Classical N-body Problem Research Articles

PRODUCTIVITY OF A LOW-BUDGET COMPUTER CLUSTER APPLIED TO OVERCOME THE N-BODY PROBLEM

The classical n-body problem in physics addresses the prediction of individual motions of a group of celestial bodies under gravitational forces and has been studied since Isaac Newton formulated his laws. Nowadays the n-body problem has been recognized in many more fields of science and engineering. Each problem of mutual interaction between objects forming a dynamic group is called as the n-body problem. The cost of the direct algorithm for the problem is O(n2) and is not acceptable from the practical point of view. For this reason cheaper algorithms have been developed successfully reducing the cost to O(nln(n)) or even O(n). Because further improvement of the algorithms is unlikely to happen it is the hardware solutions which can still accelerate the calculations. The obvious answer here is a computer cluster that can preform the calculations in parallel. This paper focuses on the performance of a low-budget computer cluster created on ad hoc basis applied to n-body problem calculation. In order to maintain engineering valuable results a real technical issue was selected to study. It was Discrete Vortex Method that is used for simulating air flows. The presented research included writing original computer code, building a computer cluster, preforming simulations and comparing the results.

Dziobek Equilibrium Configurations on a Sphere

We investigate the n-body problem on a sphere with a general interaction potential that depends on the mutual distances. We focus on the equilibrium configurations, especially on the Dziobek equilibrium configurations, which is an analogy of Dziobek central configurations of the classical n-body problem. We obtain a criterion and then reduce it to two sets of equations. Then we apply these equations to the curved n-body problem in \({\mathbb {S}}^3\). In the end, we find the derivative of the Cayley-Menger determinant.

Simulation of the Interaction of Oppositely Directed Particle Flows

A mathematical model of the early stage of the formation of interstellar clouds resulting from the interaction of oppositely directed cosmic particle flows and evolution of the clouds into planetary systems is proposed. An important role in the evolution is played by mechanical energy dissipation caused mainly by particle collisions. The simulation is based on the classical n-body problem and the Newtonian theory of instantaneous collisions of rigid bodies with a relative velocity recovery factor smaller than unity. The performed computations suggest that this model is applicable in cosmogonic theories of planetary system formation.

Results on equality of masses for choreographic solutions of n-body problems

We prove that equally spaced choreographic solutions of a large class of n-body problems, including the classical n-body problem and a subset of quasi-homogeneous n-body problems, have equal masses if the dimension of the space spanned by the point masses is n − 1, n − 2, or, if n is odd, if the dimension is n − 3. If n is even and the dimension is n − 3, then all masses with an odd label are equal and all masses with an even label are equal. Additionally, we prove that the same results hold true for any solution of an n + 1-body problem for which n of the point masses behave like an equally spaced choreography and the n + 1th point mass is fixed at the origin. Furthermore, we deduce that if the curve along which the point masses of a choreography move has an axis of symmetry, the masses have to be equal if n = 3 and that if n = 4, if three of the point masses behave as stated and the fourth mass is fixed at a point, the masses of the first three point masses are all equal. Finally, we prove for the n-body problem in spaces of negative constant Gaussian curvature that if n < 6, n ≠ 4, equally spaced choreographic solutions should have equal masses, and for n = 4, the even labeled masses are equal and the odd labeled masses are equal and that the same holds true for the n-body problem in spaces of positive constant Gaussian curvature, as long as the point masses do not move along a great circle. Additionally, we show that these last two results are also true for any solution to the n + 1-body problem in spaces of negative constant Gaussian curvature and the n + 1-body problem in spaces of positive constant Gaussian curvature, respectively, for the case that n of the point masses behave like an equally spaced choreography and the n + 1 is fixed at a point.

