17 problems

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…

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…

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…

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…

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.

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…

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…

Let $f$ be the function for the equal-mass planar $N$-body problem, and let $M(t)$ denote its Morse polynomial. Newtonian Morse-polynomial conjecture. For $A=3$, the Morse polynomi…

Let $N>4$, let $A$ be the exponent of the homogeneous potential, and let $f$ be the function whose Morse index is considered on the quotient configuration space $\mathcal{C}_N$. Fo…

Cayley–Menger factorization conjecture. The determinant $\det g^{\mu\nu}$ is homogeneous of degree $M$ in the $M$ variables $\rho_{ij}$ and factors as

Algebraicity conjecture. All symmetry operators $K_{ij}$ are algebraic differential operators in the variables $\rho_{ij}$, with coefficients linear in the $\rho_{ij}$.

Consider the planar $N$-body problem with masses $m_k>0$ under a $1/r^\alpha$ potential, where $0<\alpha<2$, and let $q$ be a central configuration. A periodic homographic motion i…

Main choreography conjecture. For every $N\geq 3$, there is a main simple choreography solution different from the trivial circle solution. The number of such distinct main simple…

Consider the Newtonian $N$-body problem, with potential corresponding to $\alpha=1$, and let $I$ be the moment of inertia. A motion has constant moment of inertia when $I$ is const…