Sur un theoreme de stabilite pour un potentiel homogene

The McGehee's study of the triple collision of the 3-body problem is here applied for the stability of an equilibrium. Let us consider the homogeneous Lagrangian: $$L = \frac{{\dot x^2 + \dot y^2 }}{2} + U(x,y)$$ whereU is polynomial, with degreek. We establish a necessary and sufficient condition onU for the stability of \(\omega (x = y = \dot x = \dot y = 0)\) .     

Merger of binary globular clusters: Case of unequal masses

Merger process of binary globular cluster is discussed for a pair of unequal-mass components. We calculated the case of mass ratio 10.5 by means of anN-body code with 6144 particles in total. We have found the followings. The mass exchange between the components takes place through the Roche-lobe overflow. In the early stages, however, the dynamical evolution is mainly governed by escape of particles from the system. As the particles escape carrying angular momentum with them, the separation between the component cluster shrinks. The time-scale of this shrinkage depends upon the size of the clusters. When a critical separation is reached, the orbital angular momentum is transferred unstably to the spins of the component clusters. This is the process of the synchronization instability which was found in a previous study on binary cluster of equal masses. As a result the component clusters merge into a single cluster. The structures of the mergers are quite similar among different cases except for the central cores which retain their initial central concentrations. In particular, the ellipticity and the rotation curve are quite close each other among models of different initial radii and of different mass ratios.     

A symplectic integrator for the symmetry reduced and regularised planar 3-body problem with vanishing angular momentum

We construct an explicit reversible symplectic integrator for the planar 3-body problem with zero angular momentum. We start with a Hamiltonian of the planar 3-body problem that is globally regularised and fully symmetry reduced. This Hamiltonian is a sum of 10 polynomials each of which can be integrated exactly, and hence a symplectic integrator is constructed. The performance of the integrator is examined with three numerical examples: The figure eight, the Pythagorean orbit, and a periodic collision orbit.     

Action-minimizing solutions of the one-dimensional <Emphasis Type="Italic">N</Emphasis>-body problem

We supplement the following result of C. Marchal on the Newtonian N-body problem: A path minimizing the Lagrangian action functional between two given configurations is always a true (collision-free) solution when the dimension d of the physical space \({\mathbb {R}}^d\) satisfies \(d\ge 2\). The focus of this paper is on the fixed-ends problem for the one-dimensional Newtonian N-body problem. We prove that a path minimizing the action functional in the set of paths joining two given configurations and having all the time the same order is always a true (collision-free) solution. Considering the one-dimensional N-body problem with equal masses, we prove that (i) collision instants are isolated for a path minimizing the action functional between two given configurations, (ii) if the particles at two endpoints have the same order, then the path minimizing the action functional is always a true (collision-free) solution and (iii) when the particles at two endpoints have different order, although there must be collisions for any path, we can prove that there are at most \(N! - 1\) collisions for any action-minimizing path.     

Bounded orbits in the elliptic restricted three-body problem

Special solutions of the planar rectilinear elliptic restricted 3-body problem are investigated for the limiting case e=1. Numerical integration is performed for primaries of equal masses. Starting values which define circular orbit solutions lead to bounded solutions if the initial radius a₀ is larger than 3.74 in units of the primaries' semimajor axis a. A comparison with the Eulerian two-fixedcentre problem is presented in order to understand qualitatively the characteristic features of bounded orbits and the transition to escape orbits.     

Numerical Exploration of Chermnykh’s Problem

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.     

Numerical simulations of granular dynamics: I. Hard-sphere discrete element method and tests

We present a new particle-based (discrete element) numerical method for the simulation of granular dynamics, with application to motions of particles on small solar system body and planetary surfaces. The method employs the parallel N-body tree code pkdgrav to search for collisions and compute particle trajectories. Collisions are treated as instantaneous point-contact events between rigid spheres. Particle confinement is achieved by combining arbitrary combinations of four provided wall primitives, namely infinite plane, finite disk, infinite cylinder, and finite cylinder, and degenerate cases of these. Various wall movements, including translation, oscillation, and rotation, are supported. We provide full derivations of collision prediction and resolution equations for all geometries and motions. Several tests of the method are described, including a model granular “atmosphere” that achieves correct energy equipartition, and a series of tumbler simulations that show the expected transition from tumbling to centrifuging as a function of rotation rate.     

