Painlevé conjecture

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

A collision singularity occurs if $\backslash mathbf\{q\}(t)$ tends to a definite limit when pseudocollision or noncollision singularity.

Paul Painlevé showed that for n = 3 any solution with a finite time singularity experiences a collision singularity. However, he failed at extending this result beyond 3 bodies. His 1895 Stockholm lectures end with the conjecture:{{cite book |first=P. |last=Painlevé |title=Lecons sur la théorie analytique des équations différentielles |url=https://archive.org/details/leonssurlathorie00pain |location=Paris |publisher=Hermann |year=1897 }}{{cite book |title=Oeuvres de Paul Painlevé |volume=Tome I |location=Paris |publisher=Ed. Centr. Nat. Rech. Sci. |year=1972 }}

{{blockquote|For n ≥ 4 the n-body problem admits noncollision singularities.}}

Edvard Hugo von Zeipel proved in 1908 that if there is a collision singularity, then $J(\backslash mathbf\{q\}(t))$ tends to a definite limit as $t\backslash rightarrow\; t^*$, where $J(\backslash mathbf\{q\})=\backslash sum\_i\; m\_i\; |\backslash mathbf\{q\}\_i|^2$ is the moment of inertia.{{cite journal |first=H. |last=von Zeipel |title=Sur les singularités du problème des corps |journal=Arkiv för Matematik, Astronomi och Fysik |volume=4 |year=1908 |pages=1–4 }} This implies that a necessary condition for a noncollision singularity is that the velocity of at least one particle becomes unbounded (since the positions $\backslash mathbf\{q\}$ remain finite up to this point).

Mather and McGehee managed to prove in 1975 that a noncollision singularity can occur in the co-linear 4-body problem (that is, with all bodies on a line), but only after an infinite number of (regularized) binary collisions.{{cite book |first1=J. |last1=Mather |first2=R. |last2=McGehee |chapter=Solutions of the collinear four-body problem which become unbounded in finite time |title=Dynamical Systems Theory and Applications |url=https://archive.org/details/dynamicalsystems00mose |url-access=limited |editor-first=J. |editor-last=Moser |editor-link=Jürgen Moser |location=Berlin |publisher=Springer-Verlag |year=1975 |pages=[https://archive.org/details/dynamicalsystems00mose/page/n576 573]–589 |isbn=3-540-07171-7 }}

Donald G. Saari proved in 1977 that for almost all (in the sense of Lebesgue measure) initial conditions in the plane or space for 2, 3 and 4-body problems there are singularity-free solutions.{{cite journal |first=Donald G. |last=Saari |title=A global existence theorem for the four-body problem of Newtonian mechanics |journal=J. Differential Equations |volume=26 |year=1977 |issue=1 |pages=80–111 |doi=10.1016/0022-0396(77)90100-0 |bibcode=1977JDE....26...80S |doi-access=free }}

In 1984, Joe Gerver gave an argument for a noncollision singularity in the planar 5-body problem with no collisions.{{cite journal |first=J. L. |last=Gerver |title=A possible model for a singularity without collisions in the five-body problem |journal=J. Diff. Eq. |volume=52 |year=1984 |issue= 1|pages=76–90 |doi= 10.1016/0022-0396(84)90136-0|bibcode=1984JDE....52...76G |doi-access=free }} He later found a proof for the 3n body case.{{cite journal |first=J. L. |last=Gerver |title=The existence of pseudocollisions in the plane |journal=J. Diff. Eq. |volume=89 |year=1991 |issue=1 |pages=1–68 |doi= 10.1016/0022-0396(91)90110-U|bibcode=1991JDE....89....1G |doi-access=free }}

Finally, in his 1988 doctoral dissertation, Jeff Xia demonstrated a 5-body configuration that experiences a noncollision singularity.

Joe Gerver has given a heuristic model for the existence of 4-body singularities.{{cite journal |first=Joseph L. |last=Gerver |title=Noncollision Singularities: Do Four Bodies Suffice? |journal=Exp. Math. |year=2003 |volume= 12|issue= 2|pages=187–198 |doi=10.1080/10586458.2003.10504491 |s2cid=23816314 |url=http://projecteuclid.org/euclid.em/1067634730 }}

In his 2013 doctoral thesis at University of Maryland, Jinxin Xue considered a simplified model for the planar four-body problem case of the Painlevé conjecture. Based on a model of Gerver, he proved that there is a Cantor set of initial conditions which lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all earlier collisions. In 2014, Xue extended his previous work and proved the conjecture for n=4.{{cite journal |last1=Xue |first1=J. |last2=Dolgopyat |first2=D. |title=Non-Collision Singularities in the Planar Two-Center-Two-Body Problem |journal=Commun. Math. Phys. |year=2016 |volume=345 |issue= 3|pages=797–879 |doi=10.1007/s00220-016-2688-6 |arxiv=1307.2645 |bibcode=2016CMaPh.345..797X |s2cid=119274578 }}

Due to the symmetry constraint, Xia's model is only valid for the 5-body problem. Gerver-Xue's model does not have such a constraint, and is likely to be generalized to the general N>4 body problem.

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