This document discusses ejection-collision orbits in celestial mechanics models with two degrees of freedom. It begins by introducing the symmetric collinear four body problem and its configuration space. It then generalizes to Hamiltonian systems with singularities representing collisions. The total collision manifold is described as invariant, with hyperbolic equilibrium points containing invariant manifolds relevant for ejection-collision orbits. For the symmetric four body problem specifically, sequences of points representing intersections of these manifolds enable the existence of ejection-collision orbits to be established for certain mass parameters.

1. Ejection-collision orbits in some models of two degrees
of freedom in Celestial Mechanics
M.´Alvarez-Ram´ırez,2
E. Barrab´es,1
M. Medina,2
M. Oll´e3
1
Universitat de Girona
2
Universidad Aut´onoma de M´exico-Iztapalapa
3
Universitat Polit`ecnica de Catalunya
CRM- HamSys 2018
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2. Outline
1 The symmetric collinear 4 body problem
2 Generalization
3 Existence of ECO in the SC4BP
4 Computation of ECO
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3. The symmetric collinear 4 body problem
Symmetric Collinear 4 Body Problem (SC4BP)
O
m1 = 1m2 = 1m4 = α m3 = 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}
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4. The symmetric collinear 4 body problem
Collisions
Single binary collision (SBC): central masses m1 and m2 collide
x = 0, y = 0
O
Double binary collision (DBC): simultaneous binary collisions between
m1 − m3 and m2 − m4.
y =
√
α x = 0
O
Quadruple collision (QC): collisions between the four bodies
x = y = 0
O
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5. The symmetric collinear 4 body problem
Energy and Hill’s region
First integral:
h = ˙x2
+ ˙y2
− U(x, y)
In a N-body problem, bounded motion can occur only if h < 0.
Configuration space
0
2
4
6
8
10
0 2 4 6 8 10
SBC
DBC
y
x
Polar coordinates:



x =
r
√
2
cos θ,
y =
r
√
2
sin θ,
{(r, θ); r ≥ 0, θα ≤ θ ≤ π/2}
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6. The symmetric collinear 4 body problem
Ejection-Collision Orbits (ECO)
Definition
An ejection-collision orbit (ECO) is a solution Γ = {γ(s)}s∈R such that
lim
s→±∞
r(s) = 0.
0
2
4
6
8
0 2 4 6 8
y
x
0
2
4
6
8
0 2 4 6 8
y
x
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7. Generalization
General setup
Consider a 2-d.o.f. Hamiltonian system with singularities (“collisions”):
H(q, p) =
1
2
pT
· A−1
· p − U(q), (q, p) ∈ D ⊂ R4
Introducing “polar” coordinates (r, θ), the hypothesis on the model are:
Configuration space D = {(r, θ); r > 0, θa < θ < θb}
The potential U(q) = V (θ)/r where
V (θ) =
c1
sin(θb − θ)
+
c2
sin(θ − θa)
+ V (θ)
0 < θb − θa < π, V (θ) > 0 is smooth and bounded.
R. Mart´ınez, (2012)
V (θ) has one non-degenerate critical point, θc, which is a minimum
“Total collision”: r = 0 “Partial collisions”: θ = θa, θb
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8. Generalization
Regularization
Change to polar-McGehee’s coordinates with a scaling of time:
(t, r, θ, v, u)
r = rv, θ = u,
v = u2
+ 1
2 v2
− V (θ), u = −1
2 uv +
dV
dθ
The flow has been extended to r = 0. The total collision manifold
C = {r = 0} is a 2-dimensional invariant manifold and the flow on C is
gradient-like with respect v.
Regularization of the singularities θ = θa, θb (Sundman, Devaney):
(s, r, θ, v, w)
w = F(θ)u,
dt
ds
= F(θ) F(θ) =
f(θ)
W(θ)
where f(θ) = sin(θ − θa) sin(θb − θ) and W(θ) = f(θ)V (θ) is positive and
bounded analytic function in [θa, θb].
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9. Generalization
Main features
For a fixed energy level h < 0:
w2
+ F(θ)2
v2
= 2f(θ) + 2rhF(θ)2
Two hyperbolic equilibrium points E+
and E−
in the collision manifold:
r = 0, θ = θc, v = ±v0, w = 0
where θc is the non-degenerate critical point of V (θ).
