For the Newtonian (gravitational) n-body problem in the Euclidean d-dimensional space, d≥2, the simplest possible periodic solutions are provided by circular relative equilibria, (RE) for short, ...namely solutions in which each body rigidly rotates about the center of mass and the configuration of the whole system is constant in time and central (or, more generally, balanced) configuration. For d≤3, the only possible (RE) are planar, but in dimension four it is possible to get truly four dimensional (RE).
A classical problem in celestial mechanics aims at relating the (in-)stability properties of a (RE) to the index properties of the central (or, more generally, balanced) configuration generating it. In this paper, we provide sufficient conditions that imply the spectral instability of planar and non-planar (RE) in R4 generated by a central configuration, thus answering some of the questions raised in 14, Page 63. As a corollary, we retrieve a classical result of Hu and Sun 11 on the linear instability of planar (RE) whose generating central configuration is non-degenerate and has odd Morse index, and fix a gap in the statement of 6, Theorem 1 about the spectral instability of planar (RE) whose (possibly degenerate) generating central configuration has odd Morse index.
The key ingredient is a new formula of independent interest that allows to compute the spectral flow of a path of symmetric matrices having degenerate starting point, and a symplectic decomposition of the phase space of the linearized Hamiltonian system along a given (RE) which is inspired by Meyer and Schmidt's planar decomposition 13 and which allows us to rule out the uninteresting part of the dynamics corresponding to the translational and (partially) to the rotational symmetry of the problem.
Abstract
In this work, we use the energy and angular momentum integrals as a resource for applying Hill’s approach to the general three-body problem. As a result, we obtain theorems on the Lagrange ...stability and Hill stability in the general three-body problem. Also, specific features of the general and restricted three-body problems are discussed.
Abstract
This paper presents an efficient CUDA-based implementation of a nonspherical discrete element method where irregular particles are described by using polyhedrons. Two strategies are employed ...to exploit the parallelism of the numerical method. One is to perform contact detection based on the contact pair level instead of the traditional particle level. The second is to reduce the computational burden of each kernel function by allocating thread blocks reasonably. Contact detection between potential contact pairs is the most complicated, time-consuming, and essential process for the polyhedral discrete element method. The linear bounding volume hierarchies are introduced to fix this issue. The hierarchies of the bounding volume tree are organized in a spatially coherent way. Such a structure can minimize branch divergence and is very suitable for parallel implementation with GPU. Two numerical examples are presented to show the performance of the code. It is found from the scenario of two sphere collision that improving the mesh resolution of polyhedral particles can reduce the computational error while slowing down the computational speed correspondingly. A trade-off must be made between accuracy and efficiency. The other example of self-gravitating aggregation demonstrates the code is convergent, stable, and highly efficient. Particularly, with a mainstream GPU, the proposed method easily performs hundreds of times faster than the serial CPU code that does the same function.
Abstract
In this paper, we present a large-scale dynamical survey of the trans-Neptunian region, with particular attention to mean-motion resonances (MMRs). We study a set of 4121 trans-Neptunian ...objects (TNOs), a sample far larger than in previous works. We perform direct long-term numerical integrations that enable us to examine the overall dynamics of the individual TNOs as well as to identify all MMRs. For the latter purpose, we apply an own-developed FAIR method that allows the semi-automatic identification of even very high-order MMRs. Apart from searching for the more frequent eccentricity-type resonances that previous studies concentrated on, we set our method to allow the identification of inclination-type MMRs, too. Furthermore, we distinguish between TNOs that are locked in a given MMR throughout the whole integration time span (10
8
yr) and those that are only temporarily captured in resonances. For a more detailed dynamical analysis of the trans-Neptunian space, we also construct dynamical maps using test particles. Observing the fine structure of the 34–80 au region underlines the stabilizing role of the MMRs, with the regular regions coinciding with the positions of real TNOs.
KAM theory owes most of its success to its initial motivation: the application to problems of celestial mechanics. The masterly application was offered by Arnold in the 60s who worked out a theorem, ...that he named the “Fundamental Theorem” (FT), especially designed for the planetary problem. However, FT could be really used at that purpose only when, about 50 years later, a set of coordinates constructively taking the invariance by rotation and close-to-integrability into account was used. Since then, some progress has been done in the symplectic assessment of the problem, and here we review such results.
We introduce an algebraic method to study local stability in the Newtonian n-body problem when certain symmetries are present. We use representation theory of groups to simplify the calculations of ...certain eigenvalue problems. The method should be applicable in many cases, we give two main examples here: the square central configurations with four equal masses, and the equilateral triangular configurations with three equal masses plus an additional mass of arbitrary size at the center. Using representation theory of finite groups, we explicitly found the eigenvalues of certain 8×8 Hessians in these examples, with only some simple calculations of traces. We also studied the local stability properties of corresponding relative equilibria in the four-body problems.
We prove that all non-degenerate relative equilibria of the planar Newtonian n–body problem can be continued to spaces of constant curvature κ, positive or negative, for small enough values of this ...parameter. We also compute the extension of some classical relative equilibria to curved spaces using numerical continuation. In particular, we extend Lagrange's triangle configuration with different masses to both positive and negative curvature spaces.
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 pre-sented research included writing original computer code, building a computer cluster, preforming simulations and comparing the results.
It has been shown by Chen that if all masses are in the same proper vector subspace of Rd(d≥2) on one of the free boundaries, local minimizers connecting the two free boundaries have no collision on ...that boundary. However, not much is known for other types of boundary configurations.
In this paper, we consider minimizers connecting two free boundaries and study possible quadruple collisions under the following four symmetric configurations: isosceles trapezoid configuration, rectangular configuration, double isosceles configuration and diamond configuration. Whenever the boundary configuration of four bodies coincides with one of the four symmetric configurations, we show that these four bodies are free of quadruple collision on that boundary set. New ideas and detailed analysis are introduced in order to construct deformed paths in some of the cases and estimate their action values. As its applications, we show the existence of several sets of periodic orbits in the planar four-body and five-body problems.
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