ICMS 2024 - Session 9: Algorithms for relative equilibria in the N-body problem

Organizers

• Marshall Hampton (UMN Duluth, USA)
  
• Anders Jensen (Aarhus University, Denmark)
  

Aim and Scope

In 1998 Smale published a list of 18 problems for the 21st century.  The sixth problem is to prove that for N points with positive masses the number of relative equilibria in the planar Newtonian N-body problem is finite up to similarity transformations. Finiteness has been proved for N<5. For N=5 the result has been proved for generic choices of masses, with possible exceptional cases of masses described by polynomial equations. For higher N much less is known.  Some results have been found for related problems including relative equilibria of point vortices. A rich variety of computational algorithms play a key role in many arguments. This session is dedicated to discussing algorithms that will help providing results on relative equilibria in the N-body problem. Methods that may be discussed include Groebner bases, numerical system solving, real algebraic geometry methods, tropical and polyhedral arguments and interval arithmetic.

Accepted Talks

A multifaceted approach to central configurations

Marshall Hampton (University of Minnesota Duluth, USA)

This talk will introduce the problem of central configurations in the N-body problem, and survey some recent results on proving the finiteness of central configurations under specific constraints using a variety of approaches including tropical geometry, computational algebra, Morse theory, and interval arithmetic. A new formulation of equations for central configurations will be presented to illustrate some of the possible connections between these approaches.

Partially rigid motions in the $n$-body problem

Rick Moeckel (University of Minnesota, USA)

A solution of the $n$-body problem in $\R^d$ is a relative equilibrium if all of the mutual distance between the bodies are constant. In other words, the bodies undergo a rigid motion. Here we investigate the possibility of partially rigid motions, where some but not all of the distances are constant. In particular, a hinged solution is one such that exactly one mutual distance varies. The goal of this paper is to show that hinged solutions don't exist when $n=3$ or $n=4$. For $n=3$ this means that if 2 of the 3 distances are constant so is the third and for $n=4$, if 5 of the 6 distances are constant, so is the sixth. These results hold independent of the dimension $d$ of the ambient space.

Central configurations - some rigorous computer assisted results

Piotr Zgliczyński (Jagiellonian University, Poland)

I will give an overview of our of recent computer assisted proofs for the rigorous count of central configurations.

Our approach is based on:

• the use of interval arithmetics methods, for example the Newton-Krawczyk operator
• a priori bounds for central configurations

This allows to obtain an rigorous listing of all central configurations when masses are away from zero and there are no bifurcation nearby in the mass space, we have done for equal masses in the planar case for $n=5,6,7$ and in the spatial case for $n=5,6$.

To extend this approach to all masses the following issues has to be solved:

• understanding of restricted N+k problems (N-big masses and k "massless" bodies) and their continuation to full problem
• the rigorous analysis of bifurcations

References:

1. M. Moczurad, P. Zgliczyński, Central configurations in planar $n$-body problem for $n=5,6,7$ with equal masses, arXiv:1812.07279, Celestial Mechanics and Dynamical Astronomy, (2019) 131: 46,
2. M.~Moczurad, P. Zgliczyński, Central configurations in spatial $n$-body problem for $n=5,6$ with equal masses, Celestial Mechanics and Dynamical Astronomy, (2020) 132:56
3. J.L. Figueras,W. Tucker, P. Zgliczynski The number of relative equilibria in the PCR4BP, Journal of Dynamics and Differential Equations (2022), https://doi.org/10.1007/s10884-022-10230-6
4. M.~Moczurad, P. Zgliczyński, Central configurations on the plane with $N$ heavy and $k$ light bodies, Communications in Nonlinear Science and Numerical Simulation, 114 (2022), 106533

On the uniqueness of the planar 5-body central configuration with a trapezoidal convex hull

Yangshanshan Liu

In order to apply Morse's critical point theory, we use mutual distances as coordinates to discuss a kind of central configuration of the planar Newtonian 5-body problem with a trapezoidal convex hull, i.e., four of the five bodies are located at the vertices of a trapezoid, and the fifth one is located on one of the parallel sides. We show that there is at most one central configuration of this geometrical shape for a given cyclic order of the five bodies along the convex hull. In addition, if the parallel side containing the three collinear bodies is strictly shorter than the other parallel side, the configuration must be symmetric, i.e., the trapezoid is isosceles, and the last body is at the midpoint of the shorter parallel side.

