Finding Brake Orbits in the (Isosceles) Three- Body Problem Rick Moeckel -- University of Minnesota Feliz cumpleaños, Ernesto

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 -- From Brake to Syzygy

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 Zero angular momentum, negative energy

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.

Hill’s Region, Symmetry 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 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

Isosceles 3BP m 1 =m 2 =1, m 3 > 0 Isosceles shape Two degrees of freedom after eliminating center of mass Jacobi variables: (ξ 1, ξ 2 ) m 1 = 1 m3m3 m 2 = 1 (ξ 1,0) (0,ξ 2 )

Size and Shape Variables Replace (ξ 1,ξ 2 ) by variables representing the size and shape of the triangle. Size: Shape: angular variable θ such that

More about the shape variable Angle θ is an infinite covering of the isosceles shapes, locally a branched double cover near the binary collision shapes (to facilitate regularization of binary collisions). -π-π/2π/2π0 Shape Potential

Regularized ODE’s Change of timescale (McGehee, Levi-Civita):

Hill’s Region and the Brake Orbit Set energy h = -1 Hill’s Region: Syzygy Lines Zero Velocity Curve

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

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

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 θ v Text Admissible masses: Choose m 3 so unstable branches of Lagrange restpoints satisfy: v>0 here v<0 here Simo numerics: 0< m 3 < 2.66 RM proof: m 3 ≈ 1

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

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

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 v r Point on periodic brake orbit !!

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.

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

Close-up of the near-triple-collision

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 ? Are there any “first-syzygy” periodic brake orbits ?

Thanks !