24 problems

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…

Arnold diffusion concerns global instability in Hamiltonian systems under arbitrarily small perturbations, with orbits undergoing large effects over time. In his example, Arnold ex…

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 $\Gamma^{\ast}(s)$, $s\geq 0$, denote a triple-collision shape curve, and let the equator circle be the locus representing the first eclipse. Triple-collision shape-curve conje…

Let $P=\Delta+V-\lambda^2$ be the Helmholtz operator on $\mathbb{R}^n$, where $\lambda^2>0$ and $V=\sum_{\alpha\in I}\pi_\alpha^*(V^\alpha)$ is a three-body potential with…

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.

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

Arnold's conjecture. The mechanism of instability based on the existence of transition chains is applicable to the general case, for example to the problem of three bodies.

Consider a charged three-body system, and call a relative equilibrium non-collinear when the three bodies are not collinear in the relative-equilibrium configuration. The Lagrange…

A Hill region is the energetically allowed region of configuration or shape space for a three-body system, and a central configuration is a configuration in which the acceleration…

The three-rotor system has a family of periodic solutions, parameterized continuously by the energy $E$, that forms choreographies up to approximately $E=5.33$. Choreography-breakd…

Consider the Manev spatial isosceles three-body problem with pairwise potential … where $B>0$, and let $C$ denote the size of the angular momentum. Non-homographic triple-collision…

In the three-body problem, let the set of oscillatory motions consist of motions whose configurations exhibit oscillatory behavior, as distinguished from the other asymptotic motio…

A non-integrable dynamical system is considered in the context of the three-body problem of celestial mechanics. Poincaré's conjecture. A non-integrable system possesses stable and…

Minimum-period conjecture. The minimum of $T$ among all figure-eight choreographic solutions is attained within the series $\alpha$; consequently,

Marchal's conjecture. All orbits pass below the minimal inertia of the Henon–Broucke orbit.

The general three-body problem concerns the equations of motion of three mutually gravitating bodies; initial conditions may be chosen so that collisions are excluded. Painlevé's c…

Consider a motion in the planar three-body problem under a homogeneous potential, with configurational measure $\mu$ constant in time. A motion is homographic when its shape is fix…

Consider the homogeneous potential with exponent $\alpha$ as above, and the moment of inertia $I$. The exceptional exponents are $\alpha=-2$ and $\alpha=2$; the equal-mass rectilin…

Let three equal masses be released from rest in a scalene triangular configuration whose side lengths satisfy … Let $r_{ij}(t)$ be the distance between masses $i$ and $j$, and let…

Assume a brake orbit segment in the three-body problem, beginning at brake time and ending at its first syzygy. Let $I(t)$ denote the total moment of inertia of the three bodies. M…

Consider three bodies with masses in the ratio $3:4:5$, placed at rest at the vertices of a $3$-$4$-$5$ triangle. Meissel's conjecture. The resulting brake orbit is periodic. Burra…

Let $t_l$ denote the Lyapunov time and $t_d$ the survival time. For the Sitnikov problem, suppose the relationship between these times is given by … where $\rho$ and $\nu$ are cons…