An index theory for asymptotic motions under singular potentials

We develop an index theory for parabolic and collision solutions to the classical n-body problem and we prove sufficient conditions for the finiteness of the spectral index valid in a large class of trajectories ending with a total collapse or expanding with vanishing limiting velocities. Both problems suffer from a lack of compactness and can be brought in a similar form of a Lagrangian System on the half time line by a regularising change of coordinates which preserve the Lagrangian structure. We then introduce a Maslov-type index which is suitable to capture the asymptotic nature of these trajectories as half-clinic orbits: by taking into account the underlying Hamiltonian structure we define the appropriate notion of geometric index for this class of solutions and we develop the relative index theory.

Fast and scalable evaluation of pairwise potentials

Pair potentials or kernels, ψ(|r|), play a critical role in a number of areas; these include biophysics, electrical engineering, fluid dynamics, diffusion physics, solid state physics, and many more. The need to evaluate these potentials rapidly for N particles gives rise to the classical N-body problem. In this paper, we present scalable parallel algorithms for evaluation of these potentials for highly non-uniform distributions. The underlying methodology for evaluating these potentials relies on the accelerated Cartesian expansion (ACE) framework that is quasi-kernel-independent with the requirement that the kernel be differentiable with known derivatives. The results presented demonstrate the accuracy control, low cost, and parallel scalability offered by this method for several example kernels and distributions of up to 5 billion particles on 16384 CPU cores. Potential applications of the algorithm include various disciplines of computational physics, engineering, machine learning, among others.

Braids with the symmetries of Platonic polyhedra in the Coulomb (N+1)-body problem

We take into account the Coulomb (N+1)-body problem with N=12,24,60. One of the particles has positive charge Q > 0, and the remaining N have all the same negative charge q < 0. These particles move under the Coulomb force, and the positive charge is assumed to be at rest at the center of mass. Imposing a symmetry constraint, given by the symmetry group of the Platonic polyhedra, we were able to compute periodic orbits, using a shooting method and continuation with respect to the value Q of the positive charge.In the setting of the classical N-body problem, the existence of such orbits is proved with Calculus of Variation techniques, by minimizing the action functional. Here this approach does not seem to work, and numerical computations show that the orbits we compute are not minimizers of the action.

Circular Non-collision Orbits for a Large Class of n-Body Problems

We prove for a large class of n-body problems including a subclass of quasihomogeneous n-body problems, the classical n-body problem, the n-body problem in spaces of negative constant Gaussian curvature and a restricted case of the n-body problem in spaces of positive constant curvature for the case that all masses are equal and not necessarily constant that any solution for which the point masses move on a circle of not necessarily constant size has to be either a regular cocircular homographic orbit in flat space, or a regular polygonal rotopulsator in curved space, under the constraint that the minimal distance between point masses attains its minimum in finite time. Additionally, we prove that the same holds true if we add an extra mass at the center of that circle and find an explicit formula for the mass of each point particle in terms of the radius of the circle. Finally, we prove that for each order of the masses there is at most one cocircular homographic orbit for the case that the masses need not be constant.

Equilibrium Solutions of the Logarithmic Hamiltonian Leapfrog for the N-body Problem

Abstract We prove that a second-order logarithmic Hamiltonian leapfrog for the classical general N-body problem (CGNBP) designed by Mikkola and Tanikawa and some higher-order logarithmic Hamiltonian methods based on symmetric multicompositions of the logarithmic algorithm exactly reproduce the orbits of elliptic relative equilibrium solutions in the original CGNBP. These methods are explicit symplectic methods. Before this proof, only some implicit discrete-time CGNBPs proposed by Minesaki had been analytically shown to trace the orbits of elliptic relative equilibrium solutions. The proof is therefore the first existence proof for explicit symplectic methods. Such logarithmic Hamiltonian methods with a variable time step can also precisely retain periodic orbits in the classical general three-body problem, which generic numerical methods with a constant time step cannot do.