Ws
(E−
), Wu
(E+
) are 2-dimensional manifolds
Wu
(E−
), Ws
(E+
) are 1-dimensional manifolds restricted to C,
ECO belong to Wu
(E+
) ∩ Ws
(E−
)
At each energy level, there exists a homothetic solution
γ(s) = (r(s), θ = θc, v(s), w = 0)
and r(s) −→
t→±∞
0 (ECO)
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10. Generalization
Total collision manifold
For a fixed energy level h < 0:
w2
+ F(θ)2
v2
= 2f(θ) + 2rhF(θ)2
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11. Generalization
Some body problems
Symmetric collinear four body problem (SC4BP)
Collinear three body problem (C3BP)
Rectangular four body Problem (R4BP)
Rhomboidal four body problem
Planar 2N equal masses on a circle with some symmetries
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12. Generalization
References
Lacomba-Sim´o
Analysis of some degenerate quadruple collisions, Celest. Mech. (1982)
Behavior of the invariant manifolds on the total collision in the CS4BP,
R4BP
Lacomba-Medina
Symbolic dynamics in the symmetric collinear four-body problem, Qual.
Theory Dyn. Syst. (2004)
Existence of ECO for specific values of the mass parameter in the SC4BP
Mart´ınez
Families of double symmetric ”Schubart-like” periodic orbits, Celest. Mech.
117 (2012, 2013)
Study of the total collision manifold in a general framework and
application to several n-body problems
Sekiguchi-Tanikawa, Sweatman, Ouyang et al. (2002-2015)
SC4BP: periodic orbits, scattering orbits, Schubart orbits, escape orbits,
....(equal masses)
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13. Generalization
Aims
To understand the mechanisms that explain the existence of ECO
(generalize the ideas of Lacomba-Medina). Main tools:
Total collision manifold
Hyperbolic equilibrium points and their invariant manifolds
To give a numerical procedure to compute ECO
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14. Existence of ECO in the SC4BP
Symmetric Collinear 4 Body Problem (SC4BP)
O
m1 = 1m2 = 1m4 = α m3 = α
x
y/
√
α
V (θ) =
1
√
2 cos θ
+
1
√
2 sin θ
+
2
√
2 α3/2
sin θ −
√
α cos θ
+
2
√
2 α3/2
sin θ +
√
α cos θ
{(r, θ) ∈ R2
; 0 ≤ r, θα ≤ θ ≤ π/2}
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15. Existence of ECO in the SC4BP
Key ingredients
Collision manifold: C = {r = 0}
Two equilibrium points E+
and E−
in the collision manifold:
r = 0, θ = θc v = ±v0, w = 0
Inv. Manif. Ws
(E+
) Wu
(E+
) Ws
(E−
) Wu
(E−
)
Dimension 1 2 2 1
θc the non-degenerate critical point (minimum) of the potential V
ECO belong to Wu
(E+
) ∩ Ws
(E−
)
Ws
C (E+
) and Wu
C (E−
) live inside C (total collision manifold)
The flow on C is gradient-like with respect v
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16. Existence of ECO in the SC4BP
The quadruple collision manifold
C = {r = 0} is an invariant manifold
W
C is at the boundary of the constant energy surface Mh on (θ, v, w).
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17. Existence of ECO in the SC4BP
Invariant manifolds on the quadruple collision manifold
The invariant manifolds Wu
C (E−
) and Ws
C (E+
) (dim 1) live in C
α = α1 = 0.09187... α1 < α = 0.25 < α2 α = α2 = 0.36142...
-4
-2
0
2
4
0.4 0.6 0.8 1 1.2 1.4
v
θ
-4
-2
0
2
4
0.6 0.8 1 1.2 1.4
v
θ
-4
-2
0
2
4
0.6 0.8 1 1.2 1.4
v
θ
Sim´o and Lacomba (1982) give a sequence of values {αk}k≥1 for which one or
both branches of Wu
C (E−
) and Ws
C (E+
) coincide (single or double
heteroclinic).
