114 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…

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 $N$-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…

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…

Let the oscillatory motions be the trajectories of the 3 Body Problem whose final motion is oscillatory, and let their set in the relevant phase space be equipped with Lebesgue mea…

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 $\mu=m_1/(m_0+m_1)$ be the mass ratio of Jupiter to the combined mass of the Sun and Jupiter, and consider the restricted planar 3-body problem in a regime where the asteroid b…

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.

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 ${\cal RS}$.

The Newtonian $n$-body problem describes the motion of $n$ massive particles under mutual gravitational attraction; in particular, the 3-body problem is the case $n=3$. A periodic…

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

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…

Arnol'd's zero-measure conjecture. The Lebesgue measure of the set $\mathcal{OS}^+\cap\mathcal{OS}^-$ is equal to zero.

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 $W^{u,s}_E$ and $W^{u,s}_C$ denote the invariant manifolds associated with the configurations $\mathcal{T}_E$ and $\mathcal{T}_C$ in the $(n+2)$-body problem, and consider thei…

1/1 secondary-resonance conjecture. The coexistence of the $lpha$ and $beta$-regimes of motion is explained by a 1/1 secondary resonance.

Let $u(t)$ denote the relevant solution of the damped Kepler problem, and let the open set in Theorem be the set of initial conditions described there. The open-set characterizatio…

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…

Critical values at infinity conjecture. The “critical values at infinity” of the energy are six values deduced from these models.

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…