Finding Brake Orbits in the (Isosceles)

Three-Body ProblemRick Moeckel -- University of Minnesota

rick@math.umn.edu

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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