In physics, the Painlevé conjecture is a theorem about singularities among the solutions to the n-body problem: there are noncollision singularities for n ≥ 4.
The theorem was proven for n ≥ 5 in 1988 by Jeff Xia and for n = 4 in 2014 by Jinxin Xue.
Contents
Background and statement
Solutions
(
q
,
p
)
{\displaystyle (\mathbf {q} ,\mathbf {p} )}
of the n-body problem
q
˙
=
M
−
1
p
,
p
˙
=
∇
U
(
q
)
{\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
t
n
{\displaystyle t_{n}}
converging to a finite
t
∗
{\displaystyle t^{*}}
where
∇
U
(
q
(
t
n
)
)
→
∞
{\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.
A collision singularity occurs if
q
(
t
)
{\displaystyle \mathbf {q} (t)}
tends to a definite limit when
t
→
t
∗
,
t
<
t
∗
{\displaystyle t\rightarrow t^{*},t<t^{*}}
. 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:
For n ≥ 4 the n-body problem admits noncollision singularities.
Development
Edvard Hugo von Zeipel proved in 1908 that if there is a collision singularity, then
J
(
q
(
t
)
)
{\displaystyle J(\mathbf {q} (t))}
tends to a definite limit as
t
→
t
∗
{\displaystyle t\rightarrow t^{*}}
, where
J
(
q
)
=
∑
i
m
i
|
q
i
|
2
{\displaystyle J(\mathbf {q} )=\sum _{i}m_{i}|\mathbf {q} _{i}|^{2}}
is the moment of inertia. This implies that a necessary condition for a noncollision singularity is that the velocity of at least one particle becomes unbounded (since the positions