40 problems

For every choice of positive masses $m_1,m_2,m_3,m_4>0$ and every prescribed cyclic ordering of four bodies, there exists exactly one strictly convex planar four-body central confi…

For every integer $n\ge 4$ and every choice of positive masses $m_1,\ldots,m_n$, consider the Newtonian $n$-body problem in $\mathbb{R}^3$. Let $\mathcal{S}_n$ be the set of collis…

Consider the spatial $N$-body problem with $N\geq 3$. Fix the center of mass at the origin. For each energy level $e$, let $H_e^{-1}(0)$ denote the corresponding energy surface, an…

Let $N$ point particles have masses $m_k>0$ and positions $\bi{q}_k$, with homogeneous potential … for $\alpha\ne0$. Define the moment of inertia and configurational measure by … A…

Let $N>3$, let $\omega\in(0,N)$, and let $\xi\in\Xi_N$. Let $q^{\omega}$ be an action minimizer of $\mathcal{A}_{\omega}$ among all loops in $\Lambda^{D_N}_{\xi}$ satisfying the co…

Let $F$ be the function defined in the paper's equation for the twisted $(2,n)$-crowns, with $n\geq 4$. Its positive zeros are denoted by $z_1$ and $z_2$. Uniqueness conjecture. Th…

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 $q(t)=(q_1(t),q_2(t),q_3(t))$ be a trajectory in the spatial Circular Restricted Three-Body Problem, in the low-energy range and near the primaries. A trajectory is bi-normal t…

Alekseev's conjecture. The set $OM$ has Lebesgue measure zero.

Alekseev's dense collision-orbit conjecture. Is there an open subset $\mathcal{U}$ of the phase space such that for a dense subset of initial conditions the associated trajectories…

Let $M_E$ be the negative-energy configuration space equipped with the Jacobi–Maupertuis metric $d_E$, let $\partial M_E$ denote its Hill boundary, and let $0$ be the origin config…

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…

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…

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…

A transition chain is a chain of hyperbolic invariant tori whose stable and unstable manifolds can be shadowed by drifting orbits in a near-integrable Hamiltonian system. Arnold's…

In the outer, retrograde configuration, let $\mathcal{D}_{{{{{{{{{\rm t} t} t} t} t} t} t} t} s}$ be the $8$-dimensional phase-space region almost completely filled with a positive…

Let ${\cal M}({\cal E})=\{({\rm r},{\rm G},{\rm g}):{\rm E}({\rm r},{\rm G},{\rm g})={\cal E}\}$ be the lifted level manifolds of the Euler integral, and let…

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…

Generalized Marchal's conjecture. For any odd number of bodies, the $n$-gon choreography and the $n$-body figure eight are in the same continuation class.

Marchal's conjecture. The three-body equilateral triangle of Lagrange and the three-body figure eight are in the same continuation class.

Assume that $\mathcal{K}$ is not central. Let $M ein\mathbb{N}$ be as in the symmetry condition, and let $(\Delta\theta)_{\min}$ be the minimum total angle compatible with…

Consider a balanced family in the three-body problem in $\mathbb{R}^4$, with functions $h$ and $k$ defined along the family. Critical-point conjecture. On each balanced family, the…