Finding Brake Orbits in the (Isosceles) Three- Body Problem. Rick Moeckel -- University of Minnesota

Transcription

1 Finding Brake Orbits in the (Isosceles) Three- Body Problem Rick Moeckel -- University of Minnesota

2 Goal: Describe an Existence Proof for some simple, periodic solutions of the 3BP Brake orbit: initial velocities are all zero Periodic Symmetric with respect to syzygy set Part of a project with R. Montgomery and A. Venturelli

3 Setting: Planar 3BP Masses m 1, m 2, m 3 Positions q 1, q 2, q 3 Velocities v 1, v 2, v 3 Question: Select initial positions and release the bodies with zero initial velocity. What can happen? --> Brake Orbits

4 Triple Collision Lagrange: equilateral shape Euler: special collinear shapes Time reversibility ==> collision-ejection orbits Very simple solutions, play an important role, but they!re not periodic. Unlike double collisions, triple collisions are not regularizeable.

5 Hill!s Region, Symmetry Lagrangian, 2 degrees of freedom: L=T(v)+U(q) Energy constant: T(v)-U(q) = h Hill s region: T(v)! 0 ---> U(q) " -h Zero velocity curve Brake orbits start on zero velocity curve Symmetry ---> can seek 1/4th of a periodic orbit. Try to hit the symmetry line orthogonally

6 Isosceles 3BP m 1 =m 2 =1, m 3 > 0 Isosceles shape Two degrees of freedom after eliminating center of mass m 3 (0,! 2 ) Jacobi variables: (! 1,! 2 ) m 1 = 1 L = T U (! 1,0) m 2 = 1 T = 1 4 ξ µ ξ 2 µ = 2m m 3 U = 1 ξ 1 + 2m 3 ξ 2 1 /4 + ξ2 2

7 Size and Shape Variables Replace (! 1,! 2 ) by variables representing the size and shape of the triangle. Size: I = r 2 = 1 2 ξ2 1 + µξ 2 2 Shape: angular variable " such that ξ 1 = 2 r 1 sin2 θ 1 + sin 2 θ ξ 2 = r µ 2 sin θ 1 + sin 2 θ

8 More about the shape variable Angle " is an infinite covering of the isosceles shapes, locally a branched double cover near the binary collision shapes (to facilitate regularization of binary collisions). -# -#/2 0 #/2 # r 8 U = 1 r V (θ) 6 4 Shape Potential Θ

9 Regularized ODE s Change of timescale (McGehee, Levi-Civita): = r 3 2 cos 2 θ r = vr cos 2 θ v = 1 2 v2 cos 2 θ w2 (1 + sin 2 θ) 2 W (θ) θ = 1 4 w(1 + sin2 θ) 2 w = W (θ) 1 2 vw cos2 θ + sin θ cos θ(2r + v w2 (1 + sin 2 θ)) W (θ) = cos 2 θ V (θ)(regularized potential) V (θ) = (1 + sin 2 θ) 1 2 cos2 θ m 3 (1 + sin 2 θ) m 3 sin 2 θ.

10 Hill s Region and the Brake Orbit Set energy h = v2 cos 2 θ w2 (1 + sin 2 θ) 2 cos 2 θv (θ) = r cos 2 θ Hill s Region: r V (θ) Syzygy Lines r Zero Velocity Curve theta

11 Idea of Proof Find the first fourth of the orbit by shooting from the zero velocity curve to meet the line "=0 orthogonally. Must cross three regions. Reach here with v=0 (r =0) Start here I II III

12 Flow in the Energy Manifold 3D projection of 1/2 of energy manifold: Eliminate shape velocity variable w > 0. r Lagrange collision orbit v " I II III Must reach here with v=0

13 Flow in the Collision Manifold r=0 Well-studied in 80 s (Devaney, Simo, Lacomba, R.M.,...) v is increasing hyperbolic restpoints (Lag. are saddles) behavior depends on m 3 L v Text L' 4 v Admissible masses: Choose m3 so unstable branches of Lagrange restpoints satisfy: 2 v>0 here " Θ Γ v<0 here L L' Γ' 2 4 Simo numerics: 0< m3 < 2.66 RM proof: m3 " 1

14 Poincaré Maps -- Region I Follow zero velocity curve Z across region to plane "=-#/2 Lagrange collision orbit Z Z I Region is positively invariant Initial curve Z from Lagrange to infinity Image curve Z I from unstable branch to infinity

15 Poincaré Maps -- Region II Follow part of Z I across region to Lagrange plane Region is negatively invariant Z I Follow surface W s (L) back to left wall Lower part of Z I is trapped below W s (L) Image curve Z II Z II L L

16 Poincaré Maps -- Region III Follow Z II across region to syzygy plane Region is positively invariant Endpoints in unstable branches in collision manifold Image curve Z III must cross v = 0 L r Point on periodic brake orbit!! v

17 More Periodic Break Orbits Recently, numerical experiments by Sean Vig have turned up more isosceles periodic brake orbits This one has multiple collisions before syzygy. r theta

18 More Periodic Break Orbits This one has passes very close to triple collision. Near collision it has two syzygies. r

19 Close-up of the near-triple-collision

20 Questions and problems for the future How do the periodic, brake orbits fit with the known dynamics of the isosceles problem, such as, triple collision orbits, orbits near infinity, etc.? Are there any stable, periodic brake orbits? Are these orbits minimizers of some variational problem? Are there nearby, periodic brake orbits of the planar 3BP without collisions?

21 THANKS!