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.

**16:00**Marshall Hampton: A multifaceted approach to central configurations**16:30**Rick Moeckel: Partially rigid motions in the $n$-body problem**17:00**Piotr Zgliczyński: Central configurations - some rigorous computer assisted results

**16:00**Yangshanshan Liu: On the uniqueness of the planar 5-body central configuration with a trapezoidal convex hull**16:30**~~Zhiqiang Wang: The uniqueness of centered co-circular central configurations for the n-body problem~~Discussion on open problems and promising directions?**17:00**Yiyang Deng: The finiteness of relative equilibria of the 1+4 coorbital problem

**16:00**Alain Albouy: Computer algebra successes and failures in several problems concerning relative equilibria, with emphasis on an inverse problem**16:30**Kuo-Chang Chen: On finiteness of central configurations for the planar six-body problem by symbolic computations**17:00**Ke-Ming Chang: Eliminations and mass relations for zw-diagrams of the planar six-body problem

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.

Our approach is based on:

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

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

- 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,
- 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
- 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
- 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

(This is a join work with Antonio Carlos Fernandes.)

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