Effectiveness Analyses of an Electrospray Sustainer Engine Installed on a Spacecraft Maneuvering in Vicinity of an Asteroid

Motion in vicinity of complex bodies is unstable due to the dynamic changing of the irregular gravitational field. Standard spacecraft power plants designed for interplanetary flights do not provide a stable orbital motion near irregular bodies. In this paper the effectiveness analysis of an electrospray engine power plant used as a sustainer engine in an asteroid exploration mission is considered. Authors describe mathematical models of the spacecraft's motion in an unstable gravitational field as the classical n-body problem. Using simple control schemes for the spacecraft's orbital motion allows basic requirements for the spacecraft's power plant to be defined: thrust level and exhaust velocity. Ion Electrospray Systems are designed to be installed on small satellites, however, effectiveness analyses shows that the characteristics of the calculated engine system is close enough to the characteristics of an electrospray system. Authors demonstrate that ion Electrospray System modules are effective enough to be installed on spacecraft in asteroid exploration missions.

Novel isochronous N-body problems featuring N arbitrary rational coupling constants

A novel class of N-body problems is identified, with N an arbitrary positive integer (N ≥ 2). These models are characterized by Newtonian (“accelerations equal forces”) equations of motion describing N equal point-particles moving in the complex z-plane. These highly nonlinear equations feature N arbitrary coupling constants, yet they can be solved by algebraic operations and if all the N coupling constants are real and rational the corresponding N-body problem is isochronous: its generic solutions are all completely periodic with an overall period T independent of the initial data (but many solutions feature subperiods T/p with p integer). It is moreover shown that these models are Hamiltonian.

Hill Stability of Configurations in the Full N-Body Problem

AbstractRigorous results on Hill Stability for the classical N-body problem are in general unknown for N ≥ 3, due to the complex interactions that may occur between bodies and the many different outcomes which may occur. However, the addition of finite density for the bodies along with a rigidity assumption on their mass distribution allows for Hill stability to be easily established. In this note we generalize results on Hill stability developed for the Full 3-body problem and show that it can be applied to the Full N-body problem. Further, we find that Hill Stability concepts can be applied to identify types of configurations which can escape and types which cannot as a function of the system energy.

A large class of solvable discrete-time many-body problems

A class of N-body problems is identified, characterized by second-order discrete-time evolution equations determining the motion in the complex z-plane of an arbitrary number N of points zn ≡ zn(ℓ), where \documentclass[12pt]{minimal}\begin{document}$\ell =0,\pm 1,\pm 2,{\kern -2.1pt}...$\end{document}ℓ=0,±1,±2,... is the discrete-time independent variable. Both these equations of motion, and the solution of their initial-value problem, only involve algebraic operations: finding the zeros of explicitly known polynomials of degree N in z, finding the eigenvectors and eigenvalues of explicitly known N × N matrices. These models feature an arbitrarily large number of arbitrary parameters (“coupling constants”).

The Lie–Poisson structure of the reduced n-body problem

The classical n-body problem in d-dimensional space is invariant under the Galilean symmetry group. We reduce by this symmetry group using the method of polynomial invariants. One novelty of our approach is that we do not fix the centre of mass but rather use a momentum shifting trick to change the kinetic part of the Hamiltonian to arrive at a new, dynamically equivalent Hamiltonian which is easier to reduce. As a result we obtain a reduced system with a Lie–Poisson structure which is isomorphic to , independently of d. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. This splitting allows us to construct a Poisson integrator for the reduced n-body problem which is efficient away from collisions for n = 3. In particular, we could integrate the figure eight orbit in 18 time steps.

Translation invariance of weak KAM solutions of the Newtonian $N$-body problem

We consider in this note the Hamilton-Jacobi equation H(x, dx u) = c, where c \geq 0, of the classical N-body problem in an Euclidean space E of dimension k \geq 2. The fixed points of the Lax-Oleinik semigroup are global viscosity solutions for the critical value of the constant (c = 0) also called weak KAM solutions. We show that all these solutions are invariant under the action of E by translations on the space of configurations. We deduce the existence of non-invariant solutions for the super-critical equations (c > 0).

