We study a variational formulation for the incompressible Euler equations.

A scaling symmetry of the action functional leads to a new conservation law.

The conserved integral relates the kinetic energy to a quantity defined in Lagrangian coordinates.

The conserved integral controls the fluid's radial deformation. It is found to be time-irreversible.

We find a non-existence result for time-periodic solutions with nonzero, finite energy.