Families of relative periodic orbits in the planar three-body problem via consecutive alignments
arXiv:2609.01585
2026
Dynamics
1 ideas extracted · analyzed Sep 2, 2026
What the math gives to ML
The paper provides a constructive event-section method for finding and continuing relative periodic orbits: instead of integrating over an unknown full period and rotation angle, it matches the state at two consecutive alignment events in a low-dimensional nonlinear system. Its transferable asset is a symmetry-aware return-map loss for recurrent or state-space neural networks, together with a Floquet-style stability test that removes neutral directions caused by phase and continuous symmetries. A practical implementation can train a model to realize repeatable latent trajectories and use continuation in the target period or rotation angle to reveal stability boundaries before long-horizon deployment.
Ideas from this paper
Unverified
2026
Train a recurrent neural network or state-space model using a Poincare-style event loss: identify two consecutive latent alignment events and require the latent position and velocity at the second event to equal a transformed version of the first. Evaluate the Jacobian of this return map and penalize unstable non-neutral Floquet multipliers, producing long-horizon trajectories that are both periodic or symmetry-periodic and locally stable.
Useful7/10
Difficulty6/10
Novelty7/10