Symmetries and Casimir invariants for perfect fluid

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Abstract
We investigate the relations between symmetries of an adiabatic inviscid fluid and Casimir invariants of Eulerian Poisson bracket. It is shown that there exist two types of inner symmetries in addition to external symmetries. Noether's conserved quantities due to these inner symmetries are Casimir invariants when they are written in terms of Eulerian fields only. We construct the most general form for the two types of such quantities that correspond to these symmetries. However, these are shown to be equivalent. The most general Casimir of the system is shown to be the well-known Casimir functional. For a zero potential vorticity fluid, helicity is also allowed for the independent Casimir.

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