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18. Existence of ECO in the SC4BP
Invariant manifolds on the quadruple collision manifold
The invariant manifolds Wu
C (E−
) and Ws
C (E+
) (dim 1) live in C
α2 < α = 0.75 < α3 α = α3 = 0.90764... α3 < α = 1.25 < α4
-6
-4
-2
0
2
4
6
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
v
θ
-6
-4
-2
0
2
4
6
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
v
θ
-8
-4
0
4
8
0.9 1 1.1 1.2 1.3 1.4 1.5
v
θ
Sim´o and Lacomba (1982) give a sequence of values {αk}k≥1 for which one or
both branches of Wu
C (E−
) and Ws
C (E+
) coincide (single or double
heteroclinic).
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19. Existence of ECO in the SC4BP
Invariant manifolds on the quadruple collision manifold
The invariant manifolds Wu
C (E−
) and Ws
C (E+
) (dim 1) live in C
α = α4 = 1.34496... α4 < α = 2 < α5 α = α5 = 2.21775...
-8
-4
0
4
8
0.9 1 1.1 1.2 1.3 1.4 1.5
v
θ
-12
-8
-4
0
4
8
12
1 1.1 1.2 1.3 1.4 1.5
v
θ
-12
-8
-4
0
4
8
12
1 1.1 1.2 1.3 1.4 1.5
v
θ
Sim´o and Lacomba (1982) give a sequence of values {αk}k≥1 for which one or
both branches of Wu
C (E−
) and Ws
C (E+
) coincide (single or double
heteroclinic).
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20. Existence of ECO in the SC4BP
Invariant manifolds on the quadruple collision manifold
-8
-4
0
4
8
0.9 1 1.1 1.2 1.3 1.4 1.5
v
θ
α ∈ (αk, αk+1) such that there are no heteroclinic connection on C and the two
branches of Wu
C (E−
) escape preforming different binary collisions
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21. Existence of ECO in the SC4BP
Existence of ECO (1)
The intersection of Wu
C (E−
) and Ws
C (E+
) with the section Σc give 4
sequences of points {u±
j }, {s±
j }
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22. Existence of ECO in the SC4BP
Existence of ECO (1)
The sequences follow a specific order (the branches do not intersect)
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23. Existence of ECO in the SC4BP
Existence of ECO (1)
The intersection of Wu
(E+
) and Ws
(E−
) (2-dim i.m.) with C give four orbits
which intersections with the section Σc give 4 sequences of points {p±
j }, {q±
j }
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24. Existence of ECO in the SC4BP
Existence of ECO (2)
We look for Wu
(E+
) ∩ Ws
(E−
)
Consider 1st intersection with Σc = {θ = θα, θ = π/2}: J, I
E
-
E
+
J
u1
+
p1
+
s1
s2
s*
s s1
-
q1
-
I
J =< u+
1 , p+
1 > I =< q−
1 , s−
1 , >
ECO of first order
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25. Existence of ECO in the SC4BP
Existence of ECO (2)
u2
+
u4
+
u1
-
u3
-
u5
-
s1
+
s3
+
s5
+
s4
-
s2
-
q1
+
u1
+
u3
+
u5
+
u2
-
u4
-
s2
+
s4
+
s1
-
s3
-
s5
-
p1
+
q1
-
p1
-
P (I )
-1
1
P (I )
-3
1
P(J )1
P (I )-2
1
P (I )-4
1
q2
-
I1
I2
J1
q= aq
q=p/2
ECO(1)
P (I )
-1
2
ECO(1)
ECO(1,1)
ECO(1,2,1,2,1)
W (E )
u +
W (E )
s -
Following the endpoints of the arches and the flow on C, ECO of higher order exists
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26. Computation of ECO
Parametrization of invariant manifolds (I)
Parametrizations of the invariant manifolds up to a certain order:
Ψ−
1 (ξ, ϕ) = E−
+ ξ (cos(2πϕ) σ1 + sin(2πϕ) σ4) ,
Ψ+
1 (ξ, ϕ) = E+
+ ξ (cos(2πϕ) σ1 + sin(2πϕ) σ3)
σ1 and σ3,4 give the slow and fast directions
The parameter ϕ ∈ (1/4, 3/4) (r > 0):
ϕ ∈ (1/4, 1/2): θ > θc, SBC
region
ϕ ∈ (1/2, 3/4): θ < θc, DBC
region
ϕ = 1/2: homothetic orbit,
slow direction
ϕ = 1/4, 3/4 fast direction
on C 0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
y
x
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27. Computation of ECO
Parametrization of invariant manifolds (II)
Time to satisfy the escape condition for each initial condition Ψ+
1 (ξ, ϕ)
0.01
0.1
1
10
100
1000
-14 -12 -10 -8 -6 -4 -2 0
se
(ϕ-0.5)10
3
0.01
0.1
1
10
100
1000
0 2 4 6 8 10 12 14
se
(ϕ-0.5)10
3
1
4
< ϕ <
1
2
1
2
< ϕ <
3
4
ECOs live in a narrow interval of values of ϕ close to ϕ = 1/2 (slow direction)
−→ a parametrization of the inv. manif. of higher order is needed.
