65 problems

In the Newtonian $N$-body problem, let the moment of inertia with respect to the center of mass be the moment of inertia measured relative to the system's center of mass, and let a…

Let a relative equilibrium (RE) in the planar Newtonian $n$-body problem be generated by a central configuration, and let $\widehat U$ denote the relevant normalized potential whos…

Herman's conjecture. The set of non-wandering points for the flow of the $N$-body problem is nowhere dense on every energy level for $N\geq 3$.

In the $N$-body problem, let $N\geq 3$. A mass is dominant when it is much larger than the other masses. Moeckel's conjecture. There is no possibility of stable motions unless ther…

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…

Let $m$ denote the number of masses on each loop, and let the corresponding periodic orbits be represented by Fourier coefficients $a_k$, normalized by $a_1=1$. Asymptotic shape co…

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…

Conjectured six-body family. There is a family of solutions of this system parametrized by $\theta(T)=\theta_1$, for values of $\theta_1$ ranging from $\frac{\pi}{6}$ to $\pi$.

Consider an $N$-body solution with total energy $h$, potential energy $U(q(t))$, and relative equilibria, which are solutions evolving by rigid rotation and satisfying $U=2h$ throu…

Infinite periodic-solutions conjecture. There exist an infinite number of possible distinct periodic solutions to the general $N$-body problem.

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

Consider generic asymptotically flat vacuum initial data whose evolution may contain several black holes. Final State Conjecture. Generic asymptotically flat vacuum data should evo…

Rigidity conjecture. There are no partially rigid solutions of the $n$-body problem in $\mathbb{\mathbb{R}}^d$. Equivalently, the only solutions such that one of the mutual distanc…

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…

Let a solution of the three-body problem in $\mathbb{R}^d$ be partially rigid if at least one, but not all, of its mutual distances remains constant. Partial rigidity conjecture. T…

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…

Generic-motion conjecture. For generic initial condition, the solution of the $N$-body problem is globally defined on $\mathbb R$. As $t\to\pm\infty$, each body approaches either a…

Growth-rate classification conjecture. Every such orbit is a combination of $\mathcal P$, $\mathcal B$, and $\mathcal{OS}$. Conversely, every logically possible combination of…

Final-motion classification conjecture. For the $N$-body problem with $N\geq 4$, every logically possible combination of

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…

Anosov's conjecture. A genuine noncollision singularity can be found in a neighborhood of the solution in Moser and Mackay's work.

Noncollision singularity conjecture. Noncollision singularities exist in the Newtonian $N$-body problem for $N>3$.

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