Finding Brake Orbits in the (Isosceles)
Three-Body ProblemRick 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
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Setting: Planar 3BP
Masses m1, m2, m3
Positions q1, q2, q3
Velocities v1, v2, v3
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 curveSymmetry ---> 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
m1=m2=1, m3 > 0
Isosceles shape
Two degrees of freedom after eliminating center of mass
Jacobi variables: (ξ1, ξ2) m1 = 1
m3
m2 = 1(ξ1,0)
(0,ξ2)
Size and Shape Variables
Replace (ξ1,ξ2) by variables representing the size and shape of the triangle.
Size:
Shape: angular variable θ such that
More about the shape variableAngle 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/2 π/2 π0
Shape Potential
Regularized ODE’sChange 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 ProofFind the first fourth of the orbit by shooting from the
zero velocity curve to meet the line θ=0 orthogonally. Must cross three regions.
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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=0Well-studied in 80’s (Devaney, Simo, Lacomba, R.M., ....) v is increasing
hyperbolic restpoints (Lag. are saddles)
behavior depends on m3
θ
vText
Admissible masses: Choose m3 so unstable branches of Lagrange restpoints satisfy:
v>0 here
v<0 here
Simo numerics: 0< m3 < 2.66RM proof:
m3 ≈ 1
Poincaré Maps -- Region I
ZI
Lagrange collision
orbit
Follow zero velocity curve Z across region to plane θ=-π/2
Region is positively invariantInitial curve Z from Lagrange to infinityImage curve ZI from unstable branch to infinity
ZI
Poincaré Maps -- Region IIFollow part of ZI across region to Lagrange plane
Region is negatively invariantFollow surface Ws(L) back to left wallLower part of ZI is trapped below Ws(L) Image curve ZII
ZI
L L
ZII
Poincaré Maps -- Region IIIFollow ZII across region to syzygy plane
Region is positively invariantEndpoints in unstable branches in collision manifoldImage curve ZIII must cross v = 0
Lv
rPoint on periodic brake orbit !!
More Periodic Break Orbits
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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.QuickTime™ and a
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Close-up of the near-triple-collision
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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 !
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