Fig. 5: Identifying conserved quantities for the double pendulum.
From: Discovering conservation laws using optimal transport and manifold learning
a The double pendulum has four degrees of freedom: angular positions θ1, θ2 and angular velocities ω1, ω2. b Sample trajectories show sample points plotted in 2D slices of the 4D phase space. c The heuristic score (with cutoff 0.6) identifies one relevant embedding component corresponding to (e) the total energy E. d However, if we restrict the embedding to trajectories with first component v1 < − 1 (i.e. low energy trajectories) and renormalize the embedding, f–h we find two conserved quantities corresponding to the energies E± of the two decoupled low energy modes. The gray points in (f–h) correspond to the high energy trajectories (first component v1 > − 1) which are not relevant when considering the low-energy non-chaotic phase of the double pendulum.
