N-body problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Fix an integer $N\ge2$, positive masses $m_1,\ldots,m_N$, and a gravitational constant $G>0$. For the Newtonian $N$-body problem in $\mathbb R^3$, the collision-free phase space is \[\mathcal P=\{(q,p)\in(\mathbb R^3)^N\times(\mathbb R^3)^N:q_i\ne q_j\ (i\ne j)\}.\] Positions $q_i$ and momenta $p_i$ evolve by Hamilton's equations for \[H(q,p)=\sum_{i=1}^N\frac{|p_i|^2}{2m_i} -G\sum_{i<j}\frac{m_i m_j}{|q_i-q_j|},\qquad \dot q_i=p_i/m_i,\quad \dot p_i=-\nabla_{q_i}H.\] For $z\in\mathcal P$, let $I_z$ be the maximal time interval of its collision-free classical solution containing time zero. The set of global initial conditions is $\mathcal G=\{z:I_z=\mathbb R\}$. At a finite endpoint of $I_z$, a collision singularity has a collision configuration as the limiting positions; a noncollision singularity is a finite-time failure of continuation without such a collision endpoint. Measure means Lebesgue measure on the full phase space, not on a fixed energy level.

Statement recorded in the supplied source.

Determine the measure of $\mathcal P\setminus\mathcal G$. In particular, for arbitrary $N$ and positive masses, does $\mathcal G$ have full Lebesgue measure in $\mathcal P$? The complementary set separates into collision and noncollision singularities; the central measure-zero question is whether initial conditions leading to noncollision singularities have Lebesgue measure zero. By time reversal, the full-measure question is equivalent to asking for almost-everywhere existence for all positive times. [S2]

Short English statement

Do almost all collision-free initial conditions of the Newtonian N-body problem generate solutions for all time? Determine the measure of the exceptional collision and noncollision singularities.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 4200002 (AMR-041-0002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Jinxin Xue, Noncollision Singularities in a Planar Four-body Problem (2014; revised 2020), arXiv:1409.0048, Section 1.1, Motivations and perspectives, Conjecture 1, p. 3. https://arxiv.org/pdf/1409.0048.