Dynamics of the symmetric collineal four body problem, focussed in orbits that are born and end in quadruple collision

1. Ejection-Collision orbits in the symmetric collinear
four body problem
E. Barrab´es 1
M.´Alvarez-Ram´ırez 2
M. Oll´e3
1
Universitat de Girona
2
Universidad Aut´onoma de M´exico-Iztapalapa
3
Universitat Polit`ecnica de Catalunya
Soria - XVI JTMC 2017
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 1 / 1

2. The symmetric collinear 4 body problem
O
m1 = 1m1 = 1m2 = α m2 = α
x
y/
√
α
H =
p2
x
4
+
p2
y
4
− U(x, y)
U(x, y) =
1
2x
+
α5/2
2y
+ 4α3/2 y
y2 − αx2
, {(x, y) ∈ R2
; 0 <
√
αx < y}
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 2 / 1

3. Energy and Hill’s region
First integral:
h = ˙x2
+ ˙y2
− U(x, y)
Proposition:
In a N-body problem, bounded motion can occur only if h < 0.
y =
√
α x
U(x, y) = −h
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 3 / 1

4. Collisions
Proposition:
For the collinear N-body problem, all initial conditions lead to
collisions (either forward or backward in time).
D. Saari
Collisions, Rings, and other Newtonian N-Body Problems
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 4 / 1

5. Collisions
Single Binary Collision (SBC): x = 0, y = 0 type: 0
O
Double Binary Collision (DBC): y =
√
α x = 0 type: 2
O
Quadruple Collsion (QC): x = y = 0
O
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 5 / 1

6. Collisions
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8
DBC
SBC
y
x
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 6 / 1

7. Regularization
Change to McGehee’s coordinates + regularization (Sundman, Devaney)
(x, y, px, py) −→ (r, θ, v, w)
Collisions: SBC: θ = π/2 DBC: θ = θα, tan(θα) =
√
α QC: r = 0
Energy relation:
f(θ)v2
+ w2
= 2rh + 2 cos(θ)(sin(θ) −
√
α cos(θ))
for a fixed value h < 0.
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 7 / 1

8. The quadruple collision manifold
C = {r = 0} is an invariant manifold
W
The flow is gradient-like with respect v.
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 8 / 1

9. Equilibrium points
Two equilibrium points E+
and E−
:
r = 0, θ = θ0 v = ±v0 w = 0
Homothetic solution: θ = θ0 (masses retain the same configuration, up to
a scale factor, all the time)
In both cases, the associated equilibrium points are real
λ4 < λ3 < 0 < λ2 < λ1
Inv. Manif. Ws
(E+
) Wu
(E+
) Ws
(E−
) Wu
(E−
)
Dimension 1 2 2 1
Wu
(E−
) and Ws
(E+
) live inside C
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 9 / 1

10. Ejection-collision (EC) orbits
ejection orbits: orbits which begin at quadruple collision
Wu
(E+
)
collision orbits: orbits which end at quadruple collision
Ws
(E−
)
Ejection-collision orbits:
Ws
(E−
) ∩ Wu
(E+
)
R. McGehee
Triple collision in the collinear three body problem, 1974
R. Devaney
Triple collision in the planar isosceles three body problem, 1980
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 10 / 1

11. Previous works
C. Sim´o, E. Lacomba
Analysis of some degenerate quadruple collisions
Celest. Mech. Dyn. Astr., 28, 1982.
W. Sweatman
The symmetrical one-dimensional newtonian four-body problem: a
numerical investigation
Celest. Mech. Dyn. Astr. , 82, 2002.
M. Skeiguchi, K. Tanikawa.
On the symmetric collinear four-body problem
Publ. Astron. Soc. Japan, 56, 2004
M. ´Alvarez-Ram´ırez, M. Medina, C. Vidal
The trapezoidal collinear four-body problem.
Astroph. Space Sci., 358, 2015
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 11 / 1

12. Numerical exploration of general dynamics
Sekiguchi - Tanikawa [ST04]
Analysis of the SC4BP using the section Σ = {θ = θ0}
Thm: any trajectory intersects (transversally) Σ at least once. Almost
true!
Proof forgets about orbits on Ws
(E−
) and Wu
(E+
)
Classify orbits depending on the number and type of double collisions (all
of them with crossings with Σ)
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 12 / 1

13. Section Σ
Initial conditions on section Σ = {θ = θ0}
w2
+ f(θ0) v2
= p(θ0) + 2rh
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5
y
x
-5
-2.5
0
2.5
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
v
w
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5
y
x
#SBC = 1
type .0
#DBC = 1
type .2
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 13 / 1

