New proposition for efficiently exploring the space of initial conditions to find more NEW solutions

[gabrielfougeron]

Observation : 1 find the same solutions over and over again all the time. Question: How to change that ? How to explore the space of solutions more efficiently?

[gabrielfougeron]

Observation: Initial conditions uniquely define the motion. Proposition: Instead of exploring the infinite dimensional space of loops as initial states for the optimizer, rather explore the 2 x ndim x nbody dimensional space of initial conditions. Enforce initial conditions in the search using Lagrange multipliers.

Plan in several steps:

• [ ] Choose an initial condition
• [ ] integrate the system for 1 period (the period is chosen beforehand, does not really matter because of scale invariance)
• [ ] Project this solution on the space of periodic solutions
• [ ] Solve the Langrangien augmented system ==> COnditionning ? What happens when (and this will happen all the time) there is no solution ?