Info Dump on Three Body Problem
1. Governing Equations (Newtonian N-Body Framework)
start with three point masses m1,m2,m3 under Newtonian gravity:
for i = 1,2,3
We can choose units to simplify:
G=1G = 1G=1 Equal masses: m1 = m2 = m3 = 1 Often centre of mass (COM) frame & net momentum = 0
This reduces to a 6D configuration space (since COM is fixed).
2. Hamiltonian Formulation
We can rewrite this as a Hamiltonian system:
Phase space dimension = 12 (positions + momenta).
Conserved quantities:
Total energy E Total momentum P Total angular momentum L Centre of mass motion.
Setting P=0 and L=0 isolates relative motion.
3. Symmetries and Reduction
Rotational & Translational Invariance:
We can reduce to relative coordinates (Jacobi vectors):
System becomes planar when total angular momentum is zero → 4D reduced phase space.
4. Periodic Orbit Search
4.1 Action Minimization
We treat orbits as critical points of the action functional:
with:
Periodic orbits correspond to stationary action under variations:
This leads to Euler-Lagrange equations, which are equivalent to Newton’s equations.
4.2 Symmetry Constraints (Topological Classification)
The trick to find pretty orbits like this image is imposing symmetry:
Reflection symmetries:
Start with configurations that are mirror-symmetric about an axis at t=0.
Time-reversal symmetry:
Choose initial velocities so that after half a period, positions and velocities mirror.
These symmetries reduce the search space drastically.
4.3 Numerical Methods
Initial guess:
Pick random initial positions/velocities satisfying COM=0, L=0.
Integrate (e.g., Runge-Kutta 8th order).
Adjust parameters:
Use a shooting method to minimize:
Refinement:
Use Newton-Raphson or gradient descent on action.
5. Stability (Floquet Theory)
Periodic solutions can be stable or unstable:
Linearize dynamics around the periodic orbit:
Solve for Floquet multipliers:
Stable if all lie on the unit circle (rare).
Unstable if any ∣μ∣>1 (most orbits in your image are unstable).
6. Examples of Famous Periodic Orbits
Figure-Eight Orbit (Chenciner & Montgomery, 2000)
Equal masses, zero angular momentum. Elegant & stable (rare!). Discovered via variational methods.
“Butterfly”, “Dragonfly”, “Moth” (Šuvakov & Dmitrašinović, 2013)
Each orbit corresponds to a different braid topology in configuration space.
7. Chaos & Lyapunov Exponents
Even though these are periodic:
Nearby initial conditions diverge exponentially:
Periodic solutions inside chaotic sea → measure 0 in phase space.