The non-existence of centre-of-mass and linear-momentum integrals in the curved N-body problem

Abstract. We provide a class of orbits in the curved N-body prob-lem for which no point that could play the role of the centre of mass isfixed or moves uniformly along a geodesic. This proves that the equa-tions of motion lack centre-of-mass and linear-momentum integrals. 1. IntroductionThe importance of the centre-of-mass and linear-momentum integrals inthe classical N-body problem and its generalization to quasihomogeneouspotentials, [3], is never enough emphasized. From integrable systems, tothe Smale and Saari conjectures, [30], [37], to the study of singularities,[4], [10], these integrals have played an essential role towards answeringthe most basic questions of particle dynamics, in general, and celestialmechanics, inparticular. But theyseem tocharacterize onlythe Euclideanspace. Away from zero curvature, they disappear, probably because of thesymmetry loss that occurs in the ambient space. We will show here thatthey don’t exist in the curved N-body problem (the natural extension ofthe Newtonian N-body problem to spaces of constant curvature).This note follows from a discussion we had with Ernesto P´erez-Chavelaand Guadalupe Reyes Victoria of Mexico City. Given the absence of theabove mentioned integrals in discretizations of Einstein’s field equations,such as those of Levi-Civita, [23], [24], Einstein, Infeld, Hoffmann, [13],and Fock, [14], we had taken this property for granted in the curved N-body problem. But as we could not immediately support this claim with

Some problems on the classical n-body problem

Our idea is to imitate Smale's list of problems, in a restricted domain of mathematical aspects of Celestial Mechanics. All the problems are on the n-body problem, some with different homogeneity of the potential, addressing many aspects such as central configurations, stability of relative equilibrium, singularities, integral manifolds, etc. Following Steve Smale in his list, the criteria for our selection are: (1) Simple statement. Also preferably mathematically precise, and best even with a yes or no answer. (2) Personal acquaintance with the problem, having found it not easy. (3) A belief that the question, its solution, partial results or even attempts at its solution are likely to have great importance for the development of the mathematical aspects of Celestial Mechanics.

Lagrange-Pizzetti theorem for the quasihomogeneous N-body problem

We consider the N-body problem given by quasihomogeneous force functions of the form C1/r a + C2/r b (C1, C2, a, b positive constants and a ⩽ b) and address the fundamentals of homographic solutions. Generalizing techniques of the classical N-body problem, we prove that if the homographic solution is not flat, then it is homothetic; if the homographic solution is flat, then it is planar. Using them, we prove an analogue of the Lagrange-Pizzetti theorem for quasihomogeneous potentials.

REBOUND: an open-source multi-purposeN-body code for collisional dynamics

REBOUND is a new multi-purpose N-body code which is freely available under an open-source license. It was designed for collisional dynamics such as planetary rings but can also solve the classical N-body problem. It is highly modular and can be customized easily to work on a wide variety of different problems in astrophysics and beyond. REBOUND comes with three symplectic integrators: leap-frog, the symplectic epicycle integrator (SEI) and a Wisdom-Holman mapping (WH). It supports open, periodic and shearing-sheet boundary conditions. REBOUND can use a Barnes-Hut tree to calculate both self-gravity and collisions. These modules are fully parallelized with MPI as well as OpenMP. The former makes use of a static domain decomposition and a distributed essential tree. Two new collision detection modules based on a plane-sweep algorithm are also implemented. The performance of the plane-sweep algorithm is superior to a tree code for simulations in which one dimension is much longer than the other two and in simulations which are quasi-two dimensional with less than one million particles. In this work, we discuss the different algorithms implemented in REBOUND, the philosophy behind the code's structure as well as implementation specific details of the different modules. We present results of accuracy and scaling tests which show that the code can run efficiently on both desktop machines and large computing clusters.

Parker-Sochacki method for solving systems of ordinary differential equations using graphics processors

The Parker-Sochacki method, which is used for solving systems of ordinary differential equations, and implementation of this method on graphics processors are described. The solution to the classical N-body problem is considered as a test. The algorithm makes it possible to effectively use massive parallel graphics processors and provides acceptable accuracy with multiple time reduction, as compared to processors of a conventional architecture.