A Continuum of Solutions?
Consider three unit point masses in a plane.
Initial conditions are for each point mass, two coördinates and their first derivatives.
12 independent reals comprise the initial conditions.
We eliminate some:
- Total momentum
- make it (0, 0).
- Center of gravity
- Keep it at (0, 0).
- Phase
- Begin at some syzygy.
- Angle
- Rotate coördinate system so that syzygy is horizontal.
- Scale
- Scale position by s, velocity by s−½ and period by s3/2.
Set diameter of initial position to 2.
This leaves us 5 numbers to specify.
We have eliminated no solutions.
If we follow Šuvakov and Dmitrašinović we choose zero angular momentum and make initial positions equally spaced; middle mass at origin.
It surprises me that they were able to find solutions with the last constraint.
The periodic solutions by Šuvakov and Dmitrašinović are thus surprising for they seem to choose only 3 reals to specify a periodic solution.
It is as if they hove found new integrals of the system.
Perhaps there is a 2D manifold of periodic solutions in the 5D space.
That would explain, more or less, the coincidence of the existence of the symmetric solutions.
If there is a 2D manifold then a modification of my program can find some.