The uniqueness of centered co-circular central configurations for the n-body problem

Zhiqiang Wang

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 $\alpha>0$ and integers $n\geq3$ satisfying $$\frac{1}{n}\sum_{j=1}^{n-1}\csc^{\alpha}\frac{j\pi}{n}\leq1+\frac{\alpha}{4},$$ the regular $n$-gon with equal masses is the unique centered co-circular central configuration for the $n$-body problem with power-law potential $U_{\alpha}$. It quickly follows that for the Newtonian $n$-body problem (in the case $\alpha=1$) and $n\leq6$, the regular $n$-gon is the unique centered co-circular central configuration.

The finiteness of relative equilibria of the 1+4 coorbital problem

Yiyang Deng (Chongqing Technology and Business University, China)

In this talk, we will discuss the relative equilibria of the 1+4 coorbital problem which is the limit case of the planar Newtonian 5–body problem when three masses tend to zero. For generic choices of the infinitesimal masses, the number of relative equilibria of the 1+4 coorbital problem is finite. Furthermore, we will discuss the linear stability of the relative equilibria of the 1+N coorbital problem for same special case.

Computer algebra successes and failures in several problems concerning relative equilibria, with emphasis on an inverse problem

Alain Albouy (Observatoire de Paris, CNRS, France)

Polynomials with integer coefficients are quickly handled by computers. Factorizations and resultants are well-defined and reproducible operations (the result does not depend on the software). They are suitable for mathematical proofs. I will recall some results about relative equilibria which were obtained by this tool. I will give some new results about the inverse problem: given a configuration, find the masses that make it a central configuration. The first step of the study accepts negative masses. I will ask for configurations that accept two nonproportional sets of masses. The 4-body case is extremely simple (MacMillan, Bartky 1932, Albouy, Moeckel 2000) but the 5-body case is complicated and not fully described (Piña 2022). I will show that there are two types of solutions, and exclude one type under the hypothesis of positive masses. I will show computer algebra attempts for other cases.
(This is a join work with Antonio Carlos Fernandes.)

On finiteness of central configurations for the planar six-body problem by symbolic computations

Kuo-Chang Chen (National Tsing Hua University, Taiwan)

In Albouy-Kaloshin’s work on finiteness of central configurations for the 5-body problems (Ann. Math. 2012), bicolored graphs called zw-diagrams were introduced for possible scenarios when the finiteness conjecture fails, and proving finiteness amounts to exclusions of central configurations associated to these diagrams. In this talk we discuss computational complexity arise in the case of six bodies, and present two algorithms for determination of possible zw-diagrams. For the planar six-body problem, our algorithms can quickly narrow down the finiteness problem to 86 zw-diagrams. This is joint work with Ke-Ming Chang.

Eliminations and mass relations for zw-diagrams of the planar six-body problem

Ke-Ming Chang (National Tsing Hua University, Taiwan)

In a recent work on finiteness of central configurations for the planar six-body problem (J. Symb. Comp. 2024), we determine possible zw-diagrams, and for each zw-diagram we determine possible orders of positions, separations, and distance variables. In particular, we show there are at most 86 zw-diagrams if there exist infinitely many central configurations. In this talk we show that 62 of them are impossible except for masses in some co-dimension 2 variety in the mass space. Apart from standard tools from elimination theory and symbolic computations, we shall demonstrate some exceptional cases which require detailed analysis for asymptotic orders of variables. This is joint work with Kuo-Chang Chen.