisting of the potential energy of elastic, gravitational, and centrifugal forces.     

按位移推导平面轴对称问题的一般性解答

根据平面轴对称问题的物理概念, 将平面轴对称问题分为轴对称应力问题和轴对称位移问题, 给出了这两种轴对称问题的基本方程, 并指出平面轴对称位移问题是平面轴对称应力问题的特例. 在此基础上, 分别按位移推导了平面轴对称应力问题和平面轴对称位移问题的一般性解答. 按位移推导平面轴对称问题, 可以考虑体力分量, 从而可避免按应力函数推导平面轴对称应力问题时不能考虑体力分量的局限性.     

Stability of triangular libration points in a planar restricted elliptic three body problem in cases of double resonances

Stability of triangular libration points in a planar restricted elliptic three-body problem is investigated for two sets of parameters corresponding to the cases of double resonances. These positions are shown to be formally stable in the case of the fourth-order double resonance. When a weak third-order and a strong fourth-order resonances occur in the system, any motion of an approximate autonomous system is found to be bounded if initial conditions are sufficiently close to the unperturbed motion. Boundedness domain estimate is obtained.     

Networks of periodic orbits in the circular restricted three-body problem with first order post-Newtonian terms

The motivation of this article is to numerically investigate the orbital dynamics of the planar post-Newtonian circular restricted problem of three bodies. By numerically integrating several large sets of initial conditions of orbits we obtain the basins of escape. Additionally, we determine the influence of the transition parameter on the orbital structure of the system, as well as on the families of simple symmetric periodic orbits. The networks and the stability of the symmetric periodic orbits are revealed, while the corresponding critical periodic solutions are also identified. The parametric evolution of the horizontal and the vertical stability of the periodic orbits are also monitored, as a function of the transition parameter.

     

Determining the material constant in the equilibrium problem for an infinite elastic plane weakened by a circular hole

The averaging method is used to determine the material functions of the linear Cosserat theory of elasticity. The proposed method is used to find the material constant in the equilibrium problem for an infinite plane weakened by a circular hole.     

Instability of the equilibrium positions of restricted mechanical systems

Moscow Aviation Institute. Translated from Prikladnaya Mekhanika, Vol. 27, No. 2, pp. 102–106, February, 1991.     

Sufficient conditions for existence of physically significant solutions in limiting equilibrium slope stability analysis

Engineering assessment of slope stability is usually performed using limiting equilibrium analysis. This framework includes a process of minimization which identifies the critical slip surface and its associated minimal safety factor. The approach makes sense only if a minimum safety factor exists, i.e. if there is a slip surface for which the safety factor is smaller than safety factors associated with all other slip surfaces. The present work establishes conditions which guarantee that slope stability problems have a physically significant minimum. The question of existence of a minimum is relevant to all slope stability formulations which satisfy equilibrium conditions without a priori assumptions with respect to the shape of potential slip surfaces. The main purpose of the present work is to “legitimatize” the approximate, but practically useful, limiting equilibrium technique by placing it on secure foundations.The present work shows that the restrictions required in order to ensure the existence of a minimum include three, well motivated, physical elements: (a) Two integral inequality constraints restricting legitimate forms of slip surfaces, and normal stress functions acting on them. These constraints represent the obvious observation that under usual conditions slopes fail by moving down and away from the main body of the slope. (b) The strength model (Mohr envelope), should imply a finite tensile strength. (c) A “cracking criterion” which specifies the consequences (crack formation) occurring when the soil’s tensile strength is fully mobilized.     

Contact problem for a circular ring

International Applied Mechanics -     

On stability of the equilibrium of a cottrell crack

The Wiener–Hopf method is used to find the exact solution to the static symmetric plane problem of elasticity for a homogeneous isotropic plate with a finite-length crack emerging from the point of intersection of two semi-infinite straight slip (dislocation) lines. An expression for the crack-tip stress intensity factor is derived. Crack initiation is described by the Cottrell mechanism. The equilibrium of the crack is analyzed for stability     

Arnold diffusion in the elliptic restricted three-body problem

In this paper, we show the existence of the Arnold diffusion in the elliptic restricted three-body problem. This gives one of the very few examples of Arnold diffusion in real physical systems. The construction is based on the transversal homoclinic orbits in the circular restricted three-body problem ([6, 7, 14]). We prove that the small perturbations to the horseshoe maps in the neighborhood of the homoclinic orbits creates the Arnold diffusion. The existence of the Arnold diffusion also shows that the elliptic restricted three-body problem is non-integrable.     

Elastoplastic equilibrium of a weighable isotropic half-plane with a circular hole

The elastoplastic state of a weighable isotropic half-plane with a circular hole is studied. Complex Kolosov-Muskhelishvili functions which describe the elastic state of the half-plane are constructed. The unknown interface between the plastic and elastic regions is studied with allowance for the single-valuedness of the elastic displacements. The problem is also solved by the small-parameter method, and the two solutions are compared. Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 93–98, March, 1999.     

The planar mechanism as a special case in the automatic computation of static equilibrium

Aspects of the planar mechanism are considered as a special case in the development of a scheme, where joint loads are calculated from equations of static equilibrium set up automatically by computer on the basis of input specifying the types of joints and link dimensions.