8 problems

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 $n$ 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…

Consider the planar equal-mass $N$-body problem with homogeneous potential exponent $A$, and let $C_N(A)$ denote the number of equal-mass central configurations. Monotonicity conje…

Consider the planar four-body problem with homogeneous potential exponent $A$, and compare its central configurations with those in the Newtonian case $A=3$. Albouy's conjecture. F…

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…

Fifth- and seventh-order nonintegrability conjecture. If $n$ is odd, then the fifth variational equation is not integrable. If $n$ is even and $n\geq 4$, then the seventh variation…

Nonintegrability conjecture. There exists $k\in\mathbb{N}^*$ such that the variational equation at order $k$ is not integrable.