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

Symmetries and conservation laws of the Euler equations in Lagrangian coordinates