18 problems

Triangular stability classification conjecture. If $0<\mu<\mu_R$, then $\operatorname{L}_4$ and $\operatorname{L}_5$ are elliptic for every $\kappa>0$ for which they exist. If…

Pitchfork bifurcation conjecture. The triangular relative equilibria $\operatorname{T}_1$ and $\operatorname{T}_2$ bifurcate from $\operatorname{L}_3$ at $\kappa=\kappa_4(\mu)$.

Triangular existence classification conjecture. There exist $\kappa_4(\mu)<\kappa_5(\mu)$ in $(0,\pi^2/4)$ such that there are two triangular relative equilibria,…

Continuation conjecture. These two relative equilibria are continuations of $\operatorname{L}_4$ and $\operatorname{L}_5$.

Existence classification conjecture. There exist $\kappa_i=\kappa_i(\mu)\in(0,\pi^2/4)$ with $\kappa_1<\kappa_2<\kappa_3$ such that the curved restricted three-body problem has exa…

Twelve-vortex branch conjecture. This branch is born stable and minimises $H$ for fixed small positive values of $\mu$.

Ten-vortex asymmetric-branch conjecture. The stable branch that minimises $H$ for small fixed values of $\mu$ is asymmetric.

Eight-vortex asymmetric-branch conjecture. An asymmetric relative equilibrium gains stability around these parameter values.

Seven-vortex branch conjecture. This branch minimises $H$ for fixed and small values of the momentum.

Five-vortex stable-branch conjecture. This branch is born stable close to the triangular bipyramid and is the global minimiser of $H$ for fixed values of $\mu>0$ close to zero.

Six-equilibrium conjecture. The maximum number of collinear relative equilibria on a rotating meridian is six.

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…

Let $\kappa$ denote the constant curvature of the space, and let the family of relative equilibria be obtained by continuing Lagrange's equilateral triangle with equal masses…

The swimmers are rigid helical rods with circular cross-sections, capped with half-spheres at both ends. Their magnetic moment has direction $\bm{\mathsf{m}}$, and the experimental…

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…

Let a balanced configuration of the three-body problem in $\mathbb{R}^4$ have angular momentum $L$ and eigenvalues $\mu_1$ and $\mu_2$. Uniqueness conjecture. For any choice of mas…

Consider the N-body problem with general positive masses and its relative equilibria, namely solutions generated by central configurations. The first excited state is the relative…

Let $m_1,\ldots,m_n$ be a fixed set of masses in the classical $n$-body problem, and identify relative equilibria under the equivalence relation described in the source. Smale's si…