The rhomboidal symmetric four-body problem

We consider the planar symmetric four-body problem with two equal masses m ₁?=?m ₃?>?0 at positions (±x ₁(t),?0) and two equal masses m ₂?=?m ₄?>?0 at positions (0, ±x ₂(t)) at all times t, referred to as the rhomboidal symmetric four-body problem. Owing to the simplicity of the equations of motion this problem is well suited to study regularization of the binary collisions, periodic solutions, chaotic motion, as well as the four-body collision and escape manifolds. Furthermore, resonance phenomena between the two interacting rectilinear binaries play an important role.     

Universal unfolding of symmetric resonances

We present the analysis of the bifurcation sequences of a family of resonant 2-DOF Hamiltonian systems invariant under spatial mirror symmetry and time reversion. The phase-space structure is investigated by a singularity theory approach based on the construction of a universal deformation of the detuned Birkhoff–Gustavson normal form. Thresholds for the bifurcations of periodic orbits in generic position are computed as asymptotic series in terms of physical parameters of the original system.     

Axially symmetric explosion in magnetogasdynamics

A point explosion in a spheroid with axially symmetric exponential density distribution is investigated by generalizing the method of Laumbach and Probstein to include the effects of a magnetic field. It is shown that the shock velocity decreases and tends to zero. Also, the elongation of the shock envelope along the axis of symmetry is much reduced and the blowout of the shock wave is removed on account of the magnetic field.     

Cylindrically symmetric cosmologies in Lyra geometry

A class of new exact solutions of the Einstein field equations have been investigated for stationary cylindrically symmetric space-time around a local cosmic string in the theory based on Lyra’s geometry in normal gauge in the presence and absence of an electromagnetic field. The cosmological solutions have been analyzed through various physical and geometrical parameters. It has also been shown that the solutions are space-time inhomogeneous and filled with charged dust.     

Numerical analysis of the symmetric methods

Aimed at the initial value problem of the particular second-order ordinary differential equations,y =f(x, y), the symmetric methods (Quinlan and Tremaine, 1990) and our methods (Xu and Zhang, 1994) have been compared in detail by integrating the artificial earth satellite orbits in this paper. In the end, we point out clearly that the integral accuracy of numerical integration of the satellite orbits by applying our methods is obviously higher than that by applying the same order formula of the symmetric methods when the integration time-interval is not greater than 12000 periods.     

Axially symmetric explosion in a spheroid

The method of Laumbach and Probstein is applied to a point explosion in a spheroid with exponential density distribution. It is shown that the shock wave propagates strongly along the direction of symmetry axis and the envelope of the shock front elongates to the same direction. The rate of elongation of the shock envelope increases with the eccentricity of the spheroid and finally the blowout of the shock wave along the polar axis occurs when the eccentricity exceeds some critical value. It is suggested that such a blowout of the shock may be connected with some galactic outburst having axial symmetry.     

Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem

We analytically prove the existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise symmetric equal mass four-body problem. This is an extension of our previous proof of the analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar fully symmetric equal mass four-body problem. We then use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar pairwise symmetric 1, m, 1, m four-body problem for m between 0 and 1. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability when 0.54 ≤ m ≤ 1, and are linearly unstable when 0 < m ≤ 0.53.     

Inhomogeneous plane symmetric models in bimetric theory

Field equations of Rosen's bimetric theory are obtained with the aid of aninhomogeneous plane symmetric metric in the presence of a perfect fluiddistribution. Models corresponding to vacuum, stiff fluid and disorderedradiation are constructed and discussed.     

The stability of spherically symmetric accretion flows

The stability of spherically symmetric accretion flows is reconsidered. An analytical proof on stability for isothermal critical flow is given and numerical stability analyses are carried out for critical accretion flows in the range of 1<<5/3. Moreover, numerical results for subsonic flows are given.     

New mass formula for spherically symmetric stellar configurations

A new formula is derived for the mass of spherically symmetric stellar configurations. An expression will be given where the mass square is connected to an integral over the pressure of gravitating matter. This formula turns out to hold for Newtonian gravity, for Einstein's GRT, for projective and bimetric scalar-tensor and further theories of gravitation.     

Charged static axial symmetric solutions and scalar structures

This paper is devoted to the study of static axially symmetric spacetime with anisotropic fluid by means of structure scalars in the presence of electromagnetic field. The structure scalars in terms of physical variables are evaluated through the Einstein-Maxwell field equations and the inhomogeneity factors are identified. We also explore analytic solutions for isotropic as well as anisotropic fluids. It is found that isotropic solution turns out to be a charged solution which has no correspondence with the Weyl metrics while the anisotropic solution has the correspondence with the Weyl metrics.     

Non-static cylindrically symmetric fields in general projective relativity

Exact solutions for the vacuum field equations of General Projective Relativity of Arcidiacono (1986) are obtained for the non-static Einstein-Rosen metric (Einstein and Rosen, 1937). The nature of singularities in these solutions is discussed.     

Static cylindrically symmetric gravitational field with cosmological constant

We show that a class of solutions for a vacuum cylindrically symmetric gravitational field with nonvanishing cosmological constant reduce to the welknown Levi-Civita metric close to the axis. All the circular geodesics in this generalized vacuum are time like, null or space like depending on the relative magnitudes of the two parameters appearing in the solutions.For other particles moving inZ=constant plane there are some cases where the trapping of timelike and null trajectories occur, while in other cases some of the particles may orbit round the axis in a fixed zone bounded by two limiting radii.     

Numerical investigation of symmetric and asymmetric periodic oscillations

Global information for the periodic solutions — symmetric and asymmetric — of the ‘gravitational’ spring-pendulum problem is given for the first time. For two different sets of the parameters of this problem, the families of symmetric periodic solutions which emanate from the equilibrium point have been determined. Further families of asymmetric and symmetric solutions which bifurcate from them have also been examined and interesting results for their behaviour have been pointed out.