  We consider the Euler equations of incompressible inviscid fluid dynamics. We
discuss a variational formulation of the governing equations in Lagrangian
coordinates. We compute variational symmetries of the action functional and
generate corresponding conservation laws in Lagrangian coordinates. We clarify
and demonstrate relationships between symmetries and the classical balance laws
of energy, linear momentum, center of mass, angular momentum, and the statement
of vorticity advection. Using a newly obtained scaling symmetry, we obtain a
new conservation law for the Euler equations in Lagrangian coordinates in
n-dimensional space. The resulting integral balance relates the total kinetic
energy to a new integral quantity defined in Lagrangian coordinates. This
relationship implies an inequality which describes the radial deformation of
the fluid, and shows the non-existence of time-periodic solutions with nonzero,
finite energy.
