Exact Brans-Dicke-Bianchi type-VIIh solutions

We discuss the Brans-Dicke-Bianchi type-VII_h field equations. Exact solutions are given for the vacuum case as well as for the stiff matter case. The derived solutions are the generalizations of the GRT-solutions first given by Doroshkevichet al. (1973) and Lukash (1974a). In addition we present a new BDT-stiff matter solution which has no analogy in the GRT.     

Comment on the Guzmán-Bianchi type-VII h solutions

It is shown that the recently presented Brans-Dicke-Bianchi type-VII _h perfect fluid solutions by Guzmán (1989) are nothing but the solutions given already by us in our previous paper (Lorenz-Petzold, 1984).     

Close binary systems before and after mass transfer

A method is given to derive absolute dimensions from data of spectroscopic binary systems, with application to the 7th Catalogue of Spectroscopic Binary Systems of Battenet al. (1978). The different samples of solutions are analysed with regard to existing solutions. Absolute dimensions are given for 62 systems, not included in the tables of Popper (1980), Lacy (1979), and Giannone and Giannuzzi (1974).This research is supported by the National Foundation of Collective Fundamental Research of Belgium (FKFO) under No. 2.9009.79.     

A note on the elliptic restricted three-body problem

This work considers periodic solutions, arc-solutions (solutions with consecutive collisions) and double collision orbits of the plane elliptic restricted problem of three bodies for =0 when the eccentricity of the primaries,e _p , varies from 0 to 1. Characteristic curves of these three kinds of solutions are given.     

Non-singular Bianchi-Kantowski-Sachs perfect fluid solutions of no-scale supergravity

We investigate the perfect fluid field equations of the Bianchi-Kantowski-Sachs models on the basis ofN=1 no-scale supergravity. A wide class of non-singular solutions is given in the stiff matter as well as in the radiation-dominated case. The vacuum solutions are included as special cases. The solutions given are of much interest in connection with the low-energy approximation of superstrings.     

Some remarks on certain special BDT-FRW solutions

We wish to point out that the BDT-FRW (k=1) special dust solution obtained by Dehnen and Obregón (1971) and the BDT-FRW (k=1) radiation solution given by Obregón and Chauvet (1978) are contained as special cases in the special perfect fluid Jordan-Brans-Dicke solution first given by Brill (1962). In addition we give a simple proof that the special BDT-FRW (k=1) vacuum solution given by Dehnen and Obregón (1972) does not exist. We finally consider the special case =0 and present some new solutions.     

An asymmetrically rotating fluid disc with applications

The rotation of a compressible inviscid fluid disc of (1) slowly varying density or (2) nonuniform density (cold gas approximation) or (3) nonuniform density (hot, but tenuous) is considered. Perturbation methods for solving the basic equation for conservation of vorticity are used. It is found that steady state conditions are realized when vortex waves and differential rotation (jet streams) coexist; special solutions for these vortex waves are obtained. For one of these solutions, a given jet stream and its associated vortex (only one vortex per jet allowed) wave can exist only at certain discrete orbital distances, given by a geometric progressionA ⁿ wheren is an integer andA is a constant. This progression is a good representation for the distances of planets and satellites, with small orbital inclinations, from their respective parent bodies. Certain other solutions for the vortex wave yield streamlines that are logarithmic spirals. Some justifications are given for applying the model to the dynamics of hurricanes and spiral galaxies. Comparisons with observations are surprisingly favorable.The possible role of the jet streams and the steady state long vortex waves (a cooperative-vortex phenomenon) in the formation and evolution of the solar system is also discussed. Comparisons are made with the von Weizsäcker (1944 and Chandrasekhar, 1946) model of turbulent eddies in the solar nebula and with the particle (asteroidal) jet streams of Alfvén and Arrhenius (1970a, b).     

On static spherical symmetric solutions of the Bach-Einstein gravitational field equations

For field equations of 4th order, follwing from a Lagrangian “Ricci scalar plus Weyl scalar”, it is shown (using methods of non-standard analysis) that in a neighbourhood of Minkowski space there do not exist regular static spherically symmetric solutions. With that (besides the known local expansions about r = o nad r = ∞ resp.) for the first time a global statement on the existence of such solutions is given. Finally, this result will be discussed in connection with Einstein's particle programme.     

Ideal Resonance Problem: the Post-Post-Pendulum Approximation

Explicit construction of the solutions of the Hamiltonian system given by H = H ₀(J) – A(J) cos (ideal resonance problem), two orders of approximation beyond the well-known pendulum approximation. The given solutions are valid for libration amplitudes of order . The procedure used is extended to allow the construction of the solutions of Hamiltonians with perturbations involving two degrees of freedom; the post-pendulum solution of an example of this kind is constructed.     

Periodic elliptic motions in a planar restricted (N+1)-body problem

LetN2 mass points (primaries) move on a collinear solution of relative equilibrium of theN-body problem; i.e. suitably fixed on a uniformly rotating straight line. Consider the motion of a massless particle in the gravitational field of these primaries with arbitrarily given masses. An existence proof for periodic solutions (i.e. closed trajectories in a rotating coordinate system) will be given, in which the particle performs nearly keplerian elliptic motions about (and close to) any one of the primaries.     

Intrinsic variational equations for conservative dynamical systems withN degrees of freedom

For a conservative dynamical system withn deg. of freedom we show that the equations of variation along an orbit may be written with respect to an orthonormal moving frame (a generalized Frenet frame) in which the tangential variation is given by a quadrature and the normal andn-2 binormal variations are solutions ofn-1 coupled second order equations of the form of Hill's equation.     

