Math Conjectures

Improbability of noncollision singularities

OPENMajorConjectureProposed c. 1984 · Standard version

Canonical statement

Fix N≥5N\ge5 and masses m1,…,mN>0m_1,\ldots,m_N>0. For qi(t)∈R3q_i(t)\in\mathbb R^3, consider
q¨i=∑j≠imjqj−qi∣qj−qi∣3,i=1,…,N, \ddot q_i=\sum_{j\ne i}m_j\frac{q_j-q_i}{|q_j-q_i|^3},\qquad i=1,\ldots,N,
and the collision-free phase space P={(q1,…,qN,q˙1,…,q˙N):qi≠qj for i≠j}⊂R6N\mathcal P=\{(q_1,\ldots,q_N,\dot q_1,\ldots,\dot q_N):q_i\ne q_j\text{ for }i\ne j\}\subset\mathbb R^{6N}. With respect to Lebesgue measure on P\mathcal P, the set of initial conditions whose maximal solution has a finite endpoint T<∞T<\infty while
min⁡i≠j∣qi(t)−qj(t)∣⟶̸0as t↑T \min_{i\ne j}|q_i(t)-q_j(t)|\not\longrightarrow0 \quad\text{as }t\uparrow T
has measure zero.
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Fix \(N\ge5\) and masses \(m_1,\ldots,m_N>0\). For \(q_i(t)\in\mathbb R^3\), consider \[ \ddot q_i=\sum_{j\ne i}m_j\frac{q_j-q_i}{|q_j-q_i|^3},\qquad i=1,\ldots,N, \] and the collision-free phase space \(\mathcal P=\{(q_1,\ldots,q_N,\dot q_1,\ldots,\dot q_N):q_i\ne q_j\text{ for }i\ne j\}\subset\mathbb R^{6N}\). With respect to Lebesgue measure on \(\mathcal P\), the set of initial conditions whose maximal solution has a finite endpoint \(T<\infty\) while \[ \min_{i\ne j}|q_i(t)-q_j(t)|\not\longrightarrow0 \quad\text{as }t\uparrow T \] has measure zero.
DYN-013:[Saari1977ImprobabilityCollisions][Simon1984FifteenProblems][Xia1992NoncollisionSingularities][Quaschner2025Improbability]
Dynamical systemsMathematical physics

Notes

In the Newtonian NN-body problem, a singularity is a solution that ceases to exist at a finite time TT; it is a noncollision singularity if the minimal interparticle distance does not tend to zero as t↑Tt\uparrow T, which forces bodies to escape to infinity in finite time. The conjecture asserts that for every N≥5N\ge5 the initial conditions leading to such behaviour form a Lebesgue-null subset of phase space. The improbability question was canonically highlighted in Simon's 1984 list of problems in mathematical physics [Simon1984FifteenProblems], following Saari's earlier work on small particle numbers [Saari1977ImprobabilityCollisions].

Existence is no longer at issue: Xia constructed noncollision singularities in a five-body problem [Xia1992NoncollisionSingularities], and the remaining four-body case of Painlevé's existence question was later settled by Xue; this problem concerns only the measure-theoretic rarity of such orbits. On that side, the case N=4N=4 is covered by Saari's global existence theorem [Saari1977ImprobabilityCollisions], and recent results show that noncollision orbits produced by Xia-type mechanisms are improbable [Quaschner2025Improbability].

What is missing is an argument covering all possible escape mechanisms at once: for general N≥5N\ge5 no measure-zero theorem is known, and the conjecture is open.

References (4)

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