Instability of an area-preserving polynomial mapping

Fixed points and eigencurves have been studied for the Hénon-Heiles mapping:x′=x+a (y?y ³),y′=y(x′?x′ ³). Eigencurves of order 21 proceed rapidly to infinity fora=1.78, but as ‘a’ decreases, they spiral around the origin repeatedly before escaping to infinity. Fixed pointsx _f on thex-axis have been located for the range 1≤a≤2.4, for ordersn up to 100. Their locations vary continuously witha, as do the eigencurves, and hyperbolic points remain hyperbolic. Forn=3 and 2.4≥a≥2.37, a very detailed study has been made of how escape occurs, with segments of an eigencurve mapping to infinity through various escape channels. Further calculations with ‘a’ decreasing to 2.275 show that this instability is preserved and that the eigencurve will spiral many times around the origin before reaching an escape channel, there being more than 34 turns fora=2.28. The rapid increase of this number is associated with the rapid decrease of the intersection angle between forward and backward eigencurves (at the middle homoclinic point), with decreasing ‘a’, this angle governing the outward motion. By a semi-topological argument, it is shown that escape must occur if the above intersection angle is nonzero. In the absence of a theoretical expression for this angle, one is forced to rely on the numerical evidence. If the angle should attain zero for a valuea=a _c>a_m,wherea _m .is the minimum value for which the fixed points exist, then no escape would be possible fora _c However, on the basis of calculations by Jenkins and Bartlett (1972) forn=6, and the results of the present article forn=3, it appears highly probable thata _c=a_m,and that escape from the neighborhood of a hyperbolic point is always possible. If there is escape from the hyperbolic fixed point forn=4,a=1.6, located atx _f=0.268, then the eigencurve must cross the apparently closed invariant curve of Hénon-Heiles which intersects thex-axis atx?±0.4, so that this curve cannot in fact be closed.     

On hierarchical cosmology

Progress in laboratory studies of plasmas and in the methods of transferring the results to cosmic conditions, together within situ measurements in the magnetospheres, are now causing a ‘paradigm transition’ in cosmic plasma physics. This involves an introduction ofinhomogeneous models with double layers, filaments, ‘cell walls’, etc. Independently, it has been discovered that the mass distribution in the universe is highly inhomogeneous; indeed,hierarchical. According to de Vaucouleurs, the escape velocity of cosmic structures is 10²–10³ times below the Laplace-Schwarzschild limit, leaving avoid region which is identified as a key problem in cosmology. It is shown that a plasma instability in the dispersed medium of the structures may produce this void and, hence, explain the hierarchical structure. The energy which is necessary may derive either from gravitation or from annihilation caused by a breakdown of cell walls. The latter alternative is discussed in detail. It leads to a ‘Fireworks Model’ of the evolution of the metagalaxy. It is questioned whether the homogeneous four-dimensional big bang model can survive in an universe which is inhomogeneous and three-dimensional.     

Étude du choc de deux petites masses en présence d'un troisième corps de masse prépondérante

The analytical study of the evolution in the rectilinear problem of three bodies, leads us to consider the collision between two bodies,M ₂ andM ₃, in the presence of the third body,M ₁. This problem, which seems to be difficult to approach in the general case, can be partly solved if the masses ofM ₂ andM ₃ are equal and can be neglected in regard toM ₁. In this particular case of the general problem, the mechanical study of a collision betweenM ₂ andM ₃, leads to two distinct types of collisions: ‘instantaneous collisions’, and ‘collisions with repetition’, according to the value of a parameter which depends on the position and the speed of the binaryM ₂ M ₃, relative toM ₁, in the collision. In the first type, the collision exchanges the speeds ofM ₂ andM ₃, while in the second type, there is a series of collisions succeeding each other.     