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28. Computation of ECO
Characterization of ECO
Let B be the set of all possible sequences of the elements 1 and 2.
P : Wu
(E+
) −→ B
Γ −→ (p1, p2, . . . , pn, . . . )
pj =
1 the j-th intersection is a SBC,
2 the j-th intersection is a DBC,
for j = 1, . . . , n.
Theorem
Let Γ ∈ Wu
(E+
). Then
1 P(Γ) is finite if and only if it corresponds to an ECO.
2 If Γ is an ECO such that P(Γ) = (p1, p2, . . . , pn), then Γ is an ECO with
P(Γ) = (pn, pn−1, . . . , p1).
3 Γ is a symmetric ECO if and only if P(Γ) is a symmetric sequence.
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29. Computation of ECO
Computation of ECO (I)
Section
Σc = {(r, v, θ, w); w = 0, θ = θα or θ = π/2},
Let Γ an orbit with initial condition given by the parametrization Ψ+
k (ξ, ϕ).
The map Pn(ϕ) = (p1, . . . , pn) codes the first n intersections with Σc.
Proposition
Let ϕ1 and ϕ2 be such that
Pn+1(ϕ1) = (p1, p2, . . . , pn, p1
n+1),
Pn+1(ϕ2) = (p1, p2, . . . , pn, p2
n+1),
with p1
n+1 = p2
n+1. Then, there exists a value ϕ ∈ (ϕ1, ϕ2) such that
P(ϕ) = (p1, p2, . . . , pn).
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30. Computation of ECO
Computation of ECO (II)
Section
Σc = {(r, v, θ, w); w = 0, θ = θα or θ = π/2},
To detect a change in pn we look for the zeros of
Fn(ϕ) = r(θ − θc)
-4
-2
0
2
4
-0.015 -0.01 -0.005 0
θ5
F5
Fn
(ϕ-0.5)10
3
-4
-2
0
2
4
-0.015 -0.01 -0.005 0
θ6
F6
Fn
(ϕ-0.5)10
3
1
4
< ϕ <
1
2
1
2
< ϕ <
3
4
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31. Computation of ECO
ECO of order 1 and 2 for SC4BP
(1) (2)
0
2
4
6
8
0 2 4 6 8
y
x
0
2
4
6
8
0 2 4 6 8
y
x
(1,1) (2,2)
0
2
4
6
8
0 2 4 6 8
y
x
0
2
4
6
8
0 2 4 6 8
y
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32. Computation of ECO
ECO of order 3 and 4 for SC4BP
(1,1,1) (2,2,2)
0
2
4
6
8
0 2 4 6 8
y
x
0
2
4
6
8
0 2 4 6 8
y
x
(1,1,1,1) (2,2,2,2)
0
2
4
6
8
0 2 4 6 8
y
x
0
2
4
6
8
0 2 4 6 8
y
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33. Computation of ECO
ECO of order 6: symmetric and non-symmetric
(1, 6)
. . .,1) (2, 6)
. . .,2)
0
2
4
6
8
0 2 4 6 8
y
x
0
2
4
6
8
10
0 2 4 6 8 10
y
x
(1,2,1,2,1,2) (1,1,2,1,2,1) (2,2,1,2,1,2)
0
2
4
6
8
0 2 4 6 8
y
x
0
2
4
6
8
0 2 4 6 8
y
x
0
2
4
6
8
0 2 4 6 8
y
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34. Computation of ECO
...
Thank you!
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