For every integer n≥ 4 and every choice of positive masses m_1,…,m_n, consider the Newtonian n-body problem in R^3. Let S_n be the set of collision-free initial conditions whose maximal forward solution exists only up to some finite time T<∞, has no collision as t uparrow T, and satisfies I(t)=∑_i=1^n m_i| q_i(t)-q_cm(t)|^2 longrightarrow∞ as t uparrow T. The problem asks whether S_n has Lebesgue measure zero in the collision-free phase space for every n≥ 4.

Improbability of non-collision singularities in the n-body problem

For every integer $n\ge 4$ and every choice of positive masses $m_1,\ldots,m_n$ , consider the Newtonian $n$ -body problem in $\mathbb{R}^3$ . Let $\mathcal{S}_n$ be the set of collision-free initial conditions whose maximal forward solution exists only up to some finite time $T<\infty$ , has no collision as $t\uparrow T$ , and satisfies $I(t)=\sum_{i=1}^n m_i\lvert q_i(t)-q_{\mathrm{cm}}(t)\rvert^2\longrightarrow\infty$ as $t\uparrow T$ . The problem asks whether $\mathcal{S}_n$ has Lebesgue measure zero in the collision-free phase space for every $n\ge 4$ .

References

Primary source

arXiv

Additional references

A general improbability result for non-collision singularities as subsystems with at most four particles — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A September 2026 preprint proves the exceptional behavior is rare for several broader four-particle cluster patterns, but the full many-body question remains open.

This problem, on Barry Simon’s list, asks whether initial conditions producing finite-time escape without collisions form a measure-zero set in every system with more than three bodies. Earlier work established important special cases but described the general statement as almost completely open.

Known results

Saari proved the four-body case by 2014, according to the contemporaneous literature.
Xue constructed a Cantor set of planar four-body initial conditions producing non-collision singularities; it has measure zero and codimension $2$ on suitable energy levels.
Xia constructed spatial five-body non-collision singularities.
Gerver constructed planar examples with sufficiently many bodies.

September 2026 cluster-decomposition result

A new preprint proves improbability when the dynamics decompose into arbitrarily many total-collision subsystems and four-particle non-collision-singular subsystems with distinct asymptotic directions. A 2023 preprint also claims to solve a Marchal–Saari conjecture in a related model, but its precise scope is unclear; neither source verifies the full general statement.

Current status (as of September 2026): The conjecture is settled in the four-body case and in the newly specified cluster configurations, while the full $n$ -body improbability problem remains open.

Sources

No solutions have been posted yet.