N-Body simulations of growth from 1 km planetesimals at 0.4 AU

We present N-body simulations of planetary accretion beginning with 1 km radius planetesimals in orbit about a 1 M_⊙ star at 0.4 AU. The initial disk of planetesimals contains too many bodies for any current N-body code to integrate; therefore, we model a sample patch of the disk. Although this greatly reduces the number of bodies, we still track in excess of 10⁵ particles. We consider three initial velocity distributions and monitor the growth of the planetesimals. The masses of some particles increase by more than a factor of 100. Additionally, the escape speed of the largest particle grows considerably faster than the velocity dispersion of the particles, suggesting impending runaway growth, although no particle grows large enough to detach itself from the power law size-frequency distribution. These results are in general agreement with previous statistical and analytical results. We compute rotation rates by assuming conservation of angular momentum around the center of mass at impact and that merged planetesimals relax to spherical shapes. At the end of our simulations, the majority of bodies that have undergone at least one merger are rotating faster than the breakup frequency. This implies that the assumption of completely inelastic collisions (perfect accretion), which is made in most simulations of planetary growth at sizes 1 km and above, is inappropriate. Our simulations reveal that, subsequent to the number of particles in the patch having been decreased by mergers to half its initial value, the presence of larger bodies in neighboring regions of the disk may limit the validity of simulations employing the patch approximation.     

Spin axis evolution of two interacting bodies

We consider the solid-solid interactions in the two body problem. The relative equilibria have been previously studied analytically and general motions were numerically analyzed using some expansion of the gravitational potential up to the second order, but only when there are no direct interactions between the orientation of the bodies. Here we expand the potential up to the fourth order and we show that the secular problem obtained after averaging over fast angles, as for the precession model of Boué and Laskar [Boué, G., Laskar, J., 2006. Icarus 185, 312-330], is integrable, but not trivially. We describe the general features of the motions and we provide explicit analytical approximations for the solutions. We demonstrate that the general solution of the secular system can be decomposed as a uniform precession around the total angular momentum and a periodic symmetric orbit in the precessing frame. More generally, we show that for a general n-body system of rigid bodies in gravitational interaction, the regular quasiperiodic solutions can be decomposed into a uniform precession around the total angular momentum, and a quasiperiodic motion with one frequency less in the precessing frame.     

Symetries du Probleme desN Corps,Regularisation des Solutions Centrales Singulieres

The newtonian problem ofn mass points bodies is invariant by several changes of spatio-temporal variables. These symmetries correspond to arbitrary choices of the referential and they are related via Noether's theorem or by its generalization to conservative quantities of the motion. Forn=2 the author has defined two families of symmetriesS ₁ andS ₂ changing the eccentricity of a solution. The family of symmetries,S ₁, is associated to the arbitrary choice of thezero level of the potential and may related unbounded and bounded solutions. The family of symmetries,S ₂, is related to a possibleaffinity of the configurations space. Via a symmetry of theS ₂ family a zero angular momentum solution is equivalent to a non-zero angular momentum solution. Via a product of two symmetries of each family, denoted byS ₁.S ₂, any solution of the two-body problem is equivalent to a circular solution. In this paper it is shown that some of these transformations may be generalized to symmetries changing the quantityC ² H in then-body problem, whereC is the angular momentum andH is the energy. The extension is easily made to central solutions of then-body problem because involving several synchroneous two-body problems. We consider for exposition then=3 case. The principal results may be resumed by the following propositions:

1. The two families of symmetriesS ₁ andS ₂ are described by a spatial transformation product of an instantaneous homothethy and an instantaneous rotation completed by a change of temporal variable.
2. TheS ₁ family of symmetries may relate unbounded and bounded central solutions of the same type, i.e. unaligned or aligned.
3. TheS ₂ family of symmetries may regularize multiple collisions among central solutions of the same type.

Therefore any central solution, via a symmetryS ₁ orS ₂ orS ₁.S ₂, is equivalent to a central circular solution of the same type. That is a form of regularization.     

On the double averaged three-body problem

The Hamiltonian of three point masses is averaged over fast variablel and ll (mean anomalies) The problem is non-planar and it is assumed that two of the bodies form a close pair (stellar three-body problem). Only terms up to the order of (a/á)⁴ are taken into account in the Hamiltonian, wherea andá are the corresponding semi-major axes. Employing the method of elimination of the nodes, the problem may be reduced to one degree of freedom. Assuming in addition that the angular momentum of the close binary is much smaller than the angular momentum of the motion of the binary around a third body, we were able to solve the equation for the eccentricity changes in terms of the Jacobian elliptic functions.     

