The Investigation of Stationary Points In Central Configuration Dynamics
A. E. Rosaev
备注：
14 pages with 3 figures, preprint version
发表信息：
Proceedings of Libration Point Orbits And Applications Conference, Parador de Aiguablava, Girona, Spain 10 - 14 June, 2002 Eds. by G. Gomez, M. W. Lo and .J.J. Masdemont, World Scientific Pub, New Jersey - London – Singapore - Hon Kong, (2003). p. 623-637
A system of N points, each having mass m, and a central mass M forming a planar central configuration, is considered. The equations of motion of a test particle are given and compared using different coordinates. For large values of N, even or odd, the equations are given too. The method used for the computation of some relevant sums can be important for practical applications. It is shown that the stationary solutions (libration points) can be determined from an algebraic equation of 5-th degree. If the mass of the particles is large, the outer libration points disappear. For the inner libration points the limit minimal distance has been detected. If m=M is small the solution obtained for the libration has as limit a solution similar to the for 3-body collinear points. We have also constructed a central configuration with 2N + 1 bodies, in which N new particles, each of mass m1= m, are placed at the libration points of our first system.
For the curved n-body problem, we show that the set of ordinary central configurations is away from most singular configurations in H^3, and away from a subset of singular configurations in S^3. We also show that each of the n!/2 geodesic ordinary central configurations for n masses has Morse index n-2. Then we get a direct corollary that there are at least (3n-4)(n-1)!/2 ordinary central configurations for given n masses if all ordinary central configurations of these masses are non-degenerate.
In this paper we generalize some results in \cite{Yu2021} concerning stacked central configurations. We can deal with the general homogeneous potential $U_α$ (containing the vortex case) in $\mathbb{R}^3$. We give the admissible set of $α$ for a convex central configuration (with respect to the Newtonian potential i.e. $α=3$). We discuss some properties of the regular $n$-gon co-circular central configurations. We also find that the stacked property is particular for central configurations by studying the S-balanced configuration case.
In an earlier paper \cite{yu2021Finiteness}, we showed that there are finitely many stationary configurations (consisting of equilibria, rigidly translating configurations, relative equilibria and collapse configurations) in the planar four-vortex problem. However, we only established finiteness of collapse configurations in the sense of prescribing a collapse constant. In this paper, by developing ideas of Albouy-Kaloshin and Hampton-Moeckel to do an analysis of the singularities, we further show that there really are finitely many collapse configurations in the four-vortex problem. This is a…
For the power-law potential $n$-body problem, we study a special kind of central configurations where all the masses lie on a circle and the center of mass coincides with the center of the circle. It is also called the centered co-circular central configuration. We get some symmetry results for such central configurations. We show that for positive numbers $α>0$ and integers $n\geq3$ satisfying $\frac{1}{n}\sum_{j=1}^{n-1}\csc^α\frac{jπ}{n}\leq1+\fracα{4}$, the regular $n$-gon with equal masses is the unique centered co-circular central configuration for the $n$-body problem with power-law pot…
Central configurations play an important role in the dynamics of the $n$-body problem: they occur as relative equilibria and as asymptotic configurations in colliding trajectories. We illustrate how they can be found as projective fixed points of self-maps defined on the shape space, and some results on the inverse problem in dimension $1$, i.e. finding (positive or real) masses which make a given collinear configuration central. This survey article introduces readers to the recent results of the author, also unpublished, showing an application of the fixed point theory.
We prove the existence of relative periodic solutions of the planar $N=\sum_{j=1}^n k_j$-body problem starting with $n$ bodies moving close to a non-degenerate central configuration and replacing each of them with clusters of $k_j$ bodies that move close to a small central configuration. We name these solutions carousel solutions. The proof relies on blow-up techniques for variational methods used in our previous work.
We consider, after Albouy-Moeckel, the inverse problem for collinear central configurations: given a collinear configuration of $n$ bodies, find positive masses which make it central. We give some new estimates concerning the positivity of Albouy-Moeckel pfaffians: we show that for any homogeneity $α$ and $n\leq 6$ or $n\leq 10$ and $α=1$ (computer-assisted) the pfaffians are positive. Moreover, for the inverse problem with positive masses, we show that for any homogeneity and $n\geq 4$ there are explicit regions of the configuration space without solutions of the inverse problem.
Central configurations and relative equilibria are an important facet of the study of the $N$-body problem, but become very difficult to rigorously analyze for $N>3$. In this paper we focus on a particular but interesting class of configurations of the 5-body problem: the equilateral pentagonal configurations, which have a cycle of five equal edges. We prove a variety of results concerning central configurations with this property, including a computer-assisted proof of the finiteness of such configurations for any positive five masses with a range of rational-exponent homogeneous potentials (…
We study the indices of the geodesic central configurations on $\H^2$. We then show that central configurations are bounded away from the singularity set. With Morse's inequality, we get a lower bound for the number of central configurations on $\H^2$.
We study the problem of planar central configurations with $N$ heavy bodies and $k$ bodies with arbitrary small masses. We derive the equation which describe the limit of light masses going to zero, which can be seen as the equation for central configurations in the anisotropic plane. Using computer rigorous computations we compute all central configurations for $N=2$ and $k=3,4$ and for the derived limit problem. We show that the results are consistent.
A stochastic optimization algorithm for analyzing planar central and balanced configurations in the $n$-body problem is presented. We find a comprehensive list of equal mass central configurations satisfying the Morse equality up to $n=12$. We show some exemplary balanced configurations in the case $n=5$, as well as some balanced configurations without any axis of symmetry in the cases $n=4$ and $n=10$.
The number of central configurations in the four body problem was proved to be finite, first by Hampton and Moeckel, then by Albouy and Kaloshin, when the masses are all positive. We prove that the four-body central configurations are finite for any four nonzero masses.
Two numerical algorithms for analyzing planar central and balanced configurations in the $(n+1)$-body problem with a small mass are presented. The first one relies on a direct solution method of the $(n+1)$-body problem by using a stochastic optimization approach, while the second one relies on an analytic-continuation method, which involves the solutions of the $n$-body and the restricted $(n+1)$-body problem, and the application of a local search procedure to compute the final $(n+1)$-body configuration in the neighborhood of the configuration obtained at the first two steps. Some exemplary …
An equilateral pentagon is a polygon in the plane with five sides of equal length. In this paper we classify the central configurations of the $5$-body problem having the five bodies at the vertices of an equilateral pentagon with an axis of symmetry. We prove that there are two unique classes of such equilateral pentagons providing central configurations, one concave equilateral pentagon and one convex equilateral pentagon, the regular one. A key point of our proof is the use of rational parameterizations to transform the corresponding equations, which involve square roots, into polynomial eq…
