Fig. 4: Identifying conserved quantities for planar gravitational dynamics.
From: Discovering conservation laws using optimal transport and manifold learning
a Planar gravitational dynamics has four degrees of freedom: position vector r and momentum vector p. b Sample trajectories show sample points plotted in 2D slices of the 4D phase space consisting of position r and momentum p. c The heuristic score (with cutoff 0.6) identifies three relevant embedding components corresponding to the three independent conserved quantities. d, e Components 1 and 2 embed the semi-major axis vector a with magnitude a = − 1/2E related to the energy and orientation given by the angle ϕ. f Component 6 corresponds to the angular momentum L.
