Andronikos Paliathanasis

Conservation laws for Newtonian systems usually require writing the equations of motion and then solving symmetry partial differential equations that depend on the potential. Here the dynamics is embedded through the Eisenhart-Duval lift into an extended metric, so the symmetry conditions become Killing or conformal Killing equations of that metric, solved by a Hamiltonian network that learns the potential from phase-space data plus a random-feature network. For the one-dimensional Ermakov-Pinney equation, the three Killing vectors of the flat extended plane and their nonlocal conservation laws are recovered; for the one-dimensional harmonic oscillator, the ten-dimensional conformal algebra is reconstructed under noise.

Conservation laws for Newtonian systems usually require writing the equations of motion and then solving symmetry partial differential equations that depend on the potential. Here the dynamics is embedded through the Eisenhart-Duval lift into an extended metric, so the symmetry conditions become Killing or conformal Killing equations of that metric, solved by a Hamiltonian network that learns the potential from phase-space data plus a random-feature network. For the one-dimensional Ermakov-Pinney equation, the three Killing vectors of the flat extended plane and their nonlocal conservation laws are recovered; for the one-dimensional harmonic oscillator, the ten-dimensional conformal algebra is reconstructed under noise.