A note on the magnetic perfect fluid Bianchi type-VIo solutions

We make some comments concerning the magnetic perfect fluid Bianchi type-VI_o solutions given recently by Royet al. (1985a).     

The massive scalar field in a closed Friedmann universe model — new rigorous results

For the minimally coupled scalar field in Einstein's theory of gravitation we look for the space of solutions within the class of closed Friedmann universe models. We prove D ≥ 1, where D ≥ is the dimension of the set of solutions which can be integrated up to t → ∞ (D > 0 was conjectured by PAGE (1984)). We discuss concepts like “the probability of the appearance of a sufficiently long inflationary phase” and argue that it is primarily a probability measure μ in the space V of solutions (and not in the space of initial conditions) which has to be applied. μ is naturally defined for Bianchi-type I cosmological models because V is a compact cube. The problems with the closed Friedmann model (which led to controversial claims in the literature) will be shown to originate from the fact that V has a complicated non-compact non-Hausdorff Geroch topology: no natural definition of μ can be given. We conclude: the present state of our universe can be explained by models of the type discussed, but thereby the anthropic principle cannot be fully circumvented.     

Solution of an inverse problem of the dynamics of a particle

The three-dimensional inverse problem of particle dynamics is studied here. The potentialU and the corresponding energyh are determined by the given family of possible trajectories. The classification of the solutions due to the geometry of the given family is obtained.     

Notes on the central forcer n

In this article we collect several results related to the classical problem of two-dimensional motion of a particle in the field of a central force proportional to a real power of the distancer. At first we generalize Whittaker's result of the fourteen powers ofr which lead to intergrability with elliptic functions. We enumerate six more general potentials, including Whittaker's fourteen potentials as particular cases (Sections 2 and 3).Next, we study the stability of the circular solutions, which are the singular solutions of the problem, in Whittaker's terminology. The stability index is computed as a function of the exponentn and its properties are explained, especially in terms of bifurcations with other families of ordinary periodic solutions (Sections 4, 5 and 7). In Section 6, the detailed solution of the inverse cube force problem is given in terms of an auxiliary variable which is similar to the eccentric anomaly of the Kepler problem.Finally, it is shown that the stable singular circular solutions of the central force problem generalize to stable singular elliptic solutions of the two-fixed-center problem. The stability and the bifurcations with other families of periodic solutions of the two-fixed-center problem are also described.     

Continuity and stability of families of figure eight orbits with finite angular momentum

Numerical solutions are presented for a family of three dimensional periodic orbits with three equal masses which connects the classical circular orbit of Lagrange with the figure eight orbit discovered by C. Moore [Moore, C.: Phys. Rev. Lett. 70, 3675–3679 (1993); Chenciner, A., Montgomery, R.: Ann. Math. 152, 881–901 (2000)]. Each member of this family is an orbit with finite angular momentum that is periodic in a frame which rotates with frequency Ω around the horizontal symmetry axis of the figure eight orbit. Numerical solutions for figure eight shaped orbits with finite angular momentum were first reported in [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and mathematical proofs for the existence of such orbits were given in [Marchal, C.: Celest. Mech. Dyn. Astron. 78, 279–298 (2001)], and more recently in [Chenciner, A. et al.: Nonlinearity 18, 1407–1424 (2005)] where also some numerical solutions have been presented. Numerical evidence is given here that the family of such orbits is a continuous function of the rotation frequency Ω which varies between Ω = 0, for the planar figure eight orbit with intrinsic frequency ω, and Ω = ω for the circular Lagrange orbit. Similar numerical solutions are also found for n > 3 equal masses, where n is an odd integer, and an illustration is given for n = 21. Finite angular momentum orbits were also obtained numerically for rotations along the two other symmetry axis of the figure eight orbit [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and some new results are given here. A preliminary non-linear stability analysis of these orbits is given numerically, and some examples are given of nearby stable orbits which bifurcate from these families.     

Numerical explorations of the rectilinear problem of three bodies

We present some results of a numerical exploration of the rectilinear problem of three bodies, with the two outer masses equal. The equations of motion are first given in relative coordinates and in regularized variables, removing both binary collision singularities in a single coordinate transformation. Among our most important results are seven periodic solutions and three symmetric triple collision solutions. Two of these periodic solutions have been continued into families, the outer massm ₃ being the family parameter. One of these families exists for all masses while the second family is a branch of the first at a second-kind critical orbit. This last family ends in a triple collision orbit.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.     

Higher-dimensional vacuum bianchi-mixmaster cosmologies

We derive some new exact 7-dimensional cosmological solutions |R⊗ I ⊗N, whereN = I, II, VI₀, VII₀, VIII and IX are the various 3-dimensional Bianchi models. The solutions given are higher-dimensional generalizations of the mixmaster cosmologies. There is a strong influence of the extra spacesN, which results in a fundamental change of the 3-dimensional cosmology.     

Relative equilibrium solutions in the four body problem

Beyond the casen=3 little was known about relative equilibrium solutions of then-body problem up to recent years. Palmore's work provides in the general case much useful information. In the casen=4 he gives the totality of solutions when the four masses are equal and studies some degeneracies. We present here a survey of solutions for arbitrary masses, discussing the manifolds of degeneracy. The ordering of restricted potentials allows a counting of the number of bifurcation sets and different invariant manifolds. An analysis of linear stability is done in the restricted and general cases. As a result, values of the masses ensuring linear stability are given.     

Comments on the comment on the Guzman-Bianchi type-VII h solutions

In this paper, we again discuss the new Brans-Dicke-Bianchi type-VII _h perfect fluid solutions, first given by us (Guzman, 1989). It is shown that the objections presented by Lorenz-Petzold (1989) are misleading. The dust case =0 is discussed.