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Ejection-Collision orbits in the symmetric collineal four body problem

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

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Ejection-Collision orbits in the symmetric collineal four body problem

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

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