The stability of vertical motion in the N-body circular Sitnikov problem

We present results about the stability of vertical motion and its bifurcations into families of 3-dimensional (3D) periodic orbits in the Sitnikov restricted N-body problem. In particular, we consider ν = N ? 1 equal mass primary bodies which rotate on a circle, while the Nth body (of negligible mass) moves perpendicularly to the plane of the primaries. Thus, we extend previous work on the 4-body Sitnikov problem to the N-body case, with N = 5, 9, 15, 25 and beyond. We find, for all cases we have considered with N ≥ 4, that the Sitnikov family has only one stability interval (on the z-axis), unlike the N = 3 case where there is an infinity of such intervals. We also show that for N = 5, 9, 15, 25 there are, respectively, 14, 16, 18, 20 critical Sitnikov periodic orbits from which 3D families (no longer rectilinear) bifurcate. We have also studied the physically interesting question of the extent of bounded dynamics away from the z-axis, taking initial conditions on x, y planes, at constant z(0) = z ₀ values, where z ₀ lies within the interval of stable rectilinear motions. We performed a similar study of the dynamics near some members of 3D families of periodic solutions and found, on suitably chosen Poincaré surfaces of section, “islands” of ordered motion, while away from them most orbits become chaotic and eventually escape to infinity. Finally, we solve the equations of motion of a small mass in the presence of a uniform rotating ring. Studying the stability of the vertical orbits in that case, we again discover a single stability interval, which, as N grows, tends to coincide with the stability interval of the N-body problem, when the values of the density and radius of the ring equal those of the corresponding system of N ? 1 primary masses.     

Dynamics of rotating triple systems

We investigate the dynamical evolution of 100 000 rotating triple systems with equal-mass components. The system rotation is specified by the parameter ω=?c²E, where c and E are the angular momentum and total energy of the triple system, respectively. We consider ω=0.1,1, 2, 4, 6 and study 20 000 triple systems with randomly specified coordinates and velocities of the bodies for each ω. We consider two methods for specifying initial conditions: with and without a hierarchical structure at the beginning of the evolution. The evolution of each system is traced until the escape of one of the bodies or until the critical time equal to 1000 mean system crossing times. For each set of initial conditions, we computed parameters of the final motions: orbital parameters for the final binary and the escaping body. We analyze variations in the statistical characteristics of the distributions of these parameters with ω. The mean disruption time of triple systems and the fraction of the systems that have not been disrupted in 1000 mean crossing times increase with ω. The final binaries become, on average, wider at larger angular momenta. The distribution of their eccentricities does not depend on ω and generally agrees with the theoretical law f(e)=2e. The velocities of the escaping bodies, on average, decrease with increasing angular momentum of the triple system. The fraction of the angles between the escaping-body velocity vector and the triple-system angular momentum close to 90° increases with ω. Escapes in the directions opposite to rotation and prograde motions dominate at small and large angular momenta, respectively. For slowly rotating systems, the angular momentum during their disruption is, on average, evenly divided between the escaping body and the final binary, whereas in rapidly rotating systems, about 80% of the angular momentum is carried away by the escaping component. We compare our numerical simulations with the statistical theory of triple-system disruption.     

Characteristics of plasma temperature variation during two-current-loop collisions

Characteristics of plasma temperature variations during two-current-loop collisions are described. It is revealed that plasma temperature has an oscillatory feature with damping amplitude and growing quasi-period in the case of anI-type collision. In the case of aY-type collision, if the initial current becomes strong enough, there also occur pulsations of the temperature. However, the temperature profile of anX-type collision is characterized by a single pulse only.     

Dynamics of Kepler problem with linear drag

We study the dynamics of Kepler problem with linear drag. We prove that motions with nonzero angular momentum have no collisions and travel from infinity to the singularity. In the process, the energy takes all real values and the angular velocity becomes unbounded. We also prove that there are two types of linear motions: capture–collision and ejection–collision. The behaviour of solutions at collisions is the same as in the conservative case. Proofs are obtained using the geometric theory of ordinary differential equations and two regularizations for the singularity of Kepler problem equation. The first, already considered in Diacu (Celest Mech Dyn Astron 75:1–15, 1999), is mainly used for the study of the linear motions. The second, the well known Levi-Civita transformation, allows to complete the study of the asymptotic values of the energy and to prove the existence of collision solutions with arbitrary energy.     

Periodic orbits of the planar collision restricted 3-body problem

Using the continuation method we prove that the circular and the elliptic symmetric periodic orbits of the planar rotating Kepler problem can be continued into periodic orbits of the planar collision restricted 3-body problem. Additionally, we also continue to this restricted problem the so called “comet orbits”. An erratum to this article can be found at