41 problems

Let $N$ point masses have prescribed positive masses, and consider their central configurations in the $N$-body problem. Chazy–Wintner–Smale conjecture. For any given masses, there…

In the Newtonian $N$-body problem, a central configuration is a configuration whose acceleration is proportional to its position relative to the center of mass; configurations are…

Near-minimal local-minima conjecture. For large numbers of particles, there are many local minima of $V$ with energies very close to the lower bound.

Let $n$ be the number of bodies and consider a non-singular collinear configuration with the Newtonian potential, whose associated inverse-problem matrix produces the resulting Pfa…

Non-degeneracy conjecture. There exists a permutation of the bodies such that $Q_n^m$ is a non-degenerate solution of the induced reduced system ${\cal RS}$.

Smale's finiteness conjecture. There are only finitely many normalized central configurations up to the rotational symmetry.

Let $n$ point masses have equal masses and interact through a homogeneous potential with exponent $\alpha$. Let $N(\alpha)$ denote the number of their central configurations. Hampt…

Consider the Newtonian $n$-body problem and configurations of $n$ bodies lying on a common circle, with the center of mass at the circle's center. A configuration is a co-circular…

Let $n$ point masses have positive masses $m_1,\ldots,m_n$, and let a central configuration be a non-collision configuration satisfying the central-configuration equations for some…

Let $N\geq 5$ positive masses be assigned to $N$ points on $S^1$, with a prescribed cyclic order. A convex planar central configuration of minimum type is a convex central configur…

Let four bodies with positive masses be arranged in a specified ordering along the boundary of their convex hull. A planar configuration is convex if no body lies inside or on the…

Let $n$ bodies have equal masses and lie on a common circle, with their center of mass coinciding with the center of the circle. A central configuration is one in which the acceler…

A convex configuration of four bodies has no body inside or on the convex hull formed by the other three bodies. A central configuration is one in which the acceleration vector of…

Let $n$ bodies have masses $m_1,dots,m_n$ and pairwise distances $r_{ij}$ satisfying the Albouy–Chenciner equations and the Cayley–Menger equations for planar configurations. A nor…

Consider planar equilateral 5-body central configurations with all masses positive in the Newtonian case. The regular pentagon and the star are the configurations with equal masses…

Let the five masses be nonzero, and suppose the potential exponent is a real number $A\geq 2$. A planar equilateral 5-body central configuration is a central configuration of five…

Wintner's conjecture. There are finitely many isometry classes of central configurations.

Let $\alpha=2$, let $m\geq 2$, and let $a_1\in E_1$ be a central configuration in the symmetric fixed-point setting. Its $2\pi/m$-nondegeneracy is equivalent to invertibility of th…

Let $\alpha\neq 2$, and let $a$ be a central configuration. The $2\pi$-nondegeneracy condition means that the relevant central-configuration orbit is a nondegenerate critical manif…

Let $m\geq 2$, and let $a_1$ be a central configuration in the symmetric setting where the relevant paths are $2\pi/m$-periodic. The configuration $a_1$ is $2\pi/m$-nondegenerate w…

Consider four-body central configurations, excluding equilateral central configurations. The degree of degeneracy is the dimension of the nontrivial kernel of the Hessian after rem…

Let $fr in \mathcal{X} \backslash \Delta$ be a central configuration, meaning a critical point of $\widetilde{\mathcal{U}}$, and let the trivial degeneracy subspace be … A central…

Consider the planar equal-mass $N$-body problem with homogeneous potential exponent $A$, and let $C_N(A)$ denote the number of equal-mass central configurations. Monotonicity conje…

Let $f$ be the function for the equal-mass planar $N$-body problem, and let $M(t)$ denote its Morse polynomial. Newtonian Morse-polynomial conjecture. For $A=3$, the Morse polynomi…

Consider the equal-mass planar six-body problem, with exponent $A$, function $f$, and Morse polynomial $M(t)$; let $P(t)$ be the Poincaré polynomial of the quotient configuration s…