A generalized Noether theorem for scaling symmetry

The recently discovered conserved quantity associated with Kepler rescaling is generalized by an extension of Noether’s theorem which involves the classical action integral as an additional term. For a free particle, the familiar Schrödinger dilations are recovered. A general pattern arises for homogeneous potentials. The associated conserved quantity allows us to derive the virial theorem. The relation to the Bargmann framework is explained and illustrated by exact plane gravitational waves.