14. Σ + 2 binary collisions
I.C. on Σ. Follow the orbit until 2 binary collisions.
-5
-2.5
0
2.5
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
v
w
1,2
3,4
7
6
5
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5
y
x
type .00
#SBC = 2
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 14 / 1

15. Σ + 2 binary collisions
I.C. on Σ. Follow the orbit until 2 binary collisions.
-5
-2.5
0
2.5
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
v
w
1,2
3,4
7
6
5
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5
y
x
type .02
#SBC = 1, #DBC=1
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 14 / 1

16. Σ + 2 binary collisions
I.C. on Σ. Follow the orbit until 2 binary collisions.
-5
-2.5
0
2.5
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
v
w
1,2
3,4
7
6
5
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5
y
x
type .22
#DBC = 2
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 14 / 1

17. Σ + 2 binary collisions
I.C. on Σ. Follow the orbit until 2 binary collisions.
-5
-2.5
0
2.5
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
v
w
1,2
3,4
7
6
5
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5
y
x
type .20
#SBC = 1, #DBC=1
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 14 / 1

18. Ws
(E−
) ∩ Σ
Limit orbits (towards r = 0) along the curve Ws
(E−
) ∩ Σ −→ ejection-capture
orbits with no intersections with Σ
-5
-2.5
0
2.5
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
v
w
1,2
3,4
5
6
7
0
1
2
3
4
5
6
7
8
0 0.5 1 1.5 2 2.5 3 3.5 4
y
x
1 2 3 4
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 15 / 1

19. Ws
(E−
) ∩ Σ
Limit orbits (towards r = 0) along the curve Ws
(E−
) ∩ Σ −→ ejection-capture
orbits with no intersections with Σ
-5
-2.5
0
2.5
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
v
w
1,2
3,4
5
6
7
0
1
2
3
4
5
6
7
8
0 0.5 1 1.5 2 2.5 3 3.5 4
y
x
5 6 7
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 15 / 1

20. Ws
(E−
) ∩ Σ + 2 binary collisions
-5
-2.5
0
2.5
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
v
w
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 16 / 1

21. Ws
(E−
) ∩ Σ + 2 binary collisions
-5
-2.5
0
2.5
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
v
w
0
1
2
3
4
5
6
7
8
0 0.5 1 1.5 2 2.5 3 3.5 4
y
x
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 16 / 1

22. Ws
(E−
) ∩ Σ + 2 binary collisions
-5
-2.5
0
2.5
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
v
w
0
1
2
3
4
5
6
7
8
0 0.5 1 1.5 2 2.5 3 3.5 4
y
x
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 16 / 1

23. Ws
(E−
) ∩ Σ + 2 binary collisions
-5
-2.5
0
2.5
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
v
w
0
1
2
3
4
5
6
7
8
0 0.5 1 1.5 2 2.5 3 3.5 4
y
x
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 16 / 1

24. Ws
(E−
) ∩ Σ + 2 binary collisions
0
1
2
3
4
5
6
7
8
0 0.5 1 1.5 2 2.5 3 3.5 4
y
x
-5
-2.5
0
2.5
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
v
w
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 16 / 1

25. Ws
(E−
) ∩ Σ1,2
-5
-2.5
0
2.5
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
v
w
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 17 / 1

26. Suitable surface of section
Between two consecutive binary collisions of the same type θ has a
maximum or a minimum. Also binary collisions correspond to max-min of
θ:
θ = w = 0
All EC orbits that do not cross θ = θ0, do cross w = 0.
“Symmetric” orbits: if v = w = 0 at time s0 then
r(s0 + s) = r(s0 − s) θ(s0 + s) = θ(s0 − s), ∀s
0
2
4
6
8
10
0 1 2 3 4 5
y
x
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 18 / 1

27. Suitable surface of section
-8
-6
-4
-2
0
2
4
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
v
θ
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 19 / 1

28. Suitable surface of section
-8
-6
-4
-2
0
2
4
6
8
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
v
θ
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 19 / 1

29. Work in progress
Parametrization of Ws
(E+
) up to an adequate order
Computation of the ejection-collision orbits using a proper section
Relation between the EC with only one type of binary collision with the
dynamics of the collision manifold
Vary α. In particular, what happen for those values of α for which there
exist heteroclinic connections inside the collision manifold.
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 20 / 1

30. Future work
Oscillatory motions:
lim sup
t→±∞
r = ∞ lim inf
t→±∞
r < ∞
Heteroclinic connections between QC - infinity: intersection between the
invariant manifolds Wu/s
(E+/−
) with the invariant manifolds of the
equilibrium points at infinity
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8
DBC
SBC
y
x
Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 21 / 1

31. Barrab´es, ´Alvarez, Oll´e (June 19, 2017) SC4BP 22 / 1