Painlevé conjecture

In physics, the Painlevé conjecture is a conjecture about singularities among the solutions to the n-body problem: there are noncollision singularities for n ≥ 4.

The conjecture has been proven for n ≥ 5 by Jeff Xia. The 4-particle case remains an open problem.

Solutions ${\displaystyle (\mathbf {q} ,\mathbf {p} )}$ of the n-body problem ${\displaystyle {\dot {\mathbf {q} }}=M^{-1}\mathbf {p} ,\;{\dot {\mathbf {p} }}=\nabla U(\mathbf {q} )}$ (where M are the masses and U denotes the gravitational potential) are said to have a singularity if there is a sequence of times ${\displaystyle t_{n}}$ converging to a finite ${\displaystyle t^{*}}$ where ${\displaystyle \nabla U\left(\mathbf {q} \left(t_{n}\right)\right)\rightarrow \infty }$. That is, the forces and accelerations become infinite at some finite point in time.

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