“Painlevé conjecture”的意思、由来-开放百科全书

The conjecture has been proven for n ≥ 5 by Jeff Xia.^[3]^[4] The 4-particle case remains an open problem.^[5]

Background and statement

Solutions

of the n-body problem

(where M are the masses and U denotes the gravitational potential) are said to have a singularity if there is a sequence of times

converging to a finite

where

. That is, the forces and accelerations become infinite at some finite point in time.

A collision singularity occurs if

tends to a definite limit when

. If the limit does not exist the singularity is called a 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 that

Development

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.^[9]

In 1984 Joe Gerver gave an argument for a noncollision singularity in the planar 5-body problem with no collisions.^[11] He later found a proof for the 3n body case.^[12]

Finally, in his 1988 doctoral dissertation, Jeff Xia demonstrated a 5-body configuration that experiences a noncollision singularity.^[3]^[4]

Joe Gerver has given a heuristic model for the existence of 4-body singularities^[13] but at present no formal proof exists.

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. ^[14]

References

1. ^¹Florin N. Diacu. Painlevé's Conjecture. The Mathematical Intelligencer. Vol 13, no. 2, 1993
2. ^Florin Diacu, Philip Holmes, Celestial Encounters: The Origins of Chaos and Stability. Princeton University Press, 1996
3. ^¹Zhihong Xia. The Existence of Noncollision Singularities in Newtonian Systems. Annals of Mathematics. Second Series, Vol. 135, No. 3 (May, 1992), pp. 411–468
4. ^¹Donald G. Saari and Zhihong (Jeff) Xia. Off to Infinity in Finite Time. Notices of the AMS, vol 42, no. 5, 1993, pp. 538–546
5. ^Edward Belbruno. Capture Dynamics and Chaotic Motions in Celestial Mechanics. Princeton University Press, 2004 p. 8
6. ^P. Painlevé, Lecons sur la théorie analytique des équations différentielles, ParisÖ Hermann, 1897
7. ^Oeuvres de Paul Painlevé, Tome I, Paris Ed. Centr. Nat. Rech. Sci., 1972
8. ^H. von Zeipel, Sur les singularités du problème des corps, Arkiv för Mat. Astron. Fys. 4, (1908), 1–4.
9. ^J. Mather and R. McGehee, Solutions of the collinear four-body problem which become unbounded in finite time, Dynamical Systems Theory and Applications (J. Moser, ed.), Berlin: Springer-Verlag, 1975, 573–589.
10. ^Donald G. Saari. A global existence theorem for the four-body problem of Newtonian mechanics, J. Differential Equations 26 (1977), 80–111.
11. ^J. L. Gerver, A possible model for a singularity without collisions in the five-body problem, J. Diff. Eq. 52 (1984), 76–90.
12. ^J. L. Gerver, The existence of pseudocollisions in the plane, J. Diff. Eq. 89 (1991), 1-68.
13. ^Joseph L. Gerver, Noncollision Singularities: Do Four Bodies Suffice?, Exp. Math. (2003), 187–198.
14. ^Xue, J. & Dolgopyat, D. Commun. Math. Phys. (2016) 345: 797. https://doi.org/10.1007/